The Biggest Ideas in the Universe | 16. Gravity

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll today's big idea I'm pretty sure it's gonna be one of your favorites it's definitely one of my favorites the idea is gravity by which we mean of course general relativity Einstein's theory of gravity we already mentioned gravity in the context of Newtonian theory now we're gonna move on to the real stuff and let me be clear about one of the reasons why this is gonna be one of your favorites not only is general relativity by itself just a wonderful theory you know I had a I wrote a textbook on general relativity space time and geometry which I opened by saying general relativity is the most beautiful physical theory ever invented I think that's absolutely true I'll try to make the case for that as much as I love quantum mechanics general relativity is prettier but the other thing is it's also easier we've been doing a bit of quantum mechanics quantum field theory stuff and I understand that a lot of it is abstract and very far removed more everyday lives and very difficult to get it's worth it and I hope that some of the nuances have come across but it can be taxing and I completely understand that whereas general relativity even though it has its own mathematical language and in some ways it is also a little bit removed from our everyday lives it is easier to understand sorry I'm hesitating because there's Ariel as complaining in the background so I'm not playing fetch with her sorry about that you gotta put up with that this is an irony of course because general relativity has a reputation for being incredibly difficult to understand the old joke is that Arthur Eddington the British astrophysicist who was a great popularizer of relativity as soon as it came out he was asked at a talk you know I've been told the questioner says there's only 12 people there's only three people in the world who understand general relativity can you tell me who the third one is after him and Einstein and Eddington hesitated and the questioner says oh don't be so modest and anything says oh no I'm trying to think of who the third person is this is not true was never the case that only two or three people understood general relativity it's it's a language of mathematics that we need to understand but we can get there we can get there faster than most people because you've listened to the previous video that we've had about geometry and apology so you know something about the geometry of curved space and curved space-time already so for us it's gonna be a breeze all right let's go so let's think about Newtonian gravity first we've already talked about Newtonian gravity professor Newton told us that the force due to gravity looks like Newton's constant G times one mass m1 times the other mass m2 over R squared the inverse square law of gravity so you have two objects there is action at a distance in between them they pull each other together proportional to their masses and inversely proportional to the distance between them and then Laplace says that you can think of this as the slope the force due to gravity can be thought of that as the slope of the gravitational potential field so you have some field Phi as a function of distance R and it looks like this since you're sitting here and the slope of that field gives you the force of gravity it's not a different theory than Newton's it's exactly the same as Newtonian gravity but it's put into the language of a field theory so that rather than simply action at a distance you have an equation that tells you how the field changes from point to point that is a purely local equation so that's a step forward however this is you know this is 1600s this is around 1800 but then Maxwell comes along with electromagnetism ok Maxwell and other people but one of the great achievements of Maxwell's electromagnetism in the mid-1800s was understanding that there's a unification of electric phenomena magnetic phenomena and also light and radiation and part of understanding the symmetries of that theory was Einstein's appreciation that the speed of light would be the same to every one that you did not need an ether pervading space-time to explain any of it and the pole theory that we now call special relativity and the reason I'm bringing these up is of course because Newtonian gravity is not compatible with special relativity so Newton's equation here for the force due to an object with a mass I'm acting on an object with mass m2 m1 and m2 there's no time in here the time parameter the time of the universe does not appear in this equation which means that if I move one object very very quickly the gravitational force on the other one changes instantaneously no matter how far away it is there's no speed of light built into this equation anywhere and when Maxwell came up with enm electricity and magnetism and then Einstein codified it in terms of special relativity the speed of light has this really important feature you cannot send signals faster than the speed of light if Newtonian gravity were true you would be able to send signals faster than the speed of light through the force of gravity in some object that you're waving back and forth so Einstein knew this he knew that you would have to modify a Newtonian gravity and I think that even before Einstein probably people appreciated it they appreciated that the symmetries of Newtonian mechanics were different than the symmetries of Maxwell's electromagnetism there's a Galilean invariance in Newtonian mechanics and there is Lorenz Ian's invariance in relativity those two things are not the same thing so I thought a lot about how to do this and there's sort of a general way that you would try to think about how to do this which is to write down a relativistic field theory in other words to update Laplace's theory in such a way it would give you all the features the gravity had people tried that it didn't really work nordstrom among other people was it was a person who tried it but it didn't really work for various reasons we're not going to go down all those blind paths we're gonna skip right to the right answer so Einstein comes along after inventing special relativity you remember just to put this in perspective Einstein didn't really invent special relativity he perfected it circa 1905 he really was the one who understood what special relativity really meant and explained it got rid of the ether all that stuff we now know as Lorentz invariance was understood at a deeper level and in that same year 1905 Einstein suggested that light could be thought of as particles which we now call photons and he did other things as well so he was thinking very much about quantum mechanical phenomena this whole time and also thermodynamics and statistical mechanics but certainly after putting relativity together special relativity in 1905 gravity was foremost on his mind and he spent around ten years 1905 to 1915 under Supes understanding how you could fit gravity into the relativistic framework before coming up with general relativity and the names aren't very good special relativity general relativity it's not really special versus general it's two very different things special relativity like quantum mechanics or classical mechanics is a framework it's a sub framework of classical mechanics everything here is classical even though it's relativity we're not doing quantum classical means non Quantum's we're classical for this whole video so relative special relativity is a different version of classical mechanics than Newtonian space-time is but it's still classical mechanics okay general relativity is a specific theory so once you have special relativity that's a framework in which you can build many theories like Maxwell's electromagnetism like relativistic quantum field theory the standard model of particle physics is a theory that lives within the framework of special relativity it's a quantum version but still the same basic framework there general relativity is also a different theory that lives within that framework that special relativity sets up so it's actually in some sense more specific than special relativity it's the theory of special relativity when you let space-time be curved and you give it a specific way of being curved and you call that gravity so his breakthrough Einstein's breakthrough came from as it often did Brian Stein thought experiments you know he was a genius clearly he's Einstein but he wasn't the most mathematically inclined physicist around even at that time he was the genius at thinking of things physically and in terms of thought experiments that gave you immense physical insight into what's going on so the thought experiment that he came up with is the following take a box a sealed box that you can't get out of but a physicist inside this box okay and they're doing experiments and you're giving him the task find out what are the conditions in which the box is embedded the bigger outside world okay so you could ask for example is there an electric field here's an electric field vector imagine there's some electric field pointing vertically both inside and outside the box could you do experiments inside the box that lets you know that there's an electric field surrounding you in all places and the answer is clearly yes and so Einstein wouldn't put it in these terms but what we can do is say look imagine there's an electron II - and there's a proton P + okay two different particles with two different electric charges the way that the sign convention works out the proton gets accelerated in the direction of the electric field so the proton gets accelerated vertically upward the electron will get accelerated vertically downward so something different happens to the electron and the proton because they have different electric charges in the presence of an electric field therefore if there were zero electric field they would just sit there if there is an electric field pointing in some direction they begin to cell accelerate in opposite senses along that direction in which the electric field is pointing so the answer to the question can you detect an electric field while you're in a sealed box is clearly yes so Einstein says okay let's play the same game for gravity okay imagine there's some gravitational attraction G pointing downward and here's your physicist okay can they do an experiment to detect the gravitational field and know they're in a sealed box they can't just look outside go yes I'm sitting on the earth okay therefore I know there's a gravitational field they're not allowed to look outside well you can say you know let me take a heavy object and a light object all right two things with different masses and therefore in some sense different couplings to the gravitational field and you can drop them but ever since Galileo we've known that a heavy object a light object will fall the same acceleration in a gravitational field as long as the air resistance is negligible and so on right so Einstein says you know okay I can drop two objects but they're unlike electricity and magnetism the objects in the gravitational field will always fall in exactly the same way and you might you might say well still they're falling therefore I've detected the gravitational field if I have an object right here and I drop it I have detected the gravitation field right but Einstein who is smarter than me or you says you don't know that what if there's not a gravitational field and what if I draw my box better but instead of that you're in a rocket ship okay here's the engine of your rocket ship and so you're accelerating upwards the same thing would happen right you're accelerating upwards you take these two objects you let them fall and they followed exactly the same rate in fact they're not really falling at all they're remaining stationary and your box is accelerating up into them and so what Einstein says is maybe this Galilean result the two different objects will accelerate at the same rate in a gravitational field extends more broadly maybe it's not just the physics of falling objects maybe it's all of physics he says maybe there is no experiment you can do in that box which could actually tell the difference between being in an accelerated box versus that being stationary on the earth for some other gravitational field okay so we're comparing stationery on earth and our usual way of thinking about it - accelerating at 1g one earth gravitational acceleration but out in outer space far away from the actual earth and we're also imagining you can't hear the rocket engine roaring or anything like that okay in that case Einstein goes on to invent