The secrets of Einstein's unknown equation – with Sean Carroll

Thanks, I don't know if you've wandered around the Royal Institution here, but you notice that on several walls there are video screens showing some sort of greatest hits clip, you can see David Attenborough and others with animals doing tremendously clever things. right here at this desk and others are doing demonstrations with extreme temperatures, high and low, or a garbage Goldberg machine with a marble running and I'm watching this before my own lecture and I think, you know, those poor audience members forced Sit through all this, Mom, when what you really want are some good old

equation

s, you've come to the right conference, we all know the importance of

equation

s in physics and we also know that they are treated like they're a little scary, right?

No? I'm supposed to tell people about them Stephen Hawkings says that every equation cuts your book sales in half. My editor didn't like that idea, but there's also a level of appreciation and understanding that comes when you're faced with a good equation. Here is a famous equation. This is a The equation that we all know Albert Einstein gave us is equal to mc^2, the energy of an object sitting there is equal to the mass of the object multiplied by the speed of light squared. It's almost an icon of inscrutability, but it's really not. difficult to understand is that you know a little about multiplication there is a superscript exponent for both but it's okay we know how to square a number it's not really that difficult nor is it what physicists think when they say the phrase Einstein's equation here it is Einstein's equation as physicists think of it, uh, this equation is the dynamical equation that tells us how the curvature of spacetime responds to energy, mass and momentum in the universe if you said it out loud R mu new minus 12 r g muu = 8 Pig t mu new, we never showed you this equation, you don't see this one for obvious reasons, it looks very intimidating, doesn't it?

There are all these symbols, some of them are big, some of them are small, some of them are Greek and it looks like it would be like that. It will be an effort to explain it to people, but I'm trying to imagine that maybe the effort will be worth it, at least on some level. You know that an equation is often related to a poem in some sense and they are actually very similar equations. short and they're powerful and they're precise every bit counts every little notation every little Dash Etc means something and it's a little bit of work to dig into what those symbols mean but it's very doable work, we have 58 minutes left.

I'm going to do it and I think it's worth doing because you know I'm not going to give you a task unless you really want to. I have good homework available, but the point is that if you understand the equation, you appreciate science on another level that you otherwise wouldn't, so it's worth the effort to at least think about it. What I'm hiding from you is that it's also fun. I think it's not just a matter of effort. You will see this and say: Why did they always hide it from me all those years? So we will gradually begin to accept the idea of ​​classical mechanics.

Sir Isaac Newton, a famous Englishman gave us classical mechanics in the 17th century. and it really boils down at the end of the day to a single equation f equals ma there's a little bit of an intimidating factor here because f and a have little arrows on them that's because they're vectors what this equation means is that the force acting on a object is proportional to its acceleration how fast it starts moving the speed at which it starts moving and the proportionality constant is once again its mass m force equals mass time acceleration this seems pretty simple, it was a big deal in time because it used to be thought that nothing could move at a constant speed forever unless you kept pushing it and Newton comes along and says no, no, no, if you don't push something, that's exactly when things move at a constant speed. , their friction and so on, that slows them down, so what's good about this equation?

¿Why is it so important? Well, it's accurate and it's everywhere. By precise we mean that it's not just a suggestion, right? It doesn't say well, you know, the harder you push, the faster things go. Wow, that's true as far as it goes, but this is numerical, quantitative, and precise, and if you want to fly a rocket to the moon, you'll need to know exactly how much fuel you need. This type of equation takes it into consideration, but everywhere this is even more important, this equation does not say that once I pushed my car I acted with a force and accelerated a little, it says that every time someone in history of the universe pushes a car or anything else, The force exerted on it will be proportional to the acceleration.

