Transcripts For CSPAN3 Sean 20240704

Healthy democracy doesnt look just like this, it looks like this. Where americans can see democracy at work. Citizens are truly influenced a, republic thrives. Get informed straight from the source and cspan. Unfiltered, unbiased, word for word, from the Nations Capital to wherever you are. This opinion that matters the most zero. This is what democracy looks like. Cspan power by you. It is an honor for me to introduce professor sean cairo today. He has recently become a full professor of natural philosophy has the radical thesis and is red many we Interesting Research papers. Its also a Popular Science writer and has been an thats all i offer author. I always find him very thoughtprovoking. I read his book when i was an undergrad in argentina and ever since then i became fascinated with the topic hes very committed to making complex physical ideas accessible to everyone and we are very unique and wise way. So today he will be telling us about his first book in his new trilogy, the biggest ideas in the universe, space, time and motion. Where he explains not only the difficult concepts but the mathematical framework behind it. In a very beautiful way. So please join me in welcoming professor sean carroll. [applause] thank you cory. Thank you harvard bookstore, one of my favorite bookstores in the whole world. Thank you harvard university. One of my favorite universities in the whole world, i got my own ph. D. Here through number years go, what i will no longer emit and public. I q over. Much of realize its a little chilly out there. I am very happy youre able to brave the elements out there to get here. When if this is a laser pointer. What do you think . Theres something i can push on . Im a little afraid. I have a book the biggest ideas in universe flying one, 19 will be about quantum physics and and going through will be about emergence. But the two minute is in the book to talk about all the muscle take a big idea hail talk about that to give you a flavor of what you would get worried to buy the book. So today i want to talk about einstein equation. And you know im thinking about ive seen up before, good familiar territory e mc2, energy, mass. Good its not gonna be completely unfamiliar. But this is not einsteins equation. Its not what a physicist would mean heavy settle yes, im thinking about einsteins equation earlier today. Heres einsteins equation as a physicist thanks to that. Were you to say it aloud could be armenia minus one half a minute makes a five t team in you. This is the field equation for the curvature of space time itself in general relativity. We never tell you about this. We might give you the words. Space time is curve and the curvature is gravity things like that. But we dont give the a question less your physics major. Shadowed to those physics major out there maybe again it. But even in most undergraduates never see this equation if they get a physics degree. Its considered to be two armed. Theres all these greek letters in their, theyre sub script, we dont know whats going on. An hour from now we will all know whats going on. I want to teach you this equation. How do we get . There we caught with classical mechanics. The whole theme a book ones classical mechanics as opposed to Quantum Mechanics. And classical mechanics, the central equation is newtons second law, and because i may. My physics teacher when i was a freshman said the only equation need to remember is axe equals m a. Thats an exaggeration but it tells you the essential important of this equation forces. Mass times acceleration. Why is this equation so interesting and important . In part because its precise. In other words, its not just a suggestion. The difference between equations and words. Its not just saying the more you push, the more the objects will accelerate. Its a very precise quantitative relationship you can use to fire rockets to the moon. Thats the kind of precision that you need. But the other thing we dont always appreciate that its always universal. What i mean by that is that its not just saying this one time i pushed a car with a certain force and accelerated by a certain amount. Its saying every time, every where in the universe that our forces exerted on an object with max and, you can use this equation to figure out how it will move, how much it will accelerate. Now where this is philosophy lecture, rather than a physics lecture, we would add, why are their relations in the universe that without precise in that universal . Happily we are lower brown today. Where physicists, we know that they, are we celebrate and we move on. We want to learn how to move this equation. If you exert a force on something, you like celery, proportional to its mass, we need to know what the forces, so newton himself talked about of famous gravity, is famous inverse square law. Have a little object with little m masses little am, so well start accelerating might be the earth and a bigger object with matt capital and might be the sun. He can ask how much force is exerted by the force of gravity. Newton says if you have capital and with a big object little am for the little one, you draw a little vector, a little thing with magnitude and direction and you see a vector labeledy itll be called a unit vector, back to where the think length doesnt change what we are asking is what is that sector, what is the force that is acting on this object with this little am . Really this equation because of the little arrows on the afghani e, this is an equation up between two numbers but between two columns of numbers. Have the vector, has a certain amount of force in the acts direction, sort amount in the wider