what he calls the principle of equivalence and in case you're wondering the equivalence here is between gravitation and acceleration he says in a small region of space-time and this is important okay in a small region of space-time sometimes you can forget that part they shouldn't no experiment can distinguish I can do better than that I always try to rush because I write very slowly and then I become sloppy and it slows me down there's some life lesson here to be learned no experiment can distinguish between gravity and uniform acceleration so obviously if your rocket ship is boosting and then stopping and boosting again you would know that it's a rocket ship the earth doesn't do that to you but uniform acceleration is just like gravity I knew that was gonna happen there we go okay I can't put a period on the sentence good it's too close to the edge all right so that's the principle of equivalence Einstein says there's something special about gravity something that is different between gravity and other forces of nature the only one he knew very well was electromagnetism but still this is a special thing you can tell whether there is an electric field around you you can't tell whether it's a gravitational field if you're in a small region of space-time now well the reason why I'm emphasizing the small region of space-time bit is that people wise guys keep coming up with proposed counter examples to the principle of equivalence and usually it's because they're ignoring the fact that there's the phrase in the small region of space-time so in a large region of space-time so let's imagine this is the earth and let's imagine that your physicist box was bigger than the earth here's your box that you're stuck in okay so there's an obvious experiment you can do to tell the earth is beneath you you can drop a ball here and a ball here on far distant parts of your box and they will move towards the center of the earth in other words it will move ever so slightly toward each other whereas in an accelerated box they'll move exactly parallel and the same thing is true here if you're truly on the earth literally here in your in your room wherever you're watching this you could drop two things and in principle they do not move on parallel lines they move closer to each other as time goes on because the earth is the center of the earth is a point toward which they're attracted but that's that effect that difference goes away in the limit as you're looking at smaller and smaller regions so already this phrase in this region of space-time it has a bit of a flavor of calculus to it doesn't it remember remember when we defined derivatives so we did derivatives in calculus we had some function of X as a function of X and we said look there was a curve there and what you want is the derivative of F at that point okay and what you do is you zoom in on it and then you draw a straight line and what you say is that as you come closer and close your ease to get to smaller and smaller parts of the curve that you're interested in you can approximate that curve no matter how curved you was originally as you go to the limit is that you're looking at smaller and smaller regions it looks more and more like a straight line so the principle of equivalence is saying something similar about space-time and in fact so you can translate the principle of equivalence pov restated or implies let's put it that way it's not exactly the same statement but the implication that einstein drew from it is that physics in small enough regions I won't keep writing the whole thing of space-time reduces to special relativity so what he's saying here is the rules of special relativity by hypothesis we just don't put gravity in there okay special relativity is not a theory of gravity but what he's gonna say is when we do include gravity we don't give up on special relativity entirely we say that special relativity applies in the limit as you go to smaller and smaller regions of space-time and just like calculus comes about by saying in a small region when you zoom in on a curve it looks like a straight line you can go from this statement that physics looks like special relativity in a tiny region to a statement that resembles differential geometry okay differential geometry is secretly what we were doing when we did geometry in previous video this is the point of what Gauss and Riemann did because rather than taking a uniform smooth geometry everywhere like you would have over a sphere they said well we can look at individual points in some wildly fluctuating geometry and describe characterize the curvature of the manifold locally at each point so this is the word where this is where the word differential comes in differential just a fancy way of saying derivative this is calculus applied to geometry so the point of differential geometry in some sense is in small regions geometry looks Euclidean so we didn't I could have set this in the geometry video but didn't say it out loud but this is the point so if you have let's say a sphere okay here's a sphere one way of thinking about what Gauss and Riemann did one way of thinking about how they managed to make sense of the geometry of different curvatures in different places was they said look if I take one point on the sphere and I zoom in on it just like the whole surface of the earth is roughly spherical but if I mean one tiny patch of it it looks flat right that's why there are people who think the whole thing is flat because they don't understand differential geometry they don't understand that the small region can look flat even though the whole region is actually quite curved and what Gauss and Riemann do is that they actually use their brains to invent a whole mathematical structure here and they're able to sort of say in the vicinity of that one point all geometry looks exactly Euclidean in the limit as you in a curved manifold to go to smaller and smaller regions you look more and more like there's no curvature at all and then you can build up how to glue together the small regions to describe the curvature globally so it's clear it should be clear it's clear to you could be you know as a leading discussion but you go from the principle of equivalence which is this thought experiment okay that's very very physical very intuitive very visualizable and then you get this implication physics in small regions reduces to special relativity which sounds exactly like the beginning of differential geometry Einstein had no idea help this he did this was not you know some clever store he told after the fact he got the principle of equivalence but he didn't know exactly what it meant he was bouncing around so between 1905 when he was stolen to clerk 1915 he was already pretty famous in between he was bouncing around he got his PhD from Zurich and then he went to Baron anyway I think to Prague and then he went back to Zurich and when he was it back in Zurich he's still thinking about this stuff and so but he ranted all his old friends and Zurich because he's got a PhD there one of his friends was Marcel Grossman who was a mathematician and Einstein described to Grossman what he's thinking about and Grossman says yeah well you need some Riemannian geometry and Einstein says well what is that I agree so Grossman taught Einstein rimoni in geometry and it's after some initial reluctance Einstein eventually says this is exactly what I need right so Einstein eventually says gravity the thing that I can get rid of in a small region of space-time is just the curvature of space-time took them a while to actually figure out the equations associated with this motto but this is the idea if in a small region of a manifold you can make the geometry look Euclidian that's equivalent to saying in a small region of space-time the geometry looks like flat space-time of special relativity and then the curvature shows up as you go to larger and larger regions and Einstein says that must be what gravity is because that's in this picture right here you see that the gravity is detectable when you can go to larger and larger regions of space-time and a small region disappears in larger regions it's very very important yet another way of saying the same thing is Einstein is saying if there is a difference between gravity and the other forces of nature okay in Newtonian physics what you would have said is that gravity is a force that lives on top of space-time and it has there's an equation for what the force is there it is inverse square law okay in some sense according to Einstein the problem the reason why gravity had been difficult to fit into special relativity is because gravity is not a force a force you could detect by doing some experiment if the force is real you should be able to detect it but gravity you can't not in a small region of space-time if you're locked in that little box so Einstein says instead of thinking of gravity as a force what you should do is think of it as a feature of space-time itself that's why you can't detect it because everything is in space-time everything is affected by the curvature of space-time in exactly the same way it's not an external force that some things can feel and some things don't it's a feature of space-time itself so that's a very very good perspective and so you know if I'm hesitating because look if you were me right after you first learned general relativity you would have been one of those people who went around making fun of other people who thought of gravity is a force because you have said no no no you're very old-fashioned gravity is not a force gravity is the curvature of space-time and you know now that I'm a little bit more mature than that I know that really the answer is who cares it doesn't matter whether you want to call gravity a force or not if Newton had been right you know if all of Newtonian mechanics had been right the idea of a force was absolutely central F equals MA right it's really an important part of how we defined the theory but he wasn't we've moved on beyond that now in quantum field theory once we get there we're not doing quantum field theory today but in quantum field theory you don't talk about forces at all there's no F equals MA in quantum field theory there's just a bunch of different fields interacting with each other so we still use the word force because it is sometimes convenient ok it is not an essential part of the ontology of nature it's a sometimes convenient concept and that's just as true in general relativity as anywhere else if someone wants to say the pen falls because the force of gravity is pulling it fine let them do that don't be it don't be a stickler about whether or not gravity is a force but you should appreciate that gravity really is the curvature of space-time because there is one shift of attitude that actually does make difference make make a big difference let's let's put it this way think of a table here's a table okay again drawing is not supposed to be the strong suit here and I'm gonna draw a coffee mug on the table maybe we're at apologists drinking our mug of coffee and not sure whether it's a coffee cup or a donut okay and the coffee cup is just sitting there on the table right and you can ask yourself is the coffee cup accelerating and this is something that Newton and Einstein would have given genuinely different answers to because Newton says no what do you mean of course the coffee cup is not accelerating it's on the table the tables on the floor presumably nothing is moving acceleration is the rate of change of velocity the velocity is zero and it stays zero the velocity is not changing therefore it's not accelerating Einstein would say yes it is accelerating because I'm stein says that the natural state of motion of objects in space-time is free fall that is to say not being pushed by not just a rocketship engine but also by a table under view or if you're sitting down right now by the chair underneath you or if you're standing that the floor underneath your feet okay he says look if you're just standing up of course you're being accelerated even if you're not running or walking if you're standing still you can feel the acceleration in your feet right that's why you feel heavy if you're sitting down you can feel the acceleration in your butt pushing you upward the natural thing so Newton would say if I move the table away the coffee cup would start accelerate and falling but it's not accelerating when it's just sitting there Einstein would say it is being accelerated by the table it's being pushed away from its natural state of motion and if you quickly remove the table the coffee cup would undergo its natural free