That's why equations are so great because they give you, even though you're sitting here thinking about it in this room right now, you suddenly have an idea of ​​how the entire universe works throughout its entire history. If things go well, it's because that it is worth contemplating what is happening here also does not tell you anything until you tell me what the force is how much force I am putting into things well, here is another famous The equation also from Sir Isaac, the story of him sitting under the tree and with the apple falling on his head is probably apocryphal and was certainly extended by Newton himself, who thought that it somewhat burnished his reputation as an inscrutable genius, um, when the apple fell from the tree the point is not that Newton discovered gravity guess what even before Newton we knew there was something called gravity what Newton realized is that universality that everyone uses about gravity the fact that his inspiration was that the apple falling from the tree is explained by the same set of rules that explains planets moving around the Sun, etc., the rules apply everywhere and in this case the rule is that the force due to gravity is inversely proportional to the distance between two objects, so that if we have a heavy object capital M light object Small M we make another small vector called e which is just the directionality where the force points, it goes from a small thing to a big thing and the force is the mass of the large object multiplied by the mass of the small object divided by the distance between them squared multiplied by a capital G constant that we call Newton's constant for obvious reasons and this pair of equations FAL Ma and FAL GMM over R 2 times the unit vector make the universe work or at least We thought they made the universe work until Professor Einstein came along, they did a pretty good job if you want to take a rocket to the moon you don't need to know about Einstein's equation, you definitely need to know about Newton's and long before Einstein there is a really important implication of this. which will be important later in the talk, but it will also show you some of the power of thinking in terms of equations because if the force acting on something is the mass times the acceleration and the force acting on something is the mass of the something else acting on it times Newton's constant times its own Mass/R 2, then the masses cancel, you can divide by Little M.

There we just took a math break to congratulate ourselves on doing some calculations, we divide it by Little M on both. sides and you instantly learn something profound and contradictory about nature the acceleration due to gravity acting on an object doesn't care how heavy that object is this seems contradictory if I take a hammer and a pen and let them fall, they will drop a different weight the different rates I'm sorry and in my brain I think that's because the hammer is heavier so here's an artist's impression of an experiment that was actually done by the Apollo astronauts, they took a hammer and a pen there upstairs, they dropped it, they had a very bad time. video cameras for some reason so this is a painting that remembers the event but the point is that on the moon the hammer and the pen fall at the same rate the real reason why the hammer and pen on Earth do not fall at the same rate The same rhythm is clearly due to air resistance, but this is a crucially important idea about how gravity works because, again, it is everywhere, it is universal, all objects accelerate in the same way, at the same speed under the force of gravity.

Universal, uh, gravity is the universal force. Keep that in mind that it's going to be The test will be done later and you know, for centuries classical mechanics was so successful that people thought they had discovered the basic framework according to which the universe works and it was just a matter of getting a good understanding of which ones. They are the forces. What are the things that act in this framework that we have, but then the 20th century arrives? Quantum mechanics, which by the way will be book two, so I'll come back here to talk about quantum mechanics later, but the other thing that happened. is relativity and relativity came in two parts one is special relativity in 1905, it's good from time to time in scientific talks to see photographs of Albert Einstein when he was young when he was thinking about all these things, you know, he went to the ruin later. aular here seems like the smartest guy in the world seems smarter than you, right, like you talked to him and you felt intimidated, someone was combing his hair, um and special relativity, there's a list of things that happen about speed of light and dilation it's all wonderful uh, we're not going to get into that, there's the book, read the book, but we don't have time for that right now, what I want to do is just use it as a stepping stone to general relativity .

General relativity was where Einstein really became a celebrity in a very real sense. Special relativity was Einstein being in the right place at the right time and laying a kind of cornerstone. He put together things that other people had already been working on in general relativity. It was his baby. but he got there reluctantly because this is the special relativity of 1905 in 1907 Herman Movi, who had been Einstein's mathematics teacher, says you know that your mathematics teachers read your famous scientific papers and Manovski said I have a better way to think about this when you talk. about special relativity and time dilation, all that, it's mathematically more elegant to imagine that what's really happening is that you have a unified space and time and he has a quote: "From now on, space alone and time alone are doomed to fade into mere shadows and only some kind of union of the two will preserve an independent reality which made scientists very poetic in those days people were skeptical they were not really immediately impressed a person who was skeptical Albert Einstein he tells his old man Professor Makowski's formulation makes rather big demands on the reader in its mathematical aspects, so if you are ever reading something and you think that the mathematics is too difficult here, imagine that you are Einstein and you think that yes, you know math is too hard here.

I'm not going to get upset about being as good as Einstein, in fact I would say that in some ways he is underrated as a physicist, but he was a physicist at heart, he wasn't a mathematician, he didn't love. mathematics by itself, learned exactly the same thing. math like he needed to know and nothing more than that, so he looks at msk's work and says: okay, you're doing some formalism, it's all very fancy, it's really of no use to me, I'm a down-to-earth guy. I want to imagine people going at the speed of light, things like that, but why did Minovsky think this?

