Physicist Sean Carroll on "The Biggest Ideas in the Universe"

hello everybody Welcome to Pioneer works I feel like every time that we're able to get together feels like a gift I don't think we take it for granted anymore do we I'm Jana Levin I'm the director of Sciences thanks so much for coming how many people are new to Pioneer works oh my God I love that uh I think it's a real honor that we're here in the Charles Atlas exhibition the mathematics of Consciousness it's such a gorgeous show I want to give a shout out to Charles who is here can we give Charles a little shout out this is one of those instances where we curate an art exhibition totally independently and it somehow creates a world in which we start to think about science and so we do science events in the context of this exhibition I think it's really exceptional and I think it's something that is a real labor of love for us and it really means a lot to us that you come out I want to do a little housekeeping we're going to have a book signing after the event and the amateur astronomers Association of New York will be out in the garden weather permitting so we can do some Sky viewing so don't race out feel welcome to linger and enjoy the evening tonight's speaker Sean Carroll is a very dear friend of mine a lifelong friend of mine it occurs to me I've learned I've known Sean longer than I've not known Sean and we used to study Quantum field Theory together you can imagine lots of Gossip there um Sean is recently become a Homewood professor of natural philosophy at Johns Hopkins University and his other title I have to quote because it's so fantastic he is fractal faculty at Santa Fe Institute uh Sean is the host of the wildly popular mindscape podcast I'm sure a lot of you have heard it mostly because I was on it that one time right um and uh he's also a New York Times best-selling author he's incredibly erudite person Sean's one of the only people I would just say hey come come claim the stage because he's so witty he's so insightful he's an unbelievable lecturer without being distant being totally engaging and thought provoking he really is a philosopher scientist which we argue about sometimes and um and and I always learn something even after all these years from listening to Sean speak Sean also has a beautiful article actually on pioneerich's broadcast which is our virtual manifestation beyond the walls it's Pioneer XP on the walls and he wrote a column just Tuesday we published it for a series we call Picture This in which a scientist describes the power of drawings and I hope you'll take a chance to read that it's really very special so my last shout out is to science sandbox of the Simons foundation for making these Live Events possible let us thank them for making this free and open to the public in New York City right and also shout out to the Sloan foundation for allowing us to make pioneerix broadcast free and open to the world and that's really just incredible generosity slim Foundation thank you and of course to you for coming and supporting us because that's really why we do this so thank you all so much and uh without another moment longer please welcome Professor Sean Carroll okay thank you Jana thank you Pioneer works thank you New York for coming out it's great to be here yes thank you New York this is my first in-person talk on this subject and as much as I do uh love Janna very much we've known each other for a long time and been friends I need to say that she lied to you just now and when she said that I am one of the few people who she would just invite to come up and take the stage and say whatever I wanted whatever what actually happened was she invited me and I said okay I would like to tell them about Einstein's equation and she said no foreign now what does that mean Einstein's equation because you've seen Einstein's equation right you've seen E equals m c squared before this is not a surprise you probably even know what it means energy is mass times the speed of light squared there's little subtleties there it actually means that mass is a kind of energy rather than energy is a kind of mass but there's nothing that is too intimidating even though it is you know symbols and there's a little exponent a little superscript you can kind of understand it so I'm here to tell you that this is not Einstein's equation this is not what physicists call Einstein's equation this is too easy we wouldn't want to get away with something this easy and give it to Professor Einstein as his namesake this is Einstein's equation and if I read it out loud it's R mu Nu minus one half r g mu Nu equals eight Pi G team you knew the E equals MC squared comes from what we call special relativity this equation is the heart of General relativity Einstein's theory of space-time and geometry and gravity and I would like to explain it to you in today's little session we have here now I know what you're thinking you're thinking wait a minute I didn't come here to hear about equations I was hoping for like some stories some jokes maybe some pictures my advice is suck it up we're gonna do the equation it's not that bad this is part of my message here it's not going to be that bad moments will be bad but the whole experience will be rewarding and useful as long as you have a study guide so you should think of yourself as a Dante there can you see this there we go this is me that's Virgil I'm guiding you through these are the equations down here they're trying to climb into your boat but I'm here to tell you they're just not that scary that's the idea okay and we can start slowly classical mechanics this is what the book is about the book is called the biggest ideas in the universe space time and motion there will be future books with more big ideas about quantum mechanics and complexity but this book is about classical mechanics bequeath to us in the 17th century by Isaac Newton and here's the equation that really sits at the heart of classical mechanics f equals m a force equals mass times acceleration the more you push the more something is going to start moving and so but let's think about what this means what's so great about this equation rather than just the words the more you push the faster something starts accelerating well for one thing it is precise I mean that's kind of hard to miss right it is a rigorous quantitative relationship it's not just telling you the more you push something the faster it goes it is a proportionality relationship that tells you the amount of force is proportional to the amount of acceleration so if I double the force on something it will accelerate twice as much precisely and there's the mass in there which means that if I push on a light object with the same force it will be accelerated more than if I push on a heavy object and it tells you by exactly how much you need that kind of precision if you want to do something like get a rocket to the moon so that's why equations are very valuable for practical purposes but the other thing is it's not just a mathematical equation it's a law of physics which means that it's Universal it's not simply telling you that this one time when I pushed something it accelerated it's telling you that every time in the universe that you act a force on something it will accelerate by exactly this much that is a non-trivial fact about the universe that there are relationships like that that there are quantitative rigorous patterns that the Universe always obeys also there's these little arrows on top of the F and on top of the a what is up with that so we got to get real here okay what's up with that is that F and a the force and the acceleration are not numbers they are vectors so they are sets of numbers okay a vector is something that has not only a size a magnitude but also a direction right when you put a force on something or when it starts accelerating you tell me not only how much force but in what direction you are pushing it so the working physicist instantly wants to know well how do I write that down how do I keep a little notebook telling me how much force I am pushing in each Direction and the answer is to think of the vector in terms of components of the vector so I go around and