fall motion so just like inertial trajectories are special in special relativity unaccelerated trajectories are special they're just as special in general relativity they seem curved to us you know the earth goes around the Sun that does not seem like a straight line like an like an unaccelerated trajectory would be but what Einstein says is that these trajectories are geodesics which is a fancy way of saying extremal distance paths we talked about this in we talked about geometry one way of defining the generalization of a straight line is to say you have two points you connect them with a whole bunch of different conceptual paths the shortest one is the straight line right in flat space if you're in curved space or curved space-time there might not be a straight line in the conventional sense but there's still an extremal distance path I would like to be able to say a shortest distance path but actually works out to be a longest time path in relativity so I'm gonna say extremal distance okay but this is the generalization of a straight line and this is what objects want to do in curved space-time objects want to move on the closest thing they can to a straight line so the motion of the earth around the Sun in space-time is the best it can do to move on a straight line it doesn't seem to be straight to us because there are no exactly straight lines because space-time itself is curved that's the lesson of general relativity so you are accelerating right now unless you just jumped off a building or something like that good so that that that concludes the truly fun part of our of our video well it depends what do you mean by fun okay this is the conceptual part you got it now you understand the conceptual beginnings of general relativity of course there's more work to be done because we need to mathematize it we need to turn into a real rigorous theory of nature how are we gonna do that okay well you need to talk about what is the amount of curvature you have if gravity is the curvature of space-time where to go gravity is the curvature of space-time geodesics are extremely distance paths you need to attach some equations to that and we know how to do that because we talked about curvature before in the video about geometry so curvature the way that we talked about it was in terms of parallel transport remember quantum chromodynamics okay remember when we did the gauge theory video we talked about quarks and how they could be thought of as living in this three dimensional space color red green blue and you might want a parallel transport a certain quark field to some other part of space-time where red green and blue might be oriented differently and you would have a connection field that tells you how to do that and then you quantize the connection field the connexion field is all over space-time and the quantum fluctuations in the connection field give rise in this case to gluons or to photons if we were doing QED etc and then there this the connection tells you how to do parallel transport and then curvature is the failure of parallel transport around a closed loop to bring you back to where you started so we mean here is that if we take a vector and we parallel transport it around a little tiny loop through space-time if the space-time in whatever thing we were moving like the cork fields or in color space if you parallel transport them around a loop if there's no curvature to the gauge field no curvature to the gluon field connection then the quarks come back exactly to where they start okay but if there is curvature to that connection field then they might be rotated a little bit and this is what curvature measures and the curvature for space-time is special and this is the beginning of why general relativity is a little bit different than all those other gauge theories general relativity is absolutely a gauge theory in some very well-defined sense it is also indubitably different than the other gauge theories and the reason why is because when we move around a quark the little vector that is the quark is pointing in a fictitious internal color space it's not pointing up or pointing left or pointing forward okay it's pointing in red green blue space which is a fake space which we make up to mathematically describe what's going on whereas in space-time you have a trajectory and the geodesic trajectory is one that parallel transports its own velocity vector or momentum vector which ever way you want to put it okay so parallel transport so let's put this way geodesic okay straight-line equivalent equals you parallel transport your own momentum or your own velocity that's saying that's why it's special because your velocity or your momentum is a vector that has a direction in space-time okay it's not in some fictitious color space it's pointing up or down or in some linear combination or something like that okay so that's why space-time is a little bit different geometrically and that's good news and bad news all right and so we don't need to keep thinking about the geometry of gauge theories cuz we're interested in space-time right now but I want to let you know that there is that difference there so in space-time just like in gauge theories II have this notion of parallel transport but there's also an alternative you also have a notion of distance between points okay that is related to the way you have a parallel transporting etc this is what is different in general relativity than in other gauge theories so alternatively this is a one definition of geodesic okay alternatively geodesic equals as we just wrote extremal distance path so what that means is we have again let's say this is a geodesic connecting two points this to you desk has a length let me move it over here a little bit and the length L well what is it I don't know what it is you could go and actually measure what it is but one way of figuring it out would be to divide it up into little segments remember differential geometry we go into tiny little regions and then we call this little tiny part you give it a length DL I'm gonna write that as a curved L because otherwise just looks like the number one okay so D lowercase squared - L is the distance along that little tiny bit of the interval and then the whole length along the path L is the sum of all of the little infant asthma length DL okay that's easy now if that part was easy the question is what is do how do we know like given some actual path what do you mean by being given some actual path you might be given coordinates okay like there's some coordinates on the sphere on the earth or in space whatever if you're NASA you need to have a coordinate system that extends throughout the solar system heliocentric coordinates if you Center them on the Sun for example geocentric have you centered them on the earth and then you might want to specify your path you want to tell your path to somebody else you need to say here are the coordinates of my path as I travel along it okay so the relationship between coordinates and the path is very important so we want is a formula for the infinitesimal distance in terms of the coordinates and the coordinates once you're in curved space-time you can't just say well choose right angles coordinates choose Cartesian coordinates okay because there aren't any like on a sphere you can't do that on some wildly curved space-time you can't do that so you need a way to do this no matter what the coordinates are doing you need a perfectly general way of seeing what is this little infinitesimal length so what we can do is basically we alluded to this and we talked about geometry we can generalize Pythagoras's theorem all right so let's imagine that here your coordinates and I'm drawing them in some curvilinear way because there's no preferred structure right angles parallelism anything like that is an arbitrary coordinate system call one of them X the other one Y what we want to do is figure out what is the distance between two points okay and so this is a small bit of X DX this is a small bit of Y dy from here to here this is DX there to there and this is DL there there so it looks like it should kind of be Pythagoras's theorem right DL squared is DX squared plus dy squared but these are not right angles and there might be other things going on so you need to be able to generalize Pythagoras's theorem and the general formula that you would write down looks like this and it's gonna be a little weird looking I'll explain why so again the Pythagoras would be Pythagoras is not right in this case but it would be DL squared is DX squared plus dy squared so we want more general and what we'll do is we'll keep the feature that DL squared scales as DX squared or dy squared okay so if the distances change you got to square everything so the more general formula is DL squared is some coefficient a times DX squared plus some coefficient B times DX times dy and this is why it's going to look weird I'm gonna write B again but then dy times DX plus a third coefficient C times dy squared okay now this is a generalization it's quadratic it either you know there's either two appearances of DX or two appearances of dy or one H of DX and dy so that makes sense good the part that doesn't make sense is why did I have DX dy and then again dy DX why didn't I just right since DX dy equals dy DX why did I just write two times B times DX dy well the answer is because this way of writing it leads to a very efficient compact way of expressing this in more general circumstances I invent what is called the metric tensor okay we have used the word tensor once before when we talked about the Riemann curvature tensor it was a way of compactly summarizing a whole bunch of different numbers so it's not a single number it's not a single vector it can be a matrix or an embedded set of matrices or something complicated like that the metric tensor is actually a pretty simple one the metric tensor g IJ which is that those indices I and J remind us that X I are the coordinates either X or Y so that's X 1 and X 2 this is not X to the first power x squared this is the first X which is X the second X which is y in this case if you imagine you had a third dimension X 3 would be Z etc and gij are the different values of G in these different directions so G X X is the thing that multiplies DX squared so it's a but let me let me before I get there let me just write G X X and then there's g x y gy x and g YY and in this case that is a and then be multiplying DX dy be again multiplying dy DX and c multiplying dy dy ok so that's why I wrote this equation in this weird form where I repeated this thing twice because I can write it as a square matrix which is symmetric G XY always equals G Y X but this just turns out to be a convenient way of representing those two numbers all right so you see it's getting a little abstract but hopefully not too much I think this is one of those things where just a little bit of work thinking about the notation is going to pay off I promise you I think I promise you I promise that I hope that it pays off how about that so the reason why this is useful because we have this formula for a tiny little distance and we can write it as d l squared equals a sum over both I and J of G IJ d X I DX j that is the compact way of writing g XX DX squared plus g x y DX dy plus g y x dy DX plus g y y dy squared it's a little bit simpler right in fact this becomes so simple i really shouldn't do this but it becomes so simple just so you get to know what you see elsewhere that people start stop writing the summation sign I think it was Einstein who gets credit for saying as long as I write g IJ a DXi dij dxj you know I'm gonna be summing over inj so I'm gonna write the summation sign anymore okay the Einstein summation convention it's called good so this is the data that we need to describe a curved manifold if I want to be able to calculate the length by the way you know I didn't because I forgot to say this out loud the point is these two definitions of oops these two definitions of geodesics are the same in general tivity you can either say i extremize the lengths along a curve or i parallel transport my own velocity vector they get me the same curves they get me the same geodesics so it doesn't matter which I use but distances are useful for other reasons so distance is sort of the the metric that tells you how distances are calculated is the thing that is the primary focus of what your attention is and general relativity everything comes from that so this is the metric metric tensor something like that but this is the metric on some two-dimensional imaginary space let's just go right to space-time what we