Because he guesses where Einstein was totally wrong and he's going to change his mind very quickly. It's the geometry of spacetime that really makes it interesting to think about. Instead of space and time separately, think about what we mean by geometry of space. There's a lot going on in space geometry or imagine here good old British geometry, tabletop geometry and it's useful to us, for example, if you have coordinates. If you have an straight but it is diagonal compared to the x and y axis AIS, the distance you travel will be the square root of x^2 + y^2, okay, the hypotenuse squared is the other two sides squared their sum, that is something that You could use it when you have this, the guy is supposed to have a little odometer here counting his steps to determine how far he's come, so it's something handy and basic.

What MOSI says is that you can think of Einstein's special relativity as exactly the same type. of the thing, but replace the distance traveled with the elapsed time, so before Einstein and Makovsky came, time was something universal according to Newton, everyone agrees on what time it is, not everyone agrees on the distance that you travel, which depends on the path you take right between two points. You can go in a straight line, you go in curved lines, different distances, but the time you thought was universal. Minkovsky says no, what you have to do is use the Pythagorean theorem for space-time, whichwhich involves a little trick, namely a minus sign, so Manovski says that the time elapsed between two events in spacetime can also be calculated in terms of a coordinate system, but now their coordinates are XYZ in the space and T in the time direction and there is a different difference between t, the universal time coordinate around the world, versus what we call the right moment, what you actually experience, you would measure it on your wrist watch and there is a formula for this and it looks a lot like the Pythagorean theorem, but not quite Pythagoras says that d^2 is x^2 + y^2 MOSI says the time you measure squared is T ^2 minus x^2 if you have ever tried to wondering which twin was the one who aged the most in the twin paradox, you know because I know I've had to wonder, just remember this equation, the minus sign means that the further you travel in space, the less time you will feel passing, that is brilliant codification of all the ideas of special relativity right there, so while manovski was saying all these mathematical niceties, Einstein was worried about gravity when we talk about classical mechanics, the first one.

What we did was we said, "Okay, let's imagine a specific force of natural gravity. It all worked out very well in the prinkipia Mathematica uh. Newton told us the rules of classical mechanics and his inverse square law for gravity, so which Einstein, quite naturally, thought had special relativity." replacing classical mechanics now needs to replace Newtonian gravity, it didn't work, you know, there's kind of an obvious set of guesses that you go through as a theoretical physicist, none of them work, you need to find something a little deeper, so Einstein got it right. About something called the equivalence principle, remember that even Newton had discovered that gravity is a universal force.

Two things fall, they will fall at the same rate, no matter what their mass is, but also no matter what they are made of, no matter what time of day it is and so on, so Einstein puts this in the form of a thought experiment, it was the best mental experimenter of the story and says that what this means is that somehow gravity is not really a force at all because I can't tell if I'm in a gravitational field, you know, I can sit here and I can do experiments. I can drop things and they will fall and I'll say aha, gravity made something fall, but he says I would say exactly the same thing if the Earth had disappeared and this.

The entire conference room was in a rocket accelerating upward at 1 G. That hasn't happened, don't be alarmed, but Einstein says there is no experiment you can do while we're locked in the conference room that can tell the difference between gravity and acceleration. We are indistinguishable from each other now you and I that is the main equivalence and you and I would say aren't we smart to have thought of that equivalence and then move on with our lives, but Einstein was Einstein and he says this means something that I had to be able to use this to find the right way to put relativity and gravity together and I suspected that what I needed was more math and I wasn't very good at math like I said, I mean I was good at it when I had to be, He didn't do it in his spare time, although fortunately one of his friends from graduate school, Marshel Grman, was an expert in what was at the time the most sophisticated mathematics and geometry available.

Here's Grman on the left, and that's what Einstein says. You have to help me with this and Marcel Grossman taught Einstein, uh, what were at that time the most modern ideas in Geometry. So what does that mean? What counts as a modern idea in Geometry? Well, the British geometry that we talked about with the Pythagorean Theorem is based, as you know, on a set of axioms and the famous and strange axiom is the parallel postulate, the idea that if you start with a small line segment, take two perpendicular rays and let them move towards infinity. The fluid says that they will remain exactly parallel at the same distance forever for many years, literally centuries, millennia.

I guess people tried to prove this feature of geometry based on other UCL aums. When I was in high school our geometry teacher gave us this as an extra credit assignment to prove the parallel postulate which he was very cruel because you can't prove the parallel postulate like they realized in the 1830s because it can be replaced with other perfectly good postulates. You can say that those initially parallel lines converge at a constant rate or maybe they diverge at a constant rate and you get perfectly good theories of geometry, different from ukan geometry, but perfectly good in their own right, if the lines start to converge at a constant rate you will get a spherical or positively curved geometry, if they diverge at a constant rate you will get a saddle or a hyperbolic or negative curve. geometry and these were researched and they were as good as British geometry, so people started saying: well, why does ukan geometry work so well in the world?