I set up coordinate systems I have a x-axis a y-axis a z-axis maybe this is not too crazy New York I'm very glad we're close to New York there's a grid on the streets and you can tell someone where to meet you on the grid and the height above ground that's a coordinate system and then if you have a vector you can think of that Vector as the part that is pointing in the X Direction a certain amount in the y direction and the Z Direction these are the components of the vector and we can think of the vector as a little column with those components in it so secretly this one equation f equals m a is three equations the force in the X direction is the mass times the acceleration in the X Direction and likewise for y likewise for Z we can even instead of the little arrow which is just sort of fun it's fun to type in the Arrow when you're typing up your slides for the talk you could also write it as indices so you write F sub I which where I could be X Y or Z the force in the X Direction the y direction or the Z Direction This is not pointless formalism this is going to become really really important very very soon in your life unless you leave I guess but as long as you're here this is going to become very very important but the physics behind it is what really matters here and the physics is clearly incomplete if I tell you f equals m a and I want to know how much an object accelerates like Newton is thinking about the planets moving around the Sun I need to tell you what the force is right how much force is being exerted is kind of an important part of this story so gravity is the first force that Newton really got right famously Newton had the inverse Square law of gravity if you have two objects with mass capital M and Little M there's a force that the Big M exerts on the Little M that is proportional to both their masses and to one over the distance between them squared the famous inverse Square law of gravity so you have another equation an equation for the force of gravity every symbol has a meaning and it is once again an equation between vectors and it's honestly not that hard to understand it is once again a proportionality it's actually pretty straightforward you can see why Newton was very happy with it the story about the Apple probably totally made up made up by Newton he told the story about himself he wanted to make himself seem a little bit more magically clever than he really was he was clever enough honestly but he told the story and the point of the story of the apple is not that he discovered gravity because an apple fell from a tree people knew about gravity already the point of the Apple story is that Newton realized that the same force that pulls the Apple from the tree explains the motion of the planets around the Sun that's the universality once again it's the same thing going on and that was a very important Leap Forward now once you have these equations f equals m a and f equals g Newton's constant of gravity M M over r squared you can ask the question how do things move in the world and you instantly discover something very profound the equations are helping you already what you discover is f equals m a and that equals GMM over r squared the Little M is the mass of the object that is being pushed around but that little M appears exactly the same way in both Expressions so it cancels you can do math namely X out the little m's and you get an equation for the acceleration of an object because of gravity that is completely independent of the object's Mass so if you drop a hammer and a feather under Gravity they will fall the same rate and the Apollo 15 astronauts actually did that they dropped a hammer and a feather they made a little movie it's too grainy to show so I'm showing you a artist's impression but they fell at the same rate okay why is that interesting and important because it was Galileo who first pointed this out that objects of different masses fall at the same rate previous generations had thought that heavier objects fall faster why would they think that because heavier objects clearly fall faster we've all seen that actually happen but what Galileo said is that's only because of air resistance and you were completely allowed to say to Professor Galileo but there's air resistance that that matters and this move of neglecting the air resistance and putting it back in later is really the birth of modern physics as I am as far as I'm concerned and for Newton it carried a lesson which he himself struggled with a little bit it means that gravity is universal it means that when you drop something under the force of gravity not only does it not matter how heavy it is it doesn't matter what color it is what it's made of how big it is nothing matters every object falls the same way under Gravity it's a completely Universal Force unlike for example the electrical force a positively charged particle and a negatively charged particle will move very differently under the force of gravity so already there's a hint something special about gravity but neither Newton or any of his contemporaries could quite put their finger on it happily time passed this guy came on the scene Albert Einstein in 1905 Einstein uh invented a whole bunch of things including the final steps in the theory of special relativity special relativity is the theory that says that motion is relative rather than absolute and also crucially importantly the speed of light is a limit an absolute limit that is the same to everyone you might think if a car is moving toward you and you're moving toward it you add up the velocities you would see the car moving at a different velocity than someone stationary on the ground Einstein's saying light doesn't work that way no matter how you think you are moving you see light moving at the same velocity and if I were giving electron special relativity we'd go into that but the point is Einstein explained the way to make sense of such a bizarre sounding statement is to change your idea of how space and time work and this is the origin of all the conventional Explorations in special relativity of length contraction time dilation all that stuff but it wasn't quite the final word on special relativity that happened two years later due to Hermann minkowski who was actually Einstein's old professor Einstein had taken classes from minkowski at University and minkowski was a mathematician not a physicist and it was he who said Albert the way to think about your theory is to imagine that there is no such thing as space or time there is only space time if you need three numbers in a coordinate system to locate yourself in space you also need one number to tell you when to get there there's a lecture at APM and Pioneer works that APM matters to locate yourself requires both locating yourself in space and time Isaac Newton could have said that what minkowski is saying is that the choice to divide space-time into space and time will be different for different kinds of people so it's better to think of it as a single unified space-time in his quote often quoted henceforth Space by itself and Time by itself are doomed to fade away into mere shadows and only a kind of Union of the two will preserve an independent reality they do not let you like write physics papers like that anymore but back in the day in 1907 you could get away with this many people thought this was a great idea one person who was skeptical Albert Einstein in a paper that he wrote the next year he says minkowski's formulation makes rather great demands on the reader in its mathematical aspects so minkowski was a mathematician but Einstein was a physicist physicist he saw minkowski's words and he thought those they were just sort of trying to dress things up in some fancy mathematical formalism he understood space and time he didn't need to glue them together to make space time he soon changed his mind by the way as we will see but what was the motivation why would minkowski say this in the first place the Insight is that time is quite a lot like space in the theory of special relativity what do I mean in space we can measure the distance when we walk down a path right that's not very surprising here's the guy with his pedometer counting