action, certain amount in the z direction, thats what it means to be a vector. As a both aside and a direction. And the direction its pointing it in this case is for so long in a sector pointing from the little mast to the big one. The size of it because began times little and times capital g which is newtons constant of gravitation divided by the distance squared. Thats why its the inverse squared law when youre very close to objects together, gravitational pull is relatively strong, when theyre far away its relatively weak. This is how it goes. So what a physicist would do is to start with some set up. Your two objects, you two masses, distance, what happens next . How do they move next . And its a very nice simplification that occurs right away at the mass of the being pulled cancels out. So f equals m a but f also equaled gm over r squared over t. So matthew can devote by little land 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 characteristics of that object. It doesnt matter what its made of, it doesnt matter how massive it is, or what day of the week it is is. Every object, the same distance from some other objects will accelerate in the same way. He can do this experimentally, he can go to the moon. So here on earth, if you drop a hammer and a feather they wont stop the same rate, its because of our resistance, not because of the laws of physics in some deep level. When you go up to the moon with no air resistance, you can ask of a drop of hammer and a feather late fall at the same rate. This was done by the apollo 15 astronauts. Hopefully they already had some faith in newtons laws of physics otherwise they wouldve not gotten to the moon to do the experiment. But happily turned out right. It made very grainy videos and earnest reconstruction of the event but two objects with very different masses fall in the same way under the force of gravity. Now 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 you are made of all accelerate under gravity in the same way. That will be crucially important. Why . Because as successful as this paradigm was, this whole set up that isaac newton gave us freedom classical mechanics, it wasnt a final word. In 1905 we had input from this guy, Albert Einstein. These days when you see a picture of albert as time its almost easily in his later years. Crumpled, the sweaters and everything, and you get the impression thats what physicists are like. This is what einstein looked like when he was driving the equations of special relativity and general relativity. His hair was calm, somebody was dressing him nicely. I dont want to say with the causal relationship was, but he was a sharp dressed young man. Thats on unison. Three of special relativity 1905 contrasts with the theory of general relativity. In special relatively theres no gravity, generally the tv there is. Well get to that. Einstein did he did not really invent the theory of special relativity, there are already various lines of reasoning that were, that and we put it all together once and for all. We wont go into the details of the special relativity, but you heard some of the jargon, some of the phrases. Motion is relative, thats where the word relativity comes from, the speed of light is cognizant, it cant cope faster than the speed limit light. And theres all these phenomenon that you talked about, length contraction, time dilation. In the book, i dont emphasize these things that much because in my quirky idiosyncratic worldview focus on things that in the plutonium days or something called length and it turns out in the relativity picture different people are not gonna agree on what it is. So rather than starting the flight saying 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 drive everything from that. But all we need to know right now is that einstein didnt actually even in 1905 put the finishing touches on the theory of relativity. That was done arguably two years later by herman murkowski who used to be einsteins professor. When kasky was a mathematician not a physicist, but he was proudly following the students progress, he knew about the theory of relativity, and was he who first said all this great work in your theory the if 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 these effects the you are talking about or manifestations of the geometry of space time. And his famous quotas space by itself in time by itself are doomed to fade into mere shadows, and only the union of the two will achieve an independent reality. A way the referees would let him get away with rhetoric like this in a physics paper today, but 100 years ago you could do it. Not everyone was impressed by this mathematical step forward including simply call i blurred einstein, wrote one of his players the murkowski formation makes rather great demands on the reader in a mathematical aspect. Let me go admiral in here, ice time is no dummy. 