What's going on? Other people said we're still making an assumption here. The assumption we are making. It is that if those lines begin to converge or diverge, they do so at a constant rate. It was Gaus. Carl Friedrich Gaus, one of the greatest mathematicians in history, who said, look, you know, once you tell me that I can have a sphere or a chair, ride or a French fry I can have whatever wild thing I want I could even have a three-dimensional space or a four-dimensional space or a 29-dimensional space and they could be diverging in some places on these lines and converging in other places as if suddenly there was a number infinite number of geometries that he could happily think of for Gaus, he had a very talented graduate student, Bernard Reman, and Reman was already a well-known mathematician, but in the German system there is an endless series of exams that you must pass before you can become in a teacher and then the last qualifying exam reman gave gaus a list of possible topics he could study and gaus pointed out the one that reman thought was the most boring the fundamentals of geometry do that one and reman shoots up and it's very funny read it I did this, you know, to write the book for the first time.

I actually looked at Remon's paper and he can't stop writing inside the paper. He says I don't like doing this, but you know they tell me to do this. It's not my natural purpose of met meter remon was to discover how to capture in absolutely perfect generality the geometric structures of a space in any number of dimensions without embedding them in any other space, just their own intrinsic character, so obviously there are many ways in which the Space could curve, twist, fold or whatever, what's the easiest way to boil that down into usable information and remanufacture because he was pretty clever, uh, he figured out that if you know the length of every curve you can draw, imagine this.

On the table you simply draw each curve, both straight lines and curved lines and everything, if you had the information in your brain about the length of each curve that would completely fix the geometry of the space, the only way those curves can fit together is if The space is flat or in this image they are spheres with small balls of thread, small lines that cross over them. The idea of ​​remon is that all you need to know is the length of each possible curve that fixes the geometry of space and how it works. It's like the Pythagorean Theorem all over again, but because things are arbitrarily twisted, you zoom in.

For those of you who know calculus, this is the traditional calculus trick when things have curves and change, you zoom in. and if you do it carefully enough, everything looks straight, flat and constant. so reman says I can take an arbitrary sinuous curve, I can zoom it in and I can use the Pythagorean theorem, so in this case this little three-dimensional diagram, the length of a little line segment is x^2 + y^2 + z ^2 the square root of that, so remon's self-appointed task is let's generalize this. The cool thing about this image we drew is that it is actually in flat space so I have drawn the X YZ coordinates if the space is not flat if I am in a sphere or a Taurus or something crazy I may not be able to use as convenient as you.

This is just for those who are familiar with the idea of ​​spherical coordinates, so you have XYZ, but you could also tell what the distance to the point is. What is the distance? angle from some line of longitude or what is the angle from the North Pole which gives you spherical coordinates which are very useful when you have spherical symmetry and here is the formula for the distance of a small infinitesimal line in terms of R the distance Theta and To those two angles, there is no need for any information about that particular equation to penetrate his brain, other than the equation becomes more complicated when the situation becomes more complicated, so what Remon needs is what is the maximum complication, what It is the most arbitrary way we can think of. this you stay looking at this expression you ask what is happening and if you are remon anyway you say okay, I think what is happening is that I have on the left side an interval, I would like to say something of distance, but in space-time it's going to be some time so we use the word interval to mean space or time some interval squared is on the left side that's what we inherit from Pythagoras on the right we have a sum of terms with all the different coordinates multiplied by some number some amount true , so d s on its own, but then the Theta term squ has an R squ outside of 5 ^ squ has r s like there's a bunch of stuff going on in the general pattern is a number times one coordinate times the other coordinate, so what the squared coordinate multiplied by some number in other words in SpaceTime, for example, where you have t x y y z, you will have a formula for the interval in spacetime just as Manowski gave us, but perhaps more general, so the squared interval will have some number * t^ 2 plus some number * T * itself in Your head, if you are that good, it quickly becomes unwieldy.