his steps if you know how long your steps are you can figure out how much distance you have traversed and if you go in a straight line you know that you will take the shortest distance between the point where you start and the point where you end if you don't know if you don't have your pedometer with you but you're walking the streets or something like that if someone has given you a coordinate system X and Y you can figure out how long you have gone with a little formula called Pythagoras's Theorem right the X and Y coordinates act like sides of a right triangle and the distance you traveled is the hypotenuse so that distance squared is just x squared plus y squared a high school geometry formula and what minkowski realized is that time is like that that we should think of time not as something absolute and out there in the universe but it's something personal as something that different people will experience in different ways the time that has elapsed on your wristwatch is like the distance you walk two people can go from the same point to the same point and walk different distances if one goes in a straight line and one goes in a curve time is like that two people can start at the same event and end at the same event but experience different amounts of time and he even came up with a formula for it and here it is and you see this formula is very similar to Pythagoras's Theorem Tau in the Greek letter tau is telling us the time that is elapsed on somebody's wristwatch T is this coordinate time that is kind of a universal thing we all agree on we say APM we all agree on what that means it's not your personal time we treat them the same your personal time in the time coordinate because most of us spend our lives moving slowly compared to the speed of light when you move close to the speed of light this becomes important but the crucial thing is there's a minus sign now there's a minus x squared so what that means is the more you move in Space X the less time you will experience that's the difference between space and time in Space the shortest distance between two points is a straight line in time the longest time between two events is a straight line and this leads to the famous twin paradox and things like that no that's a much better cleaner way to think about special relativity than what Einstein had written down it's almost like geometry but there's a minus sign how hard can it be the problem is as Einstein himself very well knew when Newton gave us classical mechanics the great Triumph was understanding gravity right but Newtonian gravity was incompatible with the new theory of special relativity so the idea was can we come up with a better Theory can we make a theory of gravity that is better than Newton's Theory and one that is compatible with relativity happily Einstein was on the scene he was thinking about it and he went back to that observation that gravity is universal right that if you drop two objects it doesn't matter what they're made of how massive they are they will fall in the same way and he promoted it to something called the principle of equivalence the equivalence of gravity and acceleration he said if you're in a box if you're in a room like this with all the windows closed you think you can tell that there's gravity because you can drop things and they will fall and you'll ha gravity but he said look if you're in a rocket ship that is accelerating far in interstellar space where there isn't any gravity if you drop things they will also fall and they will also fall in a way that is completely independent of their Mass their composition because they're not really falling at all it's the rocket that is accelerating and Einstein posited that there is nothing you can do in that room to tell whether you are really in a gravitational field or really just accelerating and this is different than other forces of nature electromagnetism for example you could just look at different kinds of electrical charge and instantly know if you're in the midst of an electrical field or a magnetic field but gravity is different because it is universal so you or I if we thought of that Insight we'd Pat ourselves on the back and we'd you know write it in our live journal or whatever the kids these days we make a tick tock I don't know Einstein being Einstein he went from the principle of equivalence to saying well if gravity is universal it must be because gravity is not a force living inside space-time but is a feature of space-time itself and the feature that he guessed it was is the curvature of space-time remember minkowski had suggested that space-time has a geometry kind of like Pythagoras Theorem but with the minus sign in there and Einstein says well maybe we need to generalize that maybe there's an even more complicated geometry that is responsible for the force of gravity the bad news was this sounded like you needed math and Einstein did not know the math he was a physicist physicist he knew exactly as much math as he needed but no more happily he was friends from University with Marcel Grossman who was a now a working mathematician and he knew this newfangled thing called differential geometry all the stuff that had been developed in the 1800s to deal with curved spaces or for that matter space time so Einstein basically got tutoring for Marshall Grossman and in fact General relativists to this day have a big international meeting called The Marshall Grossman meeting to thank Marshall Grossman for teaching geometry to Albert Einstein and it was sort of like a relatively new development at the time right like if you go back to Geometry that you learned in high school that's euclidean geometry Euclid invented this or put the final touches on it 2500 years ago or something like that and it's famously based on a postulate an axiom called The Parallel postulate if you start with a little line segment and send off two initially parallel lines from it those two initially parallel lines will stay parallel precisely forever they will stay exactly the same distance apart that seems reasonable it is what would happen if you had an infinitely big tabletop to draw lines on and people thought it was so reasonable that it was almost unimpeachable that they tried very hard to prove the parallel postulate from the other axioms of geometry turns out they didn't succeed for good reason namely you can't prove it because it might not be true so in the 1830s people loboschevsky bolyai Carl Friedrich Gauss investigated the possibility of replacing the parallel postulate with a different idea what if you say when you send out initially parallel lines they always move apart from each other or what if you say when you send out initially parallel lines they always come together it turns out these are perfectly good alternative axioms that could serve as the foundation for a different kind of geometry the diverging idea is what we call negatively curved or hyperbolic geometry it's kind of like living on a saddle or a potato chip the positive curve geometry is where the lines come together if you send up two lines from the equator lines of longitude they will meet at the North Pole people knew about that but this was the first time that mathematicians had really proposed something in the case of the negatively curved geometry that you couldn't build in your in your laboratory the negatively curved geometry cannot be embedded in ordinary three-dimensional euclidean space so this was a big breakthrough but it was also clearly extremely specialized all three of these possibilities flat negatively curved positively curved are making a huge assumption that whatever the curvature is it's exactly the same everywhere that convergence or Divergence of the parallel lines is exactly the same no matter where you start the lines and no matter how you Orient them in space so clearly there's work to be done full employment for young mathematicians let's generalize this to cases of arbitrary curvature to spaces that are lumpy and warped and twisted in different ways and that task was taken up by Bernard Riemann in the 1950s it was taken up reluctantly by the way he didn't want to do it he needed to get one more degree in the German system to be