care about right here's space-time we're gonna now have four coordinates so we're gonna start using Greek letters X mu there's an old joke that math becomes difficult when the symbols start being in the Greek alphabet rather than the Roman alphabet it's of course completely irrelevant what alphabet you use but it is somehow true that once you start running out of letters in the Roman alphabet and have to start using Greek letters things might be becoming more difficult the space-time indices coordinates are T for time XY and Z and I am setting the speed of light equal to one otherwise it would make no sense to combine time with space right relativity is where the speed of light comes in and we're certainly doing relativity and conventionally what we do is we label these X 0 X 1 X 2 and X 3 so time is the zeroth coordinate of space-time and why do we call it zero we're either sticking at the end and calling it the fourth well you know sometimes we might want to talk about more than or dimensions or fewer than four dimensions different numbers of dimensions time is always singled out so it's more convenient to call it X zero and if we get another dimension of space we would call that X 4 etc okay and then there is a metric so how about just for flat space-time for flat space-time that's what we study that's what we talked about in the video on space-time and we can summarize all of what we said in that video by one formula for the metric for flat space-time which is minus 1 0 0 0 0 plus 1 0 0 0 0 plus 1 0 0 0 0 plus 1 tada notice by the way I'm just gonna write this and I'm gonna erase it flat Euclidean space also has a metric just two dimensional tabletop right gij equals plus 1 0 0 plus 1 that's it that's all flat including geometry summarized in one little formula also from that you derive everything clearly something a little bit trickier is going on oops in space-time because there's a minus sign there what is that all about well that's reflecting the fact that video it's nice to see it pay off right it's nice to see our previous work in previous videos come come to fruition here we mentioned that there is a difference between how we calculate lengths in space and time there's a relative minus sign which means that that's why if you have a path through space the analog of a straight line is the shortest distance path through space if you're in space-time the analog is the longest time path that's because of the minus sign here in the metric and what this means is that the interval in space-time so not the physical distance but the interval in spacetime is minus DT squared plus DX squared plus dy Plus DZ squared and that should be familiar to us because when we did our space-time diagrams X T we drew light cones like this and x equals plus or minus T that's the light cone coming out of this point okay at 45-degree angles because C equals one they have the feature that D s equals zero no time nor no spatial distance is elapsed as measured by someone moving at the speed of light and that's because minus one and plus one cancel each other out in the metric all right that was just a rehash of what we've already done in the space-time video so now let's let things be curved okay so in curved space-time remember professors Gauss and Riemann told us that in curved space-time or curved space they didn't think about space-time but it's actually kind of an easy generalization but gousa riemann said if you go to small regions of space things look Euclidean Einstein says in small regions of space-time things look special relativity Anor Minkowski and if you want so this is Minkowski this is the Minkowski metric Minkowski or minkovski was Einstein's professor and again you know Einsteins slightly resistant to being too mathy so it was a Minkowski who after special relativity came along said you really should think about this in terms of four dimensional geometry and Einstein was like no don't don't make me do that but eventually he gave in so is Minkowski who actually wrote down this metric for the first time so curved space-time in small regions you can make the metric look Minkowski that's supposed to be an arrow sorry so what does that mean so at one point in curved space-time G mu nu at that particular point let's say X star equals minus one plus one plus one plus one and zeros elsewhere but elsewhere it changes elsewhere it's different so in general Jeanie nu is always symmetric so you know this element is always gonna be the same as that element whatever it is they're gonna be equal just like it was up here B equals B okay but in general the metric in an arbitrary curved space-time is not going to be minus one plus one plus one plus one it's gonna be some functions some function of T XY and Z in general all these different 4x4 matrix elements are going to be different functions of space-time okay so I can choose coordinates in which the metric looks like flat Minkowski space at one point that is the mathematical translation of the principle of equivalence the principle of equivalence says I can make gravity disappear in small regions of space differential geometry says I can make the curved space-time metric look Minkowski in in a small region of space-time that is the mathematician of the principle of equivalence so if it deviates elsewhere all the information about the curvature of space-time is in that deviation is contained in how the components these different entries in the matrix for the metric have the metric components change with space and time so for example let's do a simple example of a curved space-time the simplest example there is is actually kind of important one the expanding universe we know that cosmologically the universe looks pretty much the same on large scales in space but it's expanding in time what does that mean in terms of a metric well it means you can write G mu nu as time is just time so it's plus one so when people say is time expanding no it's just not that's a little bit unfair because the whole point of the metric is that it depends on the coordinate system you use like even in flat space if you use circular coordinates right or you know polar coordinates things like that the metric can look different so the metric does not instantly tell you the geometry of space but it can be used to tell you the geometry of space by massaging it in a particular way but it's a indirect relationship so that's why the curvature tensor becomes important the curvature tensor really does tell you the geometry of space but you derive that from the metric so the point of that was I can choose weird coordinate systems in which time changes in the metric from place to place but I don't have to I can just make it plus one whereas these plus ones sorry I like them these plus ones in the Minkowski metric okay they will necessarily have to change if the whole thing we're just the Minkowski metric then I would have flat space-time and the answer is that we generally write it as a squared as a function of time for G xx and the same thing for G YY and for gzz oops shouldn't talk and write at the same time and zeros elsewhere zero zero zero zero zero zero zero zero zero zero this is it this is I'm not fooling I'm not had anything back right this is the metric for a flat homogeneous and isotropic expanding universe cosmology what it's telling you is that time is just time time clicks off and equal intervals as the time coordinate T changes but as the spatial coordinates change or let's let's put it this way if you have some object two objects at fixed coordinates so here's an object at just call it X call it X vector sub one and here's another object at X vector sub two okay and as time goes on there's some distance between them D and as time goes on their coordinates stay the same so that by hypothesis let's say these are two galaxies I'm choosing to choose a coordinate system that stays fixed along with the galaxies so us and the galaxy that is ten billion light years away or it fixed values of the coordinates but the metric is changing as time goes on so this is now well if this is if this nice shouldn't right that that's gonna get confusing but the point is that the distance between them is proportional to a as a function of time okay this is what it means for space to expanse people will ask questions like what is space expanding into right but the whole point of differential geometry alec allison riemann is we do differential geometry intrinsically to the manifold we're not embedded in anything outside what it means that the universe is expanding is the distance between two objects at fixed coordinates gets bigger because this thing called the scale factor this is what we call it which is part of the metric gets bigger it grows over time if you were to plot a as a function of time it goes up in fact the universe is accelerating so now it actually is going like it's more like this but it's going up its first a derivative is positive that's what it means for space to expand it means that the metric is expanding in the spatial directions so even if things are staying fixed in coordinates the distance between them is expanding there's nothing that it's expanding into it's an intrinsic property of space-time itself okay good yeah I mean look by itself that is pretty mind-bending right now you've understood something about what it means to say the universe is expanding now we still have work to do okay that was nice that was talking about paths and how to measure them using the metric in space-time what we really want is an equation for the metric okay so in other words there's two aspects to a theory like general relativity one is how what is the meaning of the metric and the answer is it gives you distances time's that's how you use the metric is what you use to measure distances and times to figure out what pads are stuff like that geodesics all that stuff the other question is what is the metric what is the actual curvature of space-time okay so you need an equation for the metric and so that's what Einstein figured out that he needed and he set along how to do it and there's sort of a conceptual set up but then there's the actual work of finding the equation so remember in general relativity we have the metric which defines the connection so we can still do parallel transport right you can still take a vector and parallel and keep it as parallel to itself as it can be as we move it along curves and that's what the connection is and that tells us what the curvature is okay and now that I've given you a little bit of symbolism I'm gonna fill that in and this is sort of completely optional but the metric is written G nu nu it's just a four by four matrix in four dimensional space-time the connection is written gamma capital gamma Greek letter gamma with three indices lambda mu nu call it okay this is a function of the metric so you give me the metric all over space-time I can calculate the connection these are coefficients if you want the connection coefficients gamma lambda mu nu so this is a set of four times four times four equals 64 different numbers okay and from that I can calculate the curvature tensor are lambda Rho mu nu this is the curvature tensor let me just write tensor up there okay and we've already mentioned what are Rome you knew is what we what we said was that the curvature tensor takes in three vectors V 1 V 2 and we said if you went around a little parallelogram in this direction and then you took another vector V 3 and it as it goes around it will be rotated by that much and the little deviation which we labeled v4 was a way of measuring how much curvature there is you have to do that separately for all the different ways you could rotate a vector around a loop so for different values of V 1 and V 2 and V 3 you get different V fours and that's what our does that's what the curvature tensor does it tells you given V 1 V 2 V 3 what is V 4 it is not therefore a coincidence that when you look at the this symbolic expression for the Riemann curvature tensor there is 1 2 3 indices downstairs and one upstairs this is basically what it comes down to is these two indices right here are where you put V 1 and V 2 here is where you put V 3 and here's where you get out V 4 symbolically speaking I just want to relate this way of writing it to this way I told you before so the way I talked to you before was 3-month coverage of tensor was a map from three vectors to one vector from v1 v2 v3 to v4 here I'm saying that our lambda Rho mu nu is a collection of components which is a well what it is is a four by four matrix but instead of each element being a number it is each element is a four by four matrix