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 and he constantly tried to take the physics theories and trying to turn into mouth for its own sake. He was married thats what exactly murkowski was doing. Turns out he was wrong and one of epsteins most senior important feature was to be wrong and change his mind. He would very quickly change his mind and jump on the bandwagon. He came to believe that the space 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 isnt it geometry we should be thinking about when we think about relativity when you think about contracts with history, dont hear the word geometry that much but the essence of it is this. When you want to travel a certain distance in space is a formerly that relates the amount of distance he travel to the coordinate but in a sense role city like new york where the streets were right angle, you could figure the total intervals the, total distance between two points ago a taking to different sides of a right triangle, squaring them using five tiger assist theorem. He, squirrel coves ex squared, equals y square. It means number, when 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. There is a difference between how much distance you could travel 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. Everybody agrees with the distances, walking a Straight Line between those two points means that he personally traveled less distance and if you go off in a detour come back. Again, perfectly obvious the distance you travel the whole point of relatively is true for time rather in space time. Its been kasky that gives us the formula for figuring this. If you travel some distance in space and there is some time coordinate on the universe. Like theres some universal time that everybody agrees on, the International Bureau standards are set up we can read them and figure out what it is. Universal time corn is not the same as the time little apps on your watch. Timelapse is on your watch is something you personally experience and men kaskys point is that time is like faith. The amount of time you experience will depend on how you travel through the universe. And exactly the same way but the amount of space that you travel through depends on the path you take. Here is the formula. Its kind of like pythagoreans is theorem but its not exact of the same. Cow the greek letter cap stands for the proper time, the elapsed time, the actual amount of time you experience okay. There is a formula for. It peace corps minus x square, i sneakily set the speed of light equal to one. Theres a certainly fraction of you that are doing dimensional analysis in your head when herrera can subtract time from space or cavalierly the answer is we are using units where lengths is measured in light years and time is measured in years. How . Case of the speed of light plays a special role in this theory. But the real point is that the time you will feel a lapsing its given by this formula and for a given change in the coordinate time t, it will depend on your path through the universe how much time you actually experience. In space we all know the shortest distance between two points is a Straight Line. You can figure that out rather than going to tangle here just go straight up, he would experience less distance. But here its not peace quirk x where. This is saying that the more we move into space, a less time elapses on your watch. So the rule of relativity is not shortest distance is a Straight Line, but the longest time is a Straight Line. The more space you traverse and come back, the last time you will experience. Youve heard of the twin paradox where theres two twins one comes at the speed line comes back its always hard to remember which one remembers more time. And one moves back in then comes back always experiences last time because the longest time past is the one that just stays stationary. There you go. So this is geometry right. This is pythagorean theorem, its updated for the space time outlook and its a minus there l 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 me what initial line 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 ritchey tensor is proportional to teemu nu and if you guessed that, congratulations and einstein both. Thats what he guessed. Turns out not to work. It turns out that if that were the right answer. Energy and momentum would not be conserved in the right way. So you had to be more clever than einstein was. And you got to imagine him. 1915 sort of feverish scribbling things down because by that time einstein, bless his heart, hed given 20 talks on 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 you take the ritchie tensor and you subtract from it one half times the curvature scalar, times the metric. So those two things combined are still a two index tensor, a four by four array. You set those proportional to the energy tensor, and you get the right answer that is what physicists think of as equation. That is where we started see it was not so hard, but were not done yet. Got to tell you one more thing, because what one does, of course, one gets an equation like this has a drink, something, but how do we know whether its the right equation we have to do what is called solve thing . The equation, in other words this is an equation that because remember the riemann tensor can be calculated from the metric tensor. So if you have a metric tensor, you can calculate arm. You knew and then r and you can ask it actually equal eight pi team 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 schwarzschild and churchill died year later because he contracted terrible disease at the front. 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 the solar system. Okay, so schwarzschild didnt ask what happened inside. The sun, thats actually harder. Thats a little bit of work. You can do it. But hes just looking in empty space. Okay. And so this is the right answer. It has been tested to exquisite. Einstein used it to predict things that were shown to be correct. It was a wonderful success but you look at this closely, theres Something Weird going on and physicists love to look at things closely. Sometimes so you notice that some of the terms in the upper left hand look like one minus two. Gm over are so what happens when r equals gm . Then that zero zero term here is. And this t term sorry, this r term is one divided by zero. Youre not supposed to do that. Thats infinity thats bad. And furthermore when r equals zero, this infinity or negative. Anthony that sounds so if you plug in the numbers, you find that this is not important problem for the motion of planets around the solar system because this metric is only supposed to apply outside of the sun and r is always much, much bigger than gm outside the sun. So this apply anymore inside the sun but of course your imagination is going to go. What if we took the sun and we squeezed down to a little ball so that it was smaller than two . Gm what would happen . The answer that would happen is a black hole. That point are equals to gm is the Event Horizon of a black hole. And you can see this because remember the zeros zero component of the metric like all this pays off now this is paying off that 000 to the metric, which we said is the rate of flowing compared to some coordinate goes to zero at r equals two. 