allowed to teach in universities and his advisor was Gauss that's a pretty good advisor to have and Riemann gave him a list of possible thesis topics and Gauss came back suggesting the one that Riemann thought was the most boring the foundations of geometry and he literally you read his paper and he's complaining like I'm not very good at this he he was pretty good at it actually so what he wants to do is answer the question how do we think about arbitrary geometries in arbitrary numbers of dimensions and what he realized is if you tell me remember what the mathematician wants to do is what is the information you need what is the data what is the piece of info you need to give me to tell me the answer to the question that I'm asking the question is what's the geometry Riemann says the information you need is for every single curve or line that I can draw in this geometry you tell me how long it is you give me a way a method of calculating the length of any curve and what he argued kind of hand wavy to be honest but what he argued was if you know the length of every single curve in some space there's a unique way to knit them together to form the entire geometry so that that amount of information the length of every curve specifies the geometry uniquely that sounds like a lot of information you have to like make a list of every single curve right but that's okay there's this little thing called calculus you can read about it in my book and Riemann certainly was an expert at calculus and the The Secret of calculus is you can zoom in so Riemann says take an arbitrary kind of Wiggly curve the nice thing about smoothness in the context of geometry is no matter how Wiggly the curve is if you zoom in on it with a microscope far enough it will begin to look straight and if the background space it's in is just ordinary euclidean space then we know how to calculate the length of a straight line it's Pythagoras Theorem again just in three dimensions the distance squared is going to be x squared plus y squared plus Z squared it's not that different so what Riemann suggested is we do that but we do it for arbitrary weird geometries not just for euclidean space for example you and I know Riemann didn't know because it happened later but you and I know this is basically the information that tells us about minkowski space time right we have euclidean space which is just x squared plus y squared plus Z squared an analogous role is being played in minkowski's space-time by the time formula t squared minus x squared minus y squared minus Z squared so you want to generalize this pattern what is the pattern the pattern is you have a set of coordinates x y z or t x y z or whatever weird coordinates you want you take every combination of two coordinates and you multiply them together and you multiply that by a number and then you add up all of those combinations so x squared y squared Z squared it's all plus one in that case so you don't even notice it there's a number there but the number is plus one it sort of disappears here the number is plus one minus one minus one minus one and in all these cases it's x squared y squared x z squared but you could imagine x times y x times z t times x all that stuff you're allowed to imagine all those and in the general case they will all be there so here's the general formula this is where it gets a little tricky I know you followed everything I've said perfectly up until now take a breath and here we go there's a little formula here this is the general formula for an arbitrary tiny little interval in a space with an arbitrary geometry in arbitrary coordinates so this you know you get you got to do a little work to get this level of generality okay the interval squared it's just a generalized version of Pythagoras the interval squared is some number times t squared we're imagining we're in space time plus some number times T times x some number times T times y some number times T times Z and then you zoom over to X and do some number times x times t x squared x y and so forth four coordinates in space time you're going to have four times four is 16 terms in this equation and you're going to run out of letters pretty soon right yeah there's enough letters in the alphabet but what if you had 10 dimensional space time you'd be out of luck so this is the general formula it's a little unwieldy but this is telling you the geometry of the space in principle what you'd like is for it to be a little more clever mathematicians are great fans of being clever so they say let's invent some new notation and this is where the Greek letters come in you hearken back to the first slide where there were mu and new in Einstein's equation those are Greek letters and rather than the i j k that you might use for XYZ indices like we used for Newton's f equals m a in space time we use Greek letters to represent both time and space so you start with t x y z and you label the T coordinate X zero y not X4 well because sometimes you're going to think about two-dimensional space time or three-dimensional space time or four-dimensional space time you always have one time coordinate and some crazy number of space coordinates so let's label time as the zeroth coordinate and then you use the letter mu to stand for zero one two three or t x y z if you want to put it that way and then this interval that we invented this is the same formula as the last slide a times t squared plus b times TX we can remember what coordinates we're multiplying by and if this is t squared and T is x0 we call this core this component G zero zero lowercase G is the letter that we universally use for this idea of the interval functions the number multiplying T times x is g01 the number multiplying t y is g02 ETC and this collection of numbers how many of them are there 16 good paying attention I'm a professor I have to do this occasionally to make sure you're paying attention so four coordinates four times four sixteen numbers G mu Nu and this is given the name the metric tensor and you have right now in this moment of your life taking the first little bit of initiation into real world differential geometry the metric tensor G mu Nu tell your friends about that at the next cocktail party so let's let's bring bring it down to earth a little bit as much as we can here this is the formula we had in minkowski space for time right the time you experience squared is t squared minus x squared minus y squared minus Z squared I'm telling you that you can think of that in a compact form as components of the metric tensor and the reason why it didn't seem so hard the first time we saw it is because most of those components are zero right G zero zero or gtt is the thing multiplying t squared it's just plus one gxx gygz are all minus one and everything else is zero so you can see why this is kind of a fun useful notation the examples of metrics in curve space time will of course become much more complicated and we will need all the components that we have but in flat minkowski space time it's not so bad and all of these components mean something the spatial components the components of the metric xxxy Etc these help us calculate distances in Space the zero zero component or the TT component that's an important one if you think about it it's telling us the relationship between Tau and T so it's the rate at which time flows the rate at which your wristwatch ticks compared to this universal time coordinate and the off diagonal as we say the sort of top row and left hand column examples these are things you're going to need when time and space are twisting into each other now you might say I probably don't need that I've never seen time and space twisting into each other you are wrong if you've ever seen this picture this is the picture from the movie Interstellar that was put together by Kip Thorne and other scientists at Caltech along with the special effects crew uh making the movie and this is light rays spinning around a spinning black hole and the action of the spinning black hole is to twist time and space together you need those top and left hand side components of the metric in order to make this picture you got to do work in fact this ended up being a referee