so so it looks like this horrible messy thing where there you go 4 by 4 matrices arranged into a four by four matrix some of you might not even know what a matrix is that's okay all you need to know is sort of this rectangular array of numbers one two three four one two three four one two three four one two three four and every one of these every one of these alone trees one two three four one two three four one two three four one two three four a lot of numbers making up the Riemann curvature tensor that's all I'm trying to say okay and the reason why I have to remind you of this is because we're looking for an equation for the metric this is going to be how we get the equation for the metric okay so how do we do it so the the Riemann curvature tensor is made were made from the metric team you knew and its derivatives how much is the metric changing in the X direction the t direction the Z direction etc so there's some complicated formula for it I'm not gonna tell you what that formula is you can look it up I wrote a book remember there's also free lecture notes online if you prefer that method ok so this is what we have to work with what we want is an equation for the metric Latian for the metric is what we want well we have to work with is the Riemann curvature tensor why because the Riemann curvature tensor is the thing that tells you how curved the metric is and doesn't tell you anything else that's the important information the metric itself suffers from the flaw that if you change coordinates its values change in ways that do not reflect whether the metric is curved or not whereas the curvature tensor really cares about the curvature so the curvature tensor is what you're going to use so here are two ways of getting an equation for the metric the metric equation what you recall this is the field equation for general relativity because the metric tensor is a field it exists at every point in space-time right it has a value the values not a number the values a little 4x4 matrix but there's a different value at every point in space-time so it's a metric tensor field so these are the field equations for general relativity that we want to get way one is to guess and this was pioneered by mr. Einstein because remember he was more of a physicist than a mathematician so he had this brilliant physical insight and he just guessed the equation he did not guess it immediately though because it's still hard to get the right equation so what do you have to work with remember in Newtonian gravity remembered so remember in Newtonian gravity you have the force is G m1 m2 over R squared so you have like the force due to gravity on the left side on the right side you have the mass right you have this idea that it's the mass of the object that gives rise to a gravitation so Einstein knew that he had the generalized this somehow and so he already knew from special relativity that mass and energy are closely related and it turns out there's a way of combining them in general relativity or even in special relativity that makes that unification of matter and energy very manifest there's something called the energy momentum tensor or sometimes called the stress energy tensor I'm not going to go into details about it but it is called T mu nu so it is another four by four matrix one two three four one two three four one two three four one two three four different entries in it which tell you what is the energy density the pressure the strain or stress in different directions so if you think like you have some fluid right you have some continuum not just a point particle you have some jello or some rubber band or something like that you can stretch it you can push it you can twist it in different directions all those things you can do to it carry energy in some way carry pressure carries stretching this tension and so all of those are summarized in this four by four matrix called the energy momentum tensor so einstein's puzzle when he was trying to guess the right equation for general relativity was you have the curvature of space-time which is the characterized by the curvature tensor R lambda Rho mu nu and you have it's being you want to be sourced by energy and momentum so somehow it's related to team you knew and these muse and news don't need to be the same they're just indices telling you where you are in the matrix but there's an obvious problem this is a four by four matrix this is a four by four matrix of 4 by 4 matrices it's not the right amount of information to match up somehow happily there's a mathematical operation called contraction where basically we can pick out pieces of a tensor and combine them together giving us a smaller tensor sensor with less information but maybe the information we need so you have the Riemann curvature tensor lambda Rho you knew ok Riemann tensor and what you can do is define R Rho nu to be the sum over lambda of our lambda Rho lambda nu so in other words you can change this you can force this index mu to be equal to lambda and then you can sum over the values here's mu right there I set it equal to lambda and I sum over all possible values namely zero one two and three for the four dimensions of space-time and this is a tensor that is now not a matrix of matrices it's just a four by four matrix it is called the Ricci tensor it conveys some of the information in the Riemann tensor but not all of it and you can go further you can actually contract the Ricci tensor to get R and this is a little bit trickier but I'm gonna write it as R lambda lambda it's ashame these all use the letter R but that is a convention in relativity when you contract a tensor you keep the big name of it the same you just change the number of indices and this is just called the curvature scalar sometimes called the Ricci scalar but this is called the curvature scalar okay so the Riemann tensor is 128 different entries four times four times four times four the Ricci tensor is 16 entries four times four and the curvature scales just one number okay by the way there's redundancies here like just like the metric is symmetric you always know that G X sorry G X y equals g YX there's a lot of symmetries and anti symmetries in this complicated structure up here also but so we're not gonna worry about counting the independent components just counting the total number of components so look there's an obvious thing to guess if Einstein's method for getting a field equation for the metric is guessing there's an obvious thing to guess so you guess you look at this R Rho nu and say maybe the Ricci tensor is just proportional so there's some Kappa times T Road the indices had to match up on both sides of an equation that's a rule okay and this was a guess this is what he said in fact I'll be a little bit more conventional usually remember the the indices are just made up there they're just labels so I'm gonna call mu nu because that's what they're usually called in this context so Einstein guessed this maybe the Ricci tensor which is part of the Riemann tensor is proportional to the energy momentum tensor they both seem not as similar as both symmetric to index tensors maybe this makes sense sadly energy is not conserved if this equation is true and the Einstein again he was not necessarily intuitive mathematical genius but he was a physics genius he figured this out quite quickly so he knew this was not going to be the right equation soon after he tried to drive it so he did drive it he tried to play with it a little bit it didn't quite work what he eventually figured out was he could say well can I modify this so that energy is conserved and he can with a better gas which in fact comes out to be the correct gas that and no reason why you would see why this is the right gas but Einstein figured it out arm you knew minus 1/2 are the curvature scalar times the metric G mu nu equals was proportional to so Kappa some constant times T mu nu and this is the correct thing I'm not gonna boss it yet I'll box it soon this combination of the Ricci tensor the Ricci scalar the curricular in the metric is called the Einstein tensor because he invented it not bad for someone who didn't love math and this is the correct equation how do you know it's the correct equation well among other things you test it you do experimental tests so you can do what is called the Newtonian limit you can say this equation for general relativity is supposed to be true under all circumstances but I can consider things where G mu nu the metric is almost flat and we can make that rigorous we can specify what that means is very close to the Minkowski metric for flat space-time and the velocities of all objects that you're looking at are much much less than the speed of light which equals one okay so the Newtonian limit is both that gravity is weak compared to what general relativity is prepared to deal with but also that you're not involved in special relativity either so the speed of light is very very big compared to what you look at it and what you find is that Einstein's equation in that limit says that it says exactly what you want let me see how we wrote it down that you would get a force equal to G m1 m2 over R squared if kappa this proportionality constant right there equals 8 pi G therefore we say R mu nu minus 1/2 R G mu nu equals 8 pi G Newton's constant times T mu nu and that is Einsteins field equations for general relativity ok more details than you wanted but again you know what I'm trying to what I care about when I do all this stuff introducing you know this notation that you're never gonna use again in your life etc if I want to give you a feeling of flavor for what it's like like it's easy to say oh yeah space-time is curved and you guys that rubber sheet with a ball in it but this is what you actually do these are the equations you're actually faced with if you do general tivity for a living or even you put it to good use in your everyday life so I don't personally really do much general tivity for living but I use it everyday right I actually use it in what I do for a living cosmology and quantum gravity and stuff like that it is a it's a subject that is absolutely everywhere in astrophysics cosmology fundamental physics things like that ok and so Einstein was very happy he figured out he I think that he didn't quite get there first this is what we call Einstein's equation so now you know e equals MC squared is not anywhere close to Einsteins most important equation this is Einsteins most important equation this is the best thing I Stine ever did right here he did a lot of good things but this equation figuring out I mean the idea that space-time is curved that's good you or I maybe could have come up with that but then he figured out the equation a little bit of work and he trained himself how to do it from scratch he didn't know differential geometry he learned it and then did this with it so not too shabby but meanwhile you know he did he did you know Einstein was not like Gauss you know Gauss's rule was few but ripe he wrote very very little Einstein wrote papers like every day like he would just write everything that popped into his head and he would submit it to an Allah to physique and Dera physique and so he would he was definitely letting the world know along the way what his thoughts were going from the principle of equivalence curved space-time differential geometry to the equation you know he was very open about what he was doing so people pay attention to that and there is another way to derive Einstein equation the first play was called guessing where is it yeah guess the second way you could use the action principle the principle of least action right and this was put to work by David Hilbert so Hilbert was one of the world's best mathematicians unlike Einstein who was the world's best physicist Hilbert at the time was the world's leading mathematician and so he said you know he knew enough about physics just like Einstein knew enough about math to do differential geometry Hilbert knew enough about physics to know about the principle of least action which says you have an action which is an integral over time of a Lagrangian and if it's a field theory you're dealing with he knew that that was the integral over D for X space end time right T XY and Z of some lagrangian s'ti okay this is how you define a theory and the whole question is just what is curly L what is the Lagrangian City so unlike poor Einstein who had like all of these tensors floating around with all these different indices and you had to guess like what goes where and so forth Hilbert knew that if he just jumped to the action principle all he had to do is get one function