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 never see it cross the Event Horizon. If you instead fall into a black hole, then your personal time just keeps ticking. But youve made a terrible so the light from your cheeks is redshifted from the point of view of the outside observer. So it looks like youre very embarrassed that you have fallen into a black hole, which probably you are. This is an image slightly up image from the Event Horizon telescope, which many people here at harvard have been working on as part of the black hole initiative. The point is the point. I want to drive home is albert and Carl Schwarzschild went to their graves, not having any clue there was any such as a black hole. They werent trying to understand black holes. They certainly werent trying to predict their existence what they did was derive an equation and then solve it and the lesson is the equation cards are much smarter than we are. This is why its worth sitting through all this stuff about equations than just telling you stories and metaphors and analogies. Because theres that precision and that universe ality in the equations. And then if you extrapolate them beyond what you were trying to understand originally, you might be surprised and pleasant ways. The of a black hole lurking there inside einsteins equation. As soon as he had derived it. But it took decades for physicist realize that is what the equations were telling them these days we use einsteins equation to describe gravitational the large scale structure of the universe in the spiraling holes, the expansion of, the universe 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 equations. So as weird as they are and as much work as it is to look at them and go i dont know, theres greek letters thats greek 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 important way. Thank you. And we are going to take questions maybe the house lights could come up a little bit and there are people with microphones ready to respond. You raise your hand and have a 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 jason. Thank you for taking my question. Having all these tools to explain the fundamental workings of reality. Whats next on your hunt or what are your professional goals currently besides grading papers at Johns Hopkins . Well, you know, to be too obviously cliched about it, but what drove einstein to this great accomplishment was a fundamental income ability between, as newton understood it, and relativity that he had just invented. And we all know there is a remaining lurking incompatibility between 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 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. You got to learn it. But okay. Its very beautiful, very understandable. Quantum mechanics we dont really know what saying when we talk about Quantum Mechanics many ways. I think that the fact that we dont understand Quantum Mechanics might be related to the fact that we dont understand gravity. I think that we got lucky when we made quantum theories of Everything Else in the universe electromagnetism and the strong force, the fermions and so forth those were a simple of starting with a classical theory, quantifying it when you try to do for gravity, for general relativity, it doesnt work. So my guess is we need to understand Quantum Mechanics better if we to understand quantum gravity, whos next. Yes. Anyone yet . Thank you. Thank you very much. Hi, im steve. You touched upon the differences between physicists and mathematicians and so in riemanns paper he hypothesized about the physical implications of his work. And in einsteins time car time got very close. And im wondering if youd like to, you know, take a victory lap about the physicist, sir get into general relativity first or what it in your opinion that allowed to accomplish that when so others especially the mathematicians close. Well i do not want to take a victory lap i appreciate question and i get it but this is its harder to find a better example of when you needed both mathematicians and physicists the physicist the mathematicians would never have invented general relativity because they werent being bugged by the incompatibility, newtonian gravity and special relativity like physicists were. But some in the history of physics, theyre faced with a new phenomenon and. They need to invent new math. And physicists will sometimes do. It would have taken a long time for the physicist to invent remaining in geometry that that something that you need to be an accomplished mathematician do so i think the right way to think about it is it was the perfect synergy between the physicists way of thinking and the mathematical way of thinking. Has a go ahead peter so to my take holmes one the equations are smarter we are i love that another is that time is really weird. Yeah so given those two take home messages, you had