Journal publication where the authors were mostly people who worked at the special effects house and Kip Thorne at Caltech who later won the Nobel Prize although not for this so you do need in modern astrophysics to really have this metric tensor and let it do its work it's very important it helps us make movies and so forth it also helps us understand gravity but wait a minute Einstein's idea was not that gravity is the metric he didn't even know what a metric was at the time his idea was that gravity's curvature so clearly if Riemann is right and the metric tensor tells you everything you need to know about geometry then the metric tensor has to tell you the curvature but a lesson in mathematics is that just because it has to tell it to you doesn't mean it has to give up its Secrets easily so there is work to go from the metric which tells you the length of every curve to the curvature how do we talk about curvature well one way of thinking about it is remember Euclid's parallel postulate right start two lines initially parallel they stay parallel that's flat geometry no curvature when you have curvature the lines can either go apart or come together and what Riemann says is that in an arbitrary geometry when you shoot out those lines initially in different directions you can get different Divergence or convergence and for that matter the lines can twist around so there's a lot going on Riemann says to tell me the curvature at every point at each point you have to give me the orientation of your line segment the orientation of the initially parallel lines you shoot out from it and then you need to tell me how it twists around clearly this is a job for a tensor I didn't tell you what a tensor was I mentioned the metric tensor the short version of what a tensor is is something that has more than one index or if something has any indices at all okay G mu Nu has two indices so it's a tensor you might have thought your five minutes ago self thought wow two indices that's very intimidating and scary the Riemann tensor has four indices the Riemann tensor is what tells you how curved a space-time is are Lambda rho mu Nu it is the answer to the question you tell me the point where you start the initial segment that I'm going to shoot out lines from the direction in which I shoot out those lines and I the Riemann tensor will tell you how they converge or diverge or twist okay there's some details we're glossing over but that is the basic idea that you get there and just to sort of drive this home what's going on you know if something has one index like a vector you can think of it as a little column of quantities a number of quantities equal to the number of dimensions of space or space time or wherever you are so in four-dimensional space time the momentum Vector has four numbers the metric tensor has 16 numbers the Riemann tensor has I'm not hearing it but the answer is 256. 4 times 4 times 4 times 4 numbers yeah some people go like I knew it I just didn't want to say because I didn't want to like brag this is clearly just showing off showing you the riemont tensor no one in their right mind would ever write this down but the Riemann tensor is a four by four Matrix of four by four matrices which is just to say there's an enormous amount of information contained in this question how does space time curve around got it let's put it to work enough of this math stuff let's do some Physics Einstein's goal is to create a new equation for Gravity so this is Newton's equation for Gravity we've already canceled out the little m's right this is the equation that tells you the acceleration due to gravity depends on Newton's constant the mass of the thing causing the gravity and the distance squared so we need to replace both the right hand side and the left hand side we need to replace acceleration with something that has to do with the curvature of space-time that was already enough work that we did but you also have to replace the idea of mass because Einstein knew better than anyone else E equals m c squared mass is just a kind of energy and in fact the real Triumph of special relativity is that it unified a whole bunch of words that sound roughly energy like mass energy momentum stress various things pressure they're all part of a unified idea in relativity called the energy momentum tensor you're going to be so friendly with the tensors by the time this is done this is another two index tensor the energy momentum tensor and it combines all those things I just told you about energy pressure stress strain heat flow momentum all of those things the zero zero component is literally the energy including the mass so if you're like in the solar system if you're thinking about apples falling from trees or planets going around the Sun this is the only one you need you really can get away just with the energy aka the math s because E equals m c squared but in the universe things get more complicated and you need to think more txx tyy TZ these are telling you the pressure so the air in this room has a pressure and it has a pressure in the X Direction the y direction and the Z Direction and then you can have a bunch of wild crazy things on the other slots momentum heat flow stress Etc if you need those you're in deep do do it's very very complicated something has gone terribly wrong you should rethink your life choices but in principle everything you might ever want to need to know about energy and energy like things is summarized in this two index tensor so Einstein's task is to say okay we have the metric which tells us the geometry the metric tells us the length of any curve we can use that to calculate the Riemann tensor in case I didn't say that the nice thing about the Riemann tensor is there's a formula for it in terms of the metric if Riemann is right and the metric tells you everything it tells you the Riemann tensor among other things it's the riemont tensor that characterizes the curvature of space-time and Einstein says the curvature of space time is gravity the energy momentum tensor is telling you matter Mass momentum all that stuff so to replace Newton's equation acceleration is proportional to mass you want something like space-time curvature is proportional to the energy mentum tensor there's an immediate problem here namely the space-time curvature is characterized by a four index tensor with 256 numbers in it but the energy momentum tensor characterizing mass and all that stuff only has 16 numbers in it it's a two index tensor what are you going to do you cannot set two tensors equal to each other or proportional to each other if they don't have the same number of components you need to line them up so here's where I don't tell you everything you got to skip some steps but happily for you I wrote a book that you can buy to get all the steps filled in the short answer is you can boil down the important parts of the Riemann tensor into making smaller tensors the riemont tensor has four indices but you can distill it down to something called the Richie tensor which only has two indices named after Professor Richie carbostro and then you can boil that down to just get a number called the curvature scaler at every point in time what is the overall amount of curvature at every point in space time and then the trick if your Einstein is what how do you put these together to make something you can relate to the energy momentum tensor and you might guess well wait a minute it's pretty obvious even I can guess this this guy right in the middle the Richie tensor has two indices just like the energy momentum tensor maybe they should be proportional congratulations to you you just made the same guess that Einstein did but it's not the right one it turns out if you guess that you don't get energy conservation right and things like that so Einstein back to the drawing board he was very clever he figured it out this is the answer the answer is are mu Nu minus one half r g mu Nu equals eight Pi G team you knew the Richie tensor minus a half times the curvature scalar times the metric tensor is proportional to the energy momentum tensor and you see in that proportionality once again Newton's constant of gravity plus eight and Pi which are