You have to write too many terms. It gets boring. If there's one thing mathematicians are very good at, it's inventing notations that turn things that are unwieldy but understandable into very elegant but impenetrable. things they do, they invent something called a metric tensor and the first step here is to relabel your coordinates. You thought you were happy with TX YZ coordinates in SpaceTime, but someone reminds you that you might want to use spherical coordinates or something. complicated, so instead of TX YZ, let's call them x0 only two spatial dimensions, maybe nine spatial dimensions. if you are a string theorist but we always have a time dimension so give time the label x0 and all the space dimensions are X1 X2 X3 Etc. and then the whole thing you call it It is an index. you run 012 3 or if you want TX YZ completely equivalent x0 is the time X1 is are labeled in directions you can move in SpaceTime TX YZ, so it's really not that new or crazy and then instead of saying some time number T^2 plus some time number TX, you make up some new notation G mu new and G mu new is simply the number that is multiplied by perfect inscrutability or compactness, so this is just the sum of all those values. the indices take from G mu new * g muu x mux new is just gx02 plus g0 Etc.

Well, now Albert Einstein comes on stage who, as you can imagine, was doing his homework for Marcel Grman trying to teach him geometry and says: "This is very exhausting, writing all these signs of addition. I'm going to invent a new notation in which you write the same thing but you don't write the addition sign. He was so proud of himself that he literally told jokes. He said, "I've made a great discovery in mathematics. You don't need to write the addition sign, always." and when you have repeated indexes,you imagine we mean you should add them together.

This is called the Einstein summation convention, it's not the most important thing Einstein ever did, but he had fun with it now that we've reached total inscrutability. very compact G mu X mu X new I'm even leaving out some of the complications CU I like you, but this is what's going on. We have taken up the question of how we generalize an arbitrary geometry. Well, you need to know the length of each curve, how is that information encoded? This is how that information is encoded in the component values ​​G mu new, let's see that at work, okay, so G muu is the metric tensor, we already saw it secretly when Manovski told us about spacetime. said that the time interval it measures is related to the coordinates txy Z by total t^2 - - x^2 - y^2 - z^2 in terms of that little formula we had on the previous slide that means that g mu new is usually zero, there is no term in the equation that is x * y for example, that's why you get all zeros here on the right, but in principle the rest can be read in g0 0 or gtt is plus one 1 gxx or G11 is minus one and so on and also all these symbols have meaning.

I mean, you knew they had it, but we can even tell you what the meaning of the spacers, the just the generalization of the Pythagorean theorem, maybe in a curved spatial geometry, those are the spatial distances, what is that? It is telling you how fast time flows, now that is a completely absurd idea how fast time flows because time flows at 1 second per second, there is no other speed at which time will ever flow at two seconds per second, okay, but the time you experience can flow at different speeds with respect to the time coordinate, the time coordinate is not anything physical, it's just something we use as a convenience, so that component of the metric tells you how fast flow time compared to the coordinate that everyone has agreed to use it and then those who mix space and time literally tell you how time and space can intertwine with each other now we don't have any of that in the Manowski metric that is not happening, You could say like, when is the time? and space are going to intertwine, when will I ever see a big Hollywood blockbuster where that happens?

Well, you have to because time and space become intertwined when you're near a rotating black hole. This is a famous image from Interstellar. the movie This is the Gargantua black hole and the whole image here, all the little filaments of light were calculated numerically using a metric tensor that had a significant rotation of time and space relative to each other, it was so much work that Kip Thorne uh The winner of the Caltech Nobel Laureate, who is also the executive producer and inspiration behind Interstellar, collaborated with the special effects people here in the UK to write an article.

I mean, they also made a movie, but they wrote a bigger paper on classical quantum gravity where they explained. all the mathematical details that go into creating this image and if it weren't for the components of the metric tensor that allow time and space to bend around each other, it couldn't have happened. I hope you enjoyed that little pop culture interlude because that's it. No more of that gets serious once again, okay, so you teach Einstein that Reman figured out how to capture the information he needs to characterize a perfectly arbitrary curve geometry, now you need to put it to work, how can you make money with this metric tensioner? what you care about is not the distance along a curve, what you care about is the curvature of spacetime, what does that mean if uid says you start with two initially parallel lines and they stay perfectly parallel and Ukian geometry is plane geometry and then curved geometry?

It's not the curvature of plane geometry, it's all the different ways that those initially parallel lines can stop being perfectly parallel forever, so we have a way of characterizing this called the remon tensor much better than the metric tensor because the tensor metric only has two Greek indices the remont tensor has four greek indices that are double and the game you play is: tell me at what point you start with what line segment you start in which direction your initially parallel lines go you ask the remont tensor the question that? happens to them as I extend those lines and the remont tensor tells you how they twist or converge or diverge or whatever, so the remont tensor really captures what you care about in the geometry, namely their curvature, the gunu It's crude. metric tensor data, but what you want is the remon tensor, so just to understand this, you know in context, here we have secretly seen different types of tensors, different types of quantities that are geometric and have different numbers of components between the moment in space-time.