no indices at all right and there's a lot more sorry there's a lot fewer functions lying around then there are tensors with whole bunch of indices that's why figuring out new theories by writing down their lagrangian's is a much more sensible thing to do than by guessing the equations of motion like Einstein did in fact there's kind of only one well it's not only one but there is certainly an obvious scalar function floating around namely the curvature scalar it's not the only one because we could take the Riemann tensor and square it okay in fact that's what you have to do in other gauge theories if you do electromagnetism or QCD or whatever you take the curvature tensor and square it to get the lagrangian but in gravity there's this curvature scalar which doesn't exist in the other gauge theories because gravity is a little bit different so this is a simpler thing you could use and Hilbert figured that out immediately Hilbert knew differential geometry he knew Gauss and Riemann and all that so Hilbert's guess Hilbert says that the lagrangian equals r and then plus some lagrangian for everything else which we'll call matter right so the Gwangi is just our x matter and there's some coefficient out here and it's gonna work out to be 1 over 16 pi g let me write that bigger for you the correct coefficient to get everything right in general relativity is 16 pi G is the hole grown gene and so Hilbert guessed this immediately like overnight maybe not overnight but very quickly he figured out once einstein had explained you know what he was looking for and what the tools were what the rules of the game were he guessed this and this goes right to you you get the least action you do the calculus of variations you find out what is the path of least action what is the equation that describes it and it is exactly Einstein's equation and so this trouble that Einstein had with energy not being conserved is immediately automatically solved by the action principle that's the nice thing about the Lagrangian of the principle of least action all the symmetries all that stuff are automatically taken care of because there's not a lot of freedom in guessing a scalar function from which you can derive the equations of motion so we actually call even though this equation is the Einstein equation ok this action s equals D for X oops integral d for X I'm cheating a little bit because the D for X needs to be modified in curved space-time but you don't need to about that one over 16 PI G or plus whatever it is for matter this is called the Einstein Hilbert action and there's controversy over whether Hilbert or Einstein actually got there first with the equation I think that the answer is it at Einstein gets all the credit and deserves all the credit um certainly Hilbert was working with Einsteins ideas from the start when he did it but I think even getting the equation Einstein got it first even though Hilbert might have said it out loud first okay but I do want to emphasize it because it fits in with our previous discussion of field theories and quantum field theories working from the action principle finding Lagrangian you can do that too in general relativity and it gets you it gets you there very quickly okay I know we're going long but this is so good I gotta keep going I gotta keep telling you some of the consequences here this is general relativity all in one video ok so I almost had lecture almost in one video we're getting all of general ativy so it's very important so think about some consequences of general relativity remember we told you the consequences in terms of particles moving on geodesics etc now we have the Einstein equation for the metric we can figure out what the consequences of that are now Einstein by the way he derived this he never thought it would be easy to solve this equation or even possible to solve it because this is a short compact equation but it could it hides a lot of complication right because this are both of these ARS are contractions of the Riemann curvature tensor dream on curvature tensor is a function of the connection coefficients those gammas and those gammas are functions of the metric team you knew and its derivatives and the derivatives are everywhere and so if you wrote this out in you could write out this whole equation just in terms of the metric and it would fill up a whole page of text ok it's incredibly complicated it's not like Newton's little equation so when it comes to physics questions physics theories beauty is in the eye of the beholder you know from the point of view of when you understand a differential geometry this is a very beautiful equation a very beautiful theory but if you just wrote it out as a field equation for the metric people think he would gone mad it looks completely crazy and very difficult to solve so Einstein himself was pessimistic about ever solving it and so he was a bit surprised when in 1917 Schwarz yield Karl Schwarzschild see if I can spell his name correctly came up with an exact solution to general relativity to this equation in a very simple context so Schwartz yield said let's look at spiracle e symmetric solutions so basically which fortune was after was the metric of the gravitational field of the Sun for example this is supposed to be gravity we're talking about so you have the Sun at the center he's gonna idealize it as a point mass so forget about the size of the Sun or the interior just look at outside the Sun okay just like Newton doesn't need to worry about the force due to gravity inside the earth or the Sun he could ask you can say that outside is an inverse square law so he looks at spherical symmetry and because he's outside he looks at the vacuum in other words he looks at places where the energy momentum tensor is zero so inside the Sun it's not zero but we're gonna remove that we're not gonna look at that so we have the Sun here here's the Sun but we're gonna say let's just take out that sphere and forget it and ask what is the metric out here okay as a function of let's say R the distance from the center so we're gonna use radial coordinates polar coordinates and he finds an exact solution and even more impressive he was a soldier in the German army and he was at the front this is like literally he did this on the basis of reading Einstein's papers at a time when by the way Einstein didn't really have it right and so I'm still groping toward the final answer so Schwartz he was handicapped by that but like he was literally fighting on the front and then to pass the time he learned general relativity and then solved the equations and he was then killed soon thereafter he didn't die like a gunshot it was like pneumonia but he he died very young but he managed to do that and so his name is never gonna go away historically so in polar coordinates what does that mean that means that you have and this is just in space okay so this is X Y Z so this is not space time time is just time here but instead of using XYZ coordinates let's use our and then if you drop down this point to the XY plane this angle here is Phi and this angle here is Theta okay so you have R theta and Phi coordinates this may or may not be familiar to you but don't worry about it if it's not the point is that I'm gonna compare what the flat space-time metric is in polar coordinates to what Swart Shields metric is you can see how it actually changes so Jean you knew the metric which I said is minus one plus one plus one plus one in rectangular coordinates Cartesian coordinates in space-time Minkowski coordinates in polar coordinates it's minus one nothing happens to time plus one in the dr squared direction R squared in the D theta squared direction and R squared sine squared theta in the D Phi squared direction okay so in other words the distance in space-time is minus DT squared plus D R squared plus R squared D theta squared plus R squared sine squared theta D Phi squared this is flat space-time this is just flat space-time in a weird coordinate system okay not that weird but a little bit different than what we had before so short Shield does his work and I'm not gonna reproduce all his work and we're just gonna write to the answer so the Schwarzschild metric G mu nu is minus the quantity 1 minus 2 G M over R by the way I said C equals 1 right the speed of light is equal to 1 everywhere here so this is G's Newton's constant of gravity M is the mass of the Sun or whatever other object it is you're looking at and R is the distance between the center of the coordinates and wherever you are so you notice as R goes to infinity to GM over R goes to zero and this goes to -1 so far far away from the Sun this quantity right here goes to -1 so far away from the Sun space-time looks flat that makes sense the next thing I need to write it I need to sort of squeeze down here maybe I can move myself over a little bit mm-hmm get myself a little bit more room here I'm gonna be able to compare it to the flat space-time version the RR component the thing that multiplies dr squared is plus 1 minus 2 GM over R to the minus 1 power ok but again as R goes to infinity to GM over R goes to 0 so this just goes to plus 1 which is what we want it to do and then these components aren't even change it's just R squared and R squared sine squared theta I'll leave it to you to take this metric calculate its Riemann curvature tensor calculate the Ricci tensor and the Ricci scalar and show that it obeys Einstein's equation and thank you but it does so Einstein was very impressed that shorty was able to do that that's very good and when you can do with that that is the metric around an object in outside the object not in the interior but outside so this is the equivalent of the Newtonian gravitational field around an object this is the curved space-time metric DT squared and dr squared have components that vary with are unlike they do in just flattened space-time so we can put this to work and you know again Einstein didn't know this deine stein didn't have this metric when he first met in general relativity so he did things like the precession of mercury okay if you look up a book like mine on general relativity you will see the precession of mercury derived so here's the Sun again and then here's mercury a little planet going around the Sun and goes in an ellipse if you were Newton's laws but according to general relativity you get almost Newton's law but that's in the weak field gravity regime mercury is close enough to the Sun the gravity is pretty weak but it's a little bit not weak okay there's a deviation from Newtonian gravity and what you get is that over time this ellipse moves that's called the precession of mercury and I was to finish the sentence if you look up in a book like mine you derive the precession of mercury from the short Shield metric but originally Einstein derived it without knowing the short ship metric he just used perturbation theory kind of like the general relativity equivalent Fineman diagrams to figure out what the precession should be he got the right answer this is you know I I can't even imagine what it would have been like to be Einstein and get the right answer for the precession of mercury was already known physicists already knew that mercury was moving in a way that didn't seem to fit in didn't tony and prediction they figured that it was probably some hidden planet like it had previously worked that the outer planets right like saturn and uranus didn't move in exactly the right way and they said well if there's another planet out there perturbing it maybe that would explain it and they discovered neptune so they played the same game for mercury they imagined there was a hidden planet near the Sun they called it Vulcan it was discovered several times but never correctly people thought they saw it and they were not right and Einstein figured out that it wasn't a hidden planet at all it was actually a new theory of gravity there is a lesson there for people who are thinking about dark matter right dark matter and dark energy are gravitational effects that are purportedly due to things we don't see it's very much like the history of looking for planets in the solar system the successful discovery of Neptune was an example of where you thought that your theory of gravity did not fit so you posited a new source and then you found it the unsuccessful hypothesis of Vulcan is an example where you thought your theory of gravity didn't fit so you have to change your theory of gravity okay I think and we'll talk later but I think that in the case of dark matter and dark energy we've done that thought experiment and we know that there really is some new stuff there not a