famous numbers okay so this is it this is I don't know which is more famous eight or Pi I'm not sure they're both eight is probably more useful um this is the crowning achievement of Einstein's work on general relativity and now even though you're not yet quite uh empowered to go solving the equation you know what it means there's a matrix a four by four array of quantities on the left that characterizes the curvature of space-time in a very clear geometric way lines diverging or twisting and on the right there's another tensor that bundles up all the stuff you need to know about energy and mass and momentum and the two are proportional to each other and you go make predictions now that's the hard part because to make predictions you do need to actually solve the equation and Einstein who you know he had to learn this and he struggled through it but he did it he knew how hard it was to actually calculate this so let's say I would like to calculate a single component of the Riemann tensor in terms of the metric remember the rieman tensor depends on the metric so here's the formula for one component of the Riemann tensor written in terms of the metric again just showing off Nobody Does this Janna who introduced me and I were in the last generation of graduate students who had to calculate these things by hand today we have computers to do it you can tell by the font I didn't write this right this is a computer output Einstein looks at this and he says yeah there's just no way sorry I'm not solving this no one's solving this I can approximate it and that's what he did to make predictions he says no one's actually going to solve this exactly Carl shortshield heard him say that and said I can do it Carlos schwarchill was another German physicist remember it's 1917 it's during World War One Schwartz Shield unlike Einstein was literally in the German army he was calculating the trajectories of artillery shells to shoot at the enemy but in his time off in his short leave he went to Berlin and sat in on lectures by Einstein on general relativity and he said this he got became fascinated by this and he went back home in his spare time he tried to solve it and he did and he solved the most important question you could ask in general relativity or in any theory of gravity how much gravity does the Sun make which is actually not so surprising that you can do it because everything is spherically symmetric right this r squared and r squared sine squared theta that's just a sign that we're in spherical coordinates rather than rectangular coordinates but also outside the sun it's empty he wasn't trying to solve the equations inside the sun we have to worry about the energy momentum tensor the energy momentum tensor is just zero outside the Sun so that let him solve it and this is the payoff you've been very patient I didn't see anybody walking out it's pretty dark you could have sneaked out but you've been very patient and this is the payoff because again Einstein never thought this would happen that you would be able to write down an exact solution to his equation but the equations are unforgiving and precise once you have the equations and believe them and you solve them you're beholden to believing what the solution is trying to tell you so let's think about it it's up here in the top left corner that the interesting thing is going on right this is the TT component and the RR component of the metric tensor R is literally the distance you are away from the Sun or whatever and as R gets very very big this number 2gm over R just goes to zero because R is very very big so what are these components plus one and minus one it's just minkowski's face right there in the equation you can see that far away from the gravitating object space time is flat it's like minkowski space good but there's a weird thing happening too because if R gets close to 2gm it's then this number becomes close to one and one minus one is zero which is a weird thing for the rate of flow of time to be this number becomes one over zero which is infinity which is worse so all hell has broken loose at this quantity at this place in the universe R equals to GM what's going on with that so both short shield and Einstein and for decades after physicists said like woof this is weird we don't know what's going on what's going on of course is it's a black hole and neither schwarziel nor Einstein ever knew what a black hole was they went to their graves not knowing they certainly weren't trying to predict or understand black holes but the equations don't care about your feelings the black holes are there and you can ask what's physically going on g00 is one minus two GM over R so at large R that's just one no big deal but as R goes close to 2gm it goes to zero which means that if you visit that place in the universe if there's something that is so much mass squeezed in that it's smaller than 2gm and you visit there your clock ticks way slower than the clocks of your friends outside and you can see this in a tragic tear-jerking scene because Matthew McConaughey did this in interstellar and then he came back and his daughter was older than him nowadays we see them we see the black holes this is an image from the Event Horizon telescope of uh the light rays not given out by the black hole but by matter near the black holes supermassive black hole at the center of the Galaxy and the point I want to make is and I can't make it strongly enough this is why the equations matter because nobody wanted this but you solve the equation you have to take seriously what it what is there no Nobel prizes were given for work in general relativity in the 19 teens when Einstein invented it or in the 1920s or the 1930s or 40s or 50s there was or 60s there's one in the 70s but the last 10 Nobel prizes over half of them have been for work involving general relativity that's why the equations matter because the equations are smarter than we are our technology in the 1910s was not ready to think about black holes or gravitational waves or the Big Bang but all of those phenomena were implicit in the equation that Einstein wrote down and now we've made them frighteningly demonstrable and the nice thing about it is we're not done there are more equations to be discovered more things to be observed and along the way the universe will continue to be a lot smarter than we are thank you very much thank you and I think we have questions there's a lonely microphone sitting up here I don't know if there's a boss who's gonna Shepherd people where we just ask people to stand up or Jen asked the first question I don't know uh suggesting people come to the mic and uh and come ask questions and it's first come first serve everyone's too terrified I could try oh brother random question so you said that light can't travel double the speed towards each other what if it's just light versus light not space can't go that's the the whole stick of general relativity is of Relativity special in general is that the velocity of light will be the same as measured by anything relative to space relative to whatever else is moving by it so a velocity by construction is how much space do you travel per unit time that's what a velocity is and for light the answer is one light year per year is how I like to remember it I'm just saying if you take space out of the equation completely then there's no such thing as a velocity so again you should rethink Your Life Choices if you've taken space out of the equation I know I'm skipping ahead here please story but I have to know so have we really finally uh reconciled quantum mechanics and general relativity in a meaningful way no okay that was too easy I can elaborate yeah please do uh book two in the series of the biggest ideas in the universe we'll talk about this as did by previous book something deeply hidden yes so this is a good thing to be clear on because remember there's special relativity speed of light is constant blah blah blah but no gravity and then there's general relativity where gravity comes into the game that's this equation so special relativity is a hundred percent compatible with quantum mechanics the result of marrying special relativity in quantum mechanics is called