a slide up there where you had a tensor and there were a lot of other things that were being explained in the tensor that we didnt go into, which is totally fine, but im wondering if theres anything in those hidden dynamics about causation in of, you know, empiricists are really trying to at retro temporal causal effects but is there anything hidden in the equations given that theyre smarter than we are that would indicate that. Okay causation is also not necessarily a linear type of effect that we intuitively imagine it to be good. So the short answer is no, but i can elaborate a little bit. I would, but but i need to be careful because. Not everyone agrees with me, even though im right. Its also happens. I think the notion of causation is not fundamental. I think the notion of causation emerges at a higher level description where you have, among other things, a pronounced arrow of time that distinguish is between the past and the future. One of the 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 reflect . Im sure you will in a future book on the of time travel and parallel universes you just give me 30 seconds on what im sure is a thousand pages of thought. Sure i have to disappoint you. Im not going to reflect on that because i reflected on it in a past book. So my very first trade book is called from eternity here the quest for the ultimate theory of and i talk about both parallel universes and time travel, the introduction of general relativity, which that spacetime is curved and dynamical opens the possibility thinking about time travel in a serious scientific way. My guess is that the answer is no is not possible, but honestly we dont know. And thats kind of, you know, a thorn in our side. People like kip thorne and, also myself, have written papers about can you solve, can you find solutions to einsteins equation that would allow us to travel backward in time and the answer is you can strictly speaking them but they always seem a bit pathological. So theres no sort of clean, reliable way to twist spacetime so much that you can go back and visit yourself in the and undo your mistakes sorry. Yes. Hi. Oh, thats loud. My names alex. Thanks for taking question back when you were going through like the derivation of the equation that we were getting to in the end we went from the four component riemann tensor the reaching tensor in doing this arent we losing like i mean maybe my mathematical understanding of it isnt at a high enough level but in doing this arent losing some information, which would mean that we couldnt really do this and if we arent losing any information doing this, why do we even write it out in the more complicated way . Good. This is an excellent question. Been paying attention. Im very glad. See that . Yes but think about it. So this equation right there in the middle, on the left hand side, it involves some parts, the riemann tensor, but not all of them on the right hand side. It sets equal to the momentum tensor. So fact that the left side does not include the whole tensor allows us to imagine that if you have two space times with exactly the same configuration of energy and momentum, you might have different curvatures of spacetime and you do and theyre called gravity the way these Gravitational Waves can propagate in empty space. So one solution to einsteins equation in empty space. So 10 minutes is a zero everywhere. Theres no stuff anywhere. One solution is minkowski space spacetime, no curvature, but another solution is a gravitational wave going by. And so you dont want out the left hand side of einsteins equation two. Absolutely. The whole riemann tensor in terms of the Energy Momentum tensor want to give it to some some freedom. But it turns out you go the next layer and you and you do your math. The components of the riemann tensor that are not directly fixed by this equation are related to the components that are once you take their derivatives and things like that so its not that all hell is broken lose theres still a way to derive give the equations for Gravitational Waves from einsteins equation. Last question up at top, are there . I was wondering if you have a favorite current theologian or philosopher and and as youre thinking about trying to explain theory of Quantum Mechanics, are there any philosophers or theologians that have helped you come to ideas around the math of the physics that youre doing . Well, technically speaking, a philosopher, since i am jointly appointed in the physics department, in the philosophy department. So my favorite one is myself, but if if you exclude those im not going to answer because the philosophers that i know and admire are all my friends, sometimes my enemies. And to pick out one of them would get them jealous, you know, let me, let me. Instead of answering your question, let me say the following thing. Theres a continuum of interest and between strictly hardcore physicists, just calculate things and traditional philosophy firms who sit in their smoking jackets and talk about the meaning of life. Both of those poles of the spectrum exist. But theres a lot of activity in between. So were starting up a new program, Johns Hopkins called the forum on natural philosophy, to bring back the old days when there was not a dividing line between physics and philosophy, between science and philosophy more broadly, i think that theres plenty of science questions for which philosophy is completely irrelevant. But theres also of science questions for which philosophy is extremely helpful, not crucial. So rather than saying, who is my favorite philosopher, i would rather say, lets get everyone to be a little bit of a philosopher and a little bit of a scientist