new theory of gravity but we're not 100% sure so it's good to keep an open mind about those things okay so this fit perfectly and you know when I I figured it out it's a very precise number it's not like a qualitative feature there's an exact number that was known for how much mercury was discrepant by an Einstein's new theory predicted it perfectly and he must been very very happy when that happened that doesn't happen very often in the lifetime of a theoretical physicist but there it did another thing that you can do is the bending of light so this is an example of how general relativity is not just a cool fun theory but it's a unification it brings together not just space and time but it sort of explains what gravity is as the curvature of space-time in a way that is just more sensible than the Newtonian way of thinking about it so the procession of light says if you have a star like the Sun for example and a little light beam goes by it it will be bent okay by the gravitational field and you can calculate exactly by how much and you get the right answer and then Arthur Eddington did an eclipse expedition he claimed to measure the right answer he may have fudged his data or not it's not clear to me anyway but now we've done it very very accurately we know that Einstein again was exactly right the reason why this is impressive is not because here I mean here the number is sort of what you would guess like it's not as precise as the mercury thing oops mercury should be spelled correctly there you go what's what's impressive is that in Newtonian gravity it's not even clear what you mean like how you would do this cuz for Newton F equals MA right that's the whole motivating equation for Newtonian mechanics and but for light M equals zero there's it's a massless particle so maybe it's not accelerated at all by gravity right or maybe you say F equals G mm over R squared and you cancel the ends and so the M doesn't matter even when it's zero but you don't know is the point is just ambiguous and general relativity has no such ambiguity it gives you the answer and the answer is right and so that's very impressive okay next consequence we're gonna zoom through some consequences of general relativity another one is the gravitational redshift this is another one that Einstein came up with right away in fact actually it's an interesting history because the gravitational redshift is in some sense consequence not of general relativity full-blown but immediately of the equivalence principle you can do a little thought experiment we're not gonna do it right here but sorry let me tell you what the redshift is basically here's the earth or whatever and you have a tower and on top of the tower you build an observation deck and then you send a light ray from down here and it has some wavelength lambda 0 and it's observed up here with some wavelength lambda 1 and the prediction of general relativity is the lambda 1 is greater than lambda 0 the light has been a redshift that has it climbed out of the gravitational potential well and you can get this just from the principle of equivalence by saying imagine that the light is not climbing out of some gravitational potential well but is being sent between two rocket ships that are accelerating right and as the rocket ships accelerate of course the velocity of the rocket ship increases so there's a Doppler effect the straightforward Doppler effect says that the light that gets to the rocket ship ahead of it should be red shifted okay so you get this answer and you get it quantitatively correct even without knowing the Einstein field equations it comes right from the principle of equivalence it took a while natural 1960s that this was actually measured but again Einstein was right once again another important consequence I'm gonna say is cosmology because again much like the deflection of light where you could try to do a newtonian version and compared to general relativity but the Newtonian thing is just ill-defined it's just not clear what is going on same thing is true in cosmology and so what I mean in this case by cosmology is just the simplest thing you can imagine where you you you imagine that this is space ok space you fill it with matter uniformly this is what you're going to decide to do I'm just going to fill space with matter uniformly so there's some constant energy density Rho all throughout space dust or stars sprinkled uniformly throughout space and you ask what happens ok so from the atonium point of view you might say well what happens if I focus on one particle here and one particle here they both pull on each other right so they should come closer together that make sense gravity pulls on everything so that's one answer you can get but there's another answer you can get from the laplacian point of view you say there's a potential field and then the force due to gravity is the derivative of fine in the spatial direction but here because the energy density is constant everywhere the potential field should also be constant and therefore there is no derivative the derivatives are zero the potential is constant everywhere so this is saying that well yeah this this point is being pulled by that one but it's also being pulled by one on the other side and therefore nothing happens so there's in one way of thinking about the problem all the particles come together in Newtonian gravity and another way of thinking about it everything just sits there because all the forces exactly cancel and there's no obvious way of answering that question you can argue into it whatever answer you want but you can all argue the opposite way also nothing like that is true in general relativity in general relativity there's an unambiguous answer and there is this scale factor as a function of time and then we get an equation the Hubble constant equals 8 pi G over 3 Rho where the Hubble constant H is the expansion rate of the universe technically H equals 1 over a times the derivative a respect to time where a is that scale factor ok so there's an unambiguous equation saying it you sprinkle energy density evenly throughout the universe that's Rho then there will be an expansion rate or a contraction rate because H could be negative h squared and this is what led Einstein he was able to do this in his head and he said oh ok so the universe must need to be expanding or contracting and then this is where I think it's a bit of a bad rap because there's a story that floats around that says that Einstein derived that in general relativity if you had a uniform distribution of stuff spatially the be expanding or contracting and he had some philosophical predisposition to think that the universe was static so he got upset by this and he invented the cosmological constant which you can add to this plus lamda basically there might be some factors I think it's land over 3 and if lambda is no it's minus lambda over 3 how do I know that because the point is that a positive lambda can overcome can compensate for the energy density because lambda pushes the universe apart so this is the cosmological constant and so Einstein said well I can change my equations by adding a cosmological constant and that wasn't such a bad thing to do like the cows Marshall constant as we would now say represents the energy density empty space so it should be there anyway in some sense but the point is Einstein did not have some philosophical conviction that space was not expanding in fact when he predicted that space should either be expanding or contracting there was a moment when he was kind of excited like maybe that's true right and he asked his astronomer friends and we're talking now 19:17 or there abouts okay and as astronomer friends in 1917 it like space isn't expanding no because all they knew about was that's where the stars in the Milky Way and on average the stars in the Milky Way are not moving away from us or toward us there's some individual motions but they average out to out zero and so these drummer friends told him space is static so what he did was the correct scientific thing to do in the face of data that did not fit his theory he came up with a better theory right now later ten years later in 1927 Hubble and his friends show the universe is expanding the nine steinfeldt that he'd missed his chance like if he had ignored his observational astronomer friends and just said space must either be expanding or contracting he would have had another wonderful prediction but he didn't that's okay I don't blame him you know he didn't know that we were only looking at stuff in the galaxy okay two more quick consequences of Einstein's equation here is what we're gonna do gravitational time dilation we did time dilation for special relativity that is to say if you zip out near the speed of light and zip back you're moving on a non geodesic path you need to accelerate out and back okay so you will general experience less time than you would if you stayed home so that's where you get the twin paradox from in special relativity but now we get it even if you're stationary you can get time dilation and that's because of the Schwarzschild metric let's look back up here okay let's see if I can do this always kind of an adventure cutting and pasting in this program copy so we have the Schwarzschild metric there it is it worked with some extra pieces and what we notice is that this first term this term up here what is that that is the part of the space-time metric that is multiplying DT squared the change in the time coordinate okay so what that means is that constant R so let's imagine that you're on the surface of a planet or you're orbiting a planet okay you so you can move in theta and Phi but you're at a constant distance from the center of the planet then look what this means is that the time that you feel the change in time experienced is equal to well let's say I'll write it this way change in time experience squared that's the DS squared that we had before equals 1 minus 2 GM over R times Delta T the coordinate so this is the time coordinate remember in relativity coordinates are different than what you experienced that's the whole point of having the metric and being able to transform back and forth sorry everything here is squared but I'm gonna remove the squares let's get rid of the squared so we can just see what is going on directly and the minus sign disappeared because because how to say this D s squared this ace time interval is G mu nu DX mu DX nu but the proper time the actual time elapsed follows D tau squared equals minus D s squared so that I mean this is not supposed to be too surprising the amount of time you experience is a real number not at an imaginary number so if you get that you know delta T squared is minus something you better there better be a minus sign in there that fixes it this is where it is this is where that minus sign is so that minus sign goes away and you get this equation oh sorry I didn't finish it this part isn't squared this is the square root of this there we go oh okay this is the real equation so even if you didn't follow all that very very quick running out of time in my life explanation this is the final answer okay this is the relationship between the amount of time you experience the amount of time you would actually measure on your clock and the time coordinate at constant R in the short sealed metric and if you were to plot this to make a little bit more vivid what do we have here as a function of R this is the square root of 1 minus 2 GM over R so there's obviously something special going on at R equals 2 GM let's not get into that right now but at R equal to GM this is 1 minus 1 so this is 0 and then as you go to R equals infinity get larger and larger R to GM over R becomes smaller and smaller this becomes more and more close to the square root of 1 which is 1 so if we put one right here the curve that we have it looks like this so you asymptote to 1 okay so what this is saying is if you're sitting let's say you're sitting here at 3 GM R equals 3 GM okay then this coefficient is a smaller number than it is out here at 8 GM or whatever okay so as you get closer and closer to the gravitating object for a given change in T the time coordinate there is less and less change in your time experience do you experience less and time when you're down there in a gravitational field this is that is a correct way of saying it it'd be even more correct to compare two trajectories that start at the same point in space-time and ended at the same point but the slightly misleading way that this is sometimes translated as saying clocks run more slowly in a gravitational field clocks run at