Quantum field Theory and it's the center of all our best current understanding of the fundamental laws of nature marrying general relativity with quantum mechanics hasn't been completed yet we can do it in approximate ways in certain certain circumstances but the general full Theory isn't there yet many very bright people are working on it also I'm working on it you know we're trying to get there if I knew that I would I would tell you that I would not hide it yeah equation to the general public and the follow-up is how did you get a publisher to agree with you here's my publisher Stephen you want to come up and ask answer the uh okay um you know I think that I mean there's the there's two answers to this two-part question the honest answer is we had a pandemic and so to amuse myself and the rest of the world when went to lockdown I started making some YouTube videos I thought I would make you know the biggest ideas in the universe time space matter energy whatever and make little 15-20 minute YouTube videos where I talk about them but then I discovered I could hook my iPad up to the computer and write equations so they grew into hour and a half long videos where I did all the equations and what I discovered there is there's a huge gap between how we think about educating physics students and how we think about educating the rest of the world with the physics students we presume we have them locked up for between four and ten years right and they're going to have to do the homework sets and they're gonna have to learn everything and so we can take our time and with people who are not going to be experts we just throw up our hands and say well it's just too hard for you folks to understand this stuff and I think that there is a judicious middle ground and you can get there by not trying to teach people to solve the equations well as we leave there will be problem sets available if you want to solve the equations but all you want to do here is understand what the equations mean and if that's your only goal if you're not trying to make new professional physicists but just trying to make them understand what it means it's doable in a relatively short book why is it important because we're human beings and we're endlessly curious and this is how the world works and I want to share with everybody I can foreign again keep asking that mu and new are the Greek indices going zero one two three they mean what coordinate are we referring to in space time okay so the first one space or time maximum space both of them take on let me get this I can get it back both of them take on all the values right so mu when you have index notation I'm very happy that you asked this because I can't you know some people get it immediately some people it takes years some people never do but the idea is that rather than writing out all four x0x1 X2 X3 we just write X mu and mu could stand in for zero or one or two or three okay thank you sure hey my question is actually related to the indices so this is great good um sort of transformation between the Riemann tensor to the Richie tensor to the scalar it sounds like it was just serendipitous or so I wonder what was the chronology of these events to go from four indices to two to eventually being able to solve the overall equation yeah I mean this is an excellent question because they no one ever tells you the answer to this even if you're learning it in class because physicists and mathematicians would just as soon forget their own history so I I learned it while writing this book I've written a whole textbook on general relativity and I never knew the answer to this question so I for the first time in my life while writing this book I read riemann's original paper and there's no tensors in it I'm like what do you mean you're the dude who's named the Riemann curvature tensor after but his Insight in the paper was really that idea that if you knew the length of every curve you would know the geometry and you could get that by zooming in all of this was done later especially by Italians for some reason but there were some Germans who were involved people named christophel La Vista Vida Richie uh um there were others and basically I mean I I don't know what to say except that they were it's full employment for mathematicians here's a new way of thinking about the world what's the best way to do it carton was another one and they developed a lot of things much more than this you know a whole machinery for dealing with it having no idea it would be relevant to physics right that's what's Charming about it they're just like geometry this is great you know what can we do in this particular case what I didn't go into is that you can't just stick indices on things and call them tensors some things act like like tensors There Are Rules other things don't and so the fact that you could go from the Riemann tensor to the Richie tensor was a non-trivial step and another non-trivial step to get the curvature scaler so kudos to them for doing it they got their name on some tensors then they live forever yeah thank you quick follow-up question is it reversible no because there's far less information in the scaler than there is in the whole tensor so you lose information when you flow down those arrows sure hi Sean to go from The History of Science to maybe current problems and maybe even your last book a little bit I wanted to ask you to say more about a variety in many worlds and your your ideas around it and whether you you know your ideas are on collapse series and grw theory for instance sure I mean we should charge you extra for this you're asking for a whole nother lecture but happily I wrote another book last time called something deeply hidden Quantum worlds in the emergence of space-time which concentrated on quantum mechanics which we didn't talk about today there's only three equations in that whole book so you'll be disappointed now um but the idea that I try to explain in that book is called as as you said the many worlds interpretation of quantum mechanics and the reason why that's important or why you would invent it is quantum mechanics which came along and Einstein played a huge role in this in the 19 teens and 1920s one of its things that it says is you can't predict exactly what observational outcome you will have okay no other Theory of physics ever said that but in quantum mechanics there's a rule specially designed for observational outcomes and end rule is unpredictable it's stochastic it's random if I I can have a cat let us say in a superposition of being awake and asleep and it really is both until I observe it if you believe the usual Quantum story and so Hugh Everett who's a grad student in the 50s said well what if we treat The Observer as quantum mechanical also and we don't invoke any weird rules we just let the ordinary equations do their job and he says look the answer is that the Universe evolves into the cats awake and someone saw the cat awake plus the cat's asleep and someone saw the cat asleep all you have to do is treat those as two separate worlds and you fit all the data and explain everything why you could treat those as two separate worlds took longer to figure out but we've now figured it out so I think it's the simplest most straightforward interpretation of quantum mechanics as you allude to there are people who haven't gotten that yet so they develop other interpretations good for them I'm very slow at writing papers so I want the smallest amount of competition possible hey Sean thank you so much for coming I really appreciate what you said earlier about trying to promote the education of a kind of curious public in the middle ground so I'm sure many of us here you know we're fortunate enough to make it through eighth grade geometry but are not on track to be a PhD in physics yeah um and so your work is a very invaluable in um for people like me who are members of that curious Middle Ground um I was wondering if there are other writers or thinkers or scientists or um maybe in journalists or philosophers who um you know might be professors like yourself but also have kind of more uh work that's aimed towards this middle ground that you read or you might want to