one second per second no matter where they are but if you compare clocks at different distances from the gravitational field they will appear to run at different rates if you can compare them in some sensible way this is where that comes from now you know it now you know the formula for it it comes from the Schwarzschild metric it comes from the fact that in the metric describing the curvature around a spherically symmetric object the coefficient of time depends on where you are in space it depends on how close you are to the gravitating object and therefore the rate at which time flows from the point of view of someone very far away depends on how close you are to the gravitational field that's a pretty cool thing finally file conclude final consequence is that there are these things called black holes okay oops I thought one more consequence just now at Elte at the end why are there black holes well look we we saw here there's clearly in the short field metric something going on at R equals 2 GM right at R equals 2 GM this quantity goes to zero so it looks like time isn't flowing that's weird but worse this quantity goes to infinity because it's zero to the minus 1 power that's definitely weird it turns out this is just because all of these coordinates are bad at R equals 2 GM R equals 2 GM is not something where it's a true singularity nothing is really blowing up or an infinite coordinates are tough and Einstein and struggle with this himself for a long time like coordinates a bad choice of coordinates can lead you to think terrible things are happening in general tivity when really they're not okay so the Schwarzschild metric in these coordinates just doesn't work at R equals 2 GM there's really nothing special there's no nothing singular about it so there it does exist it is an interesting place R equals 2 GM is called the Schwarzschild and I was gonna do that eventually and typically for something like the Sun or the earth the size of the object is way bigger than the Schwarzschild radius so the short shield metric only works outside the object so what happens at article 2 GM is completely irrelevant for the Sun or the earth because you're always inside the object at R equals 2 GM but there are things called black holes where you take the whole mass of the Sun and shrink it within its short shield radius and then this is where the event horizon is of the black hole and we can now finally this is that if you if you follow me this long if you're still watching you deserve the following pay off you deserve the following reward this is like one of those things when you teach general relativity the class is you get to this in like week eight or something like that and it is you know it makes the students so happy when we get to this fact what's going on and R equals to G M well we can plot it in a spacetime diagram okay so imagine this is X Y T so space-time diagram so and here is R equals to G M Schwarzschild radius so R is the radial coordinate here R and let's just draw this sort of Schwarzschild radius at all times good like that what's happening is if you look at the form of the short shield metric as you go to large R everything just looks like flat Minkowski space-time so we can actually characterize that by drawing the light cones we just draw it down here a little bit so if you start from a point you just draw the light beams going out of 45 degrees and a physical trajectory has to stay inside the light cone it can't move faster than the speed of light but if you follow if you did this at home I do encourage you to do it's kind of fun draw curves in the R T plane so forget about theta and Phi who cares about those but draw curves in the RT plane and draw them so that D s squared equals zero that's what the light cone is the light cone is d s squared equals zero speed of light okay as you go closer to the black hole to the Schwarzschild radius what happens is the light cones tilt so what happens is the light cones begin to tilt toward the black hole so you can it's easier if you like to travel in the direction of the black hole and harder to move away from it you still can but it becomes harder and the event horizon is exactly R equals 2 GM is exactly where the light cones start looking like this they're tilted so much that now every physically allowed trajectory goes to smaller and smaller values of R okay going more slowly than the speed of light means going towards smaller values of R that's what's happening inside the event horizon not outside and that continues all the way in once you're inside the black hole yeah light cones look like this and you're just forced to go all the way to are I do this yeah I can do this because I can rotate my pad to R equals zero that is called R equals zero is the singularity and once you are inside the black hole and your light cones are tilting like that there's nothing you can do other than be driven all the way to R equals zero okay so that's fact number one that is important about black holes the first fact is space-time is so curved that simply the requirement that you have to move slower than the speed of light becomes the requirement that you have to stay that you have to go to decreasing values of R all the way to our equals zero the second fact is that that location R equals zero is not a point in space okay it is a moment in time we think about decreasing R as moving toward the center right but that's just because we are used to coordinates in which R equals zero is the center the if you if you go back to the Schwarzschild metric when R is less than 2 GM this number here in front of DT squared is positive even though there's a minus sign out front if R is less than this is one - a number bigger than one so this is a negative number inside the parentheses times a times a negative number is a positive number so this is a positive number in front of DT squared and this number is negative because R is greater than two GM so this is bigger than one so one minus a negative number is a negative number to the -1 power is still a negative number so inside the event horizon R becomes what we call the time like coordinate and the T becomes the space like a coordinate so R equals zero is not a point at the center of the black hole it is a moment in the future the reason why once you're inside the black hole you cannot help but hit the singularity is because you cannot help but age forward in time it's a bad coordinate choice like I said to use R and T but if you want to do that you can just keep in mind that R goes to zero is like going into the future you can no more avoid hitting the singularity then you could avoid hitting tomorrow and that's why black holes are cool and that's why the event horizon is such a cool thing they memorize it in the singularity in fact there's a better way very very quickly there's a better way better coordinate system called Kruskal coordinates to draw the black hole in which call it capital R and capital T and the event horizon looks like this this is R equals little articles to GM looks like a light cone which is true because it is a light cone because that's that's the light cone is on the event horizon the event rise in R equals constant R equals to GM but it's a null trajectory and the singularity is this it's a moment in the future this is R equals zero so you travel in and you're gonna hit the future that's what it is the singularity is a moment in your future okay that's what if that's what black holes are all about Einstein had no idea about any of this because it wasn't figured out I know more about general relativity even Einstein did not because I'm anywhere near as smart as Einstein but because we have learned a lot about general relativity in the years since and that's exciting the last consequence is gravitational waves I'm not gonna write anything down but I do want to let you know we discovered gravitational waves just a few years ago 2016 at Lego but there were another prediction of general relativity right from the start another prediction that was hard to make sure whether it was real because of coordinate issues we didn't know whether it was just the coordinates that were varying or truly the curvature of space-time eventually we figured it out but the point is the gravitational waves take us all the way back to here remember Newtonian gravity I said we started this video by saying you could to send information faster than the speed of light by moving something very quickly in its gravitational field would change instantly throughout the universe in general relativity that's not true in general relativity the story is exactly equivalent to the story we told about electromagnetic radiation electromagnetic waves you take an electron and move it up and down the electric field vibrates and those vibrations spread out at the speed of light in general relativity exactly the same thing is true nothing - enough to draw the picture but if you take let's say two objects two black holes or two stars and let the more but each other so they're moving their gravitational field changes as a function of time but it doesn't change instantaneously throughout the universe it ripples out at the speed of light so Einstein's equation made that prediction and this is why Einstein equation is so awesome this is this is the great thing about really powerful theories of physics is that they predict more than you start with like Einstein was just trying to satisfy the principle of equivalence right he was just trying to say like let's do some math that is compatible with the idea that in small regions of space-time physics looks like special relativity and energy momentum is conserved and all that stuff all the things that we care about here's the unique equation I come up with and out of that you predict the expansion of the universe and black holes and gravitational waves and everything and it's still true like this is a 19-15 equation no experiment we've done more than a hundred years later has shown any deviation from this equation at all ah there's a an industry that says well maybe we should modify Einstein's equation even in cosmology yeah I guess this is the last thing that's a good ending point we know of course we should modify Einstein's equation it's a classical equation there's no quantum mechanics in there at all right so Einstein's equation as good as it is even though it's it's passed all the experimental tests we've done is probably not almost certainly not the final answer of how gravity works in nature because it's not quantum mechanical at all however we can think of general relativity as an effective field theory when we talk about effective field theories we said that you could put a cutoff at short distances and say that you don't know what physics is going on that distance is smaller than that okay but you do know what physics is going on a distance is larger than that and that is generally the way you would expect field theories to work in the real world so even though we don't expect on a science equation to be absolutely understand correct in the world we do expect that it should be correct above a distance is longer than we've tested it and we've tested at distances about that big about a tenth of a millimeter to be honest okay so everywhere above the tenth to the millimeter Einstein's equation seems to work and from a theoretical perspective from our theoretical bias from our priors and a Bayesian sense there's no reason to think that it should be modified or changed in any way you're allowed to try I have tried others try people still try to modify Einstein's equations that'll be very surprising it would not be surprising to see variations of Einstein's equations at short distances but seeing them in long distances even though Einstein's equation works really really well here in the solar system is exactly the opposite of what we would expect and indeed we've seen zero evidence for it so far so my guess is as far as the larger universe is concerned this equation of Professor Einsteins is the correct equation for gravity and now you know what it is

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Views: 521,681

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Id: 353lEB63iyg

Length: 109min 32sec (6572 seconds)

Published: Tue Jul 07 2020

Reddit Comments

[u/Grnoyes]

I just realized that if you remove some letters, General relativity spells gravity

[u/Valdagast]

He says this will be easy because he's talked about topology and geometry. Very bold of him to assume that just because we watched it, we understand it.

[u/BlackEyeRed]

Does he talk about his theory of combining QM and general relativity?