recommend because uh yeah yeah you've been very invaluable in that nope it's just me that's like that's all there is I'm gonna keep saying that until the book has sold out and then uh no there are there are absolutely other um people who do things in their own way and that's great and I love it because there's a vibrant ecosystem out there I mean before I even technically answer your question let me point out there are online courses where you can learn all this stuff right I mean not just my YouTube videos but like actually professionally produced courses from MIT in Stanford and places where they really try to teach you these things you can download the lecture notes you can even do the problem sets if you want to do that um there are two very very good examples of books similar in spirit to this one one is Roger Penrose uh who's visited Pioneer works I believe wrote a book called The Road to Reality it is a giant book with all the equations in the universe um the primary reason why I wrote my book anyway even though penrose's book exists is because Roger Penrose is much smarter than us but he assumes that you're just as smart as he is so no one understands what Penrose says in that book I'm pretty convinced but they're there the other one is the theoretical minimum books by Lenny suskind um and that's an attempt it's closer in spirit to what I'm doing but it's also more like a conventional course right where they were literally a court they were again based on YouTube videos that he taught as an extension course at Stanford where he had a group of of non-experts but you know he's teaching you the nitty-gritty in a way that uh slows you down so neither Penrose nor suskin gets you to the Riemann tensor on page 228. thanks gen 11 pretty good but she's too afraid to put equations in her books yet maybe this will be a challenge for her but equations in the books yeah Carlo well I don't know I mean again if you if the question was are there people who are brilliant writers about science then oh my goodness yes Janna and Carlo and Roger Penrose and Kip Thorne and Lisa Randall and a whole bunch of people who were professional journalists not scientists including one to whom I am married so believe me I'm a huge fan of many people in many different modes writing about science and I think that we're in a embarrassment of riches in the current world about reading about good science therefore when I want to write about it I have to do it in a slightly different way and so this is a slightly different thing than those others have done uh thank you so much for your lecture tonight um I have a two-part question um I feel like it's been sort of asked in different ways so far about quantum mechanics but specifically I wondered about your thoughts on string theory and sort of the New Frontiers of physics that questions can't answer and secondly also I feel like you know your lecture was driven by equations and I wonder if um paradigms of science are usually or your thoughts on if paradigms of science are usually led by the available equations and what sort of what what happens when scientific exploration and Discovery needs to necessitate a new equation being written I.E like our equations the foundation of our understanding or is our understanding greater and we're using equations to try to describe what we can see yeah no these are both great questions in fact I can sort of combine the answers so string theory is what I would call and I think is the consensus the leading candidate for reconciling quantum mechanics and gravity that we have right now but it's not in by any stretch the consensus it's the consensus that it's leading but it's not the consensus that it's right uh there are absolutely competitors to it that are doing a very good job and it's a good example because as I explained it here more or less historically accurately general relativity came into existence because of a thought experiment the principle of equivalence they had the idea the concept and Einstein had to go out and find the equation to fit it string theory was exactly the opposite they literally uh Giovanni veneziano wrote down the equation without yet knowing what it was about and they realized that it had some similarities to the strong interactions the strong nuclear force and they looked at it more and more carefully and they realized that unfortunately he kept predicting the existence of gravity and they didn't want gravity and then they realized wait a minute gravity exists maybe it's a theory of gravity also and since then it's taken off so there's no rule for how new theories of physics are invented sometimes it's Concepts first sometimes it's equations first so you talked about time and you gave us so grasp about it there was a direction of time and there was a singular Direction Why is there that era of time why is it in One Direction do I have the book for you I talk about this in the most recent book but I wrote a whole another book called from eternity to hear on exactly that question and the answer is not in anything I talked about today everything we talked about today is the pristine beautiful fundamental equations of physics where there is no direction to time it works equally well forward and backward the directionality of time comes in because the universe is messy because there's stuff in the universe and that stuff started 14 billion years ago right after the big bang in a very very special unusual State one that we characterized by saying it had a low entropy an entropy has been increasing ever since so the laws of physics are perfectly reversible but the configuration of the universe picks out a direction of time nice job um so in that equation there is a black hole in the very left-hand Corner yeah um even though they didn't know that black holes existed it was there so my question is uh kind of related to his math something outside of us that exists that we interface with or is a language that we just use to find out you know I can't remember that right no I know it I know exactly what you're asking I have not written a book about this one uh but Pioneer works is is uh maybe they can explain but they're organizing a whole thing around it this is a very good question is math externally real do we discover it this is the idea called mathematical platonism that their math is real in a way that is objective and true just like the physical world is real but different there's also the countervailing point of view called nominalism which is mathematics is just a nice way to compactly talk about all the stuff in the physical world and I'm closer to being a nominalist than a realist about mathematics or a platonist but I do not have very strong opinions I think these are very good questions that I just don't have the answer to because it was wild that the black hole was there well good so but but let's be clear there are the laws of physics the patterns that exist that that nature seems to inviably stick to and there's the mathematical expression of those laws so I could imagine a point of view not that I'm advocating it but imagine a point of view where the universe keeps obeying the laws of physics even though math is not objectively real hi hi Sean I'm Jana we love these questions and uh I hope people will come line up for the book signing and ask Sean questions directly because otherwise this will go on for power and um also just because Sean brought up platonism and constructivism even though you said nominalism I did tolerate your philosophical twist um we're gonna do a show in Charles's exhibition on whether math is real there you go whether it's invented or discovered so join us for that in November and Consciousness in October and join Sean at the book signing line and thank you everybody because you know that was easy [Laughter] [Applause]

Channel: Pioneer Works

Views: 28,826

Keywords: sean carroll, pioneer works, red hook brooklyn, author talks, The Biggest Ideas in the Universe, Space Time and Motion, Mindscape podcast, johns hopkins, From Eternity to Here, The Particle at the End of the Universe, The Big Picture, Something Deeply Hidden, physics

Id: CH39SDlxon4

Length: 70min 54sec (4254 seconds)

Published: Fri Dec 16 2022