In ordinary Newtonian mechanics, momentum is a three-dimensional vector, but guess what in space-time it is four-dimensional because you also have a Tim-like component. Have you seen the 4x4 metric tensor, so 16 components, the remont tensor has four indices, so it's a 4x4 matrix of 4x4 matrices and I swear if anyone says they see a typo in this, I'm going to hit them again . This is one of those slides that you're not supposed to get any knowledge from. impressed and wow, there are a lot of remont tensors out there, okay, and if you're a physics grad student, sorry in my day, you would calculate the remont tensor these days, they have their fancy computers and everything and they don't know it . how is that so this was a lot so let me capture what you're supposed to get out of this.

Okay, spacetime geometry is what you need to talk about in general relativity. This is Einstein's vision. If gravity is a universal phenomenon, then. It is not like electromagnetism or other forces, it is a characteristic of space-time itself, what characteristic it is, it is the geometry of space-time, he needed mathematics to describe it, which he was taught, it is Ranian geometry or differential geometry which can be arbitrarily complicated, but let's keep it straight. Boil it down to a small piece of usable information on how to calculate the length of each infinitesimal curve and from that you can build the geometry as a whole.

That information in turn is captured in the 4x4 metric tensor in four space-time dimensions and that tensor. has a curvature characterized by the remon tensor, so in other words, the metric tensor and the remon tensor are not completely separated. The whole point of the metric tensor is that it tells you everything you need to know as long as you can extract it so that the remon The tensor is defined in terms of the metric tensor, if you work as a general relativist your day job is usually to discover the remon tensor from of the metric tensor or some vice versa version of that.

The next question, if you are Einstein, is how are you? you're going to use that technology to come up with a physical theory, a theory of gravity, well you knew what Newton did remember, this wasn't just bragging when we talked about this before, this is Newton's way of telling you the acceleration due to the gravity. Well, the acceleration depends on Newton's constant multiplied by the mass of the Earth or whatever is causing gravity divided by the distance between you and the Earth to square the center of the Earth in this case, so Einstein you need to replace both sides of this equation on the left side the acceleration due to gravity you need to have some measure of the curvature of spacetime on the right side the source of gravity for Newton the source is the mass of the gravitating object what will it be for Einstein in this relativistic context?

I already know that it won't just be the mass because Einstein told you that E is equal to mc^s, which is an indication that energy and mass are somehow unified, somehow they are part of the same thing, so What mass is not the most important thing was for Newton in relativity mass, energy, heat, momentum, tension and pressure. All of these things will somehow be sources of gravity. What we need. I must ask if anyone can guess. Do you know what we need? We need a tensioner and. we have one that I didn't give you time to guess, sorry, uh, I didn't want someone to come up with me guessing, so the energy momentum tensor is basically another 4x4 matrix that tells you all those things, it's called new T and it encapsulates all of them. the different forms that energy, momentum and other related quantities can take and once again we can look at it component by component, there is a z00 component or TT component if you will, which tells you the energy including the mass, that's what We usually think of it as the amount of energy in the old Newtonian sense, these diagonal components txx and yzz.

That's pressure, so we're thinking here not just of a single object like Newton would have thought, but we're thinking of maybe a fluid or a solid, so you take a small region of it and there's pressure in the X direction. , the y direction, the Z direction, those different directions are the diagonal components of the energy moment tensor and then there are off-diagonal components that you can sometimes ignore, not always, but they tell you how the energy changes and moves from from place to place, heat flow, momentum, etc., okay, all the different ways you can think about energy are grouped into this tensor, this will definitely be something that we're going to be using as we build the new equation for the gravity Einstein is looking for.

We have the metric. The metric defines the Remont tensor that characterizes the curvature of spacetime. Mass is a type of energy, so we are starting to see the left and right side. On the left side we want to replace Newton's equation, okay there is an obvious assumption that it won't last long on the left side, acceleration should perhaps be replaced by the curvature of spacetime which as we know is characterized by the remon R Lambda tensioner. new row mu and the right side of the equation that has the mass in Newton's equation should have the energy momentum tensor T muu, so we have the curvature of spacetime at the moment of energy on the left and mass, and so on to the right, obviously doesn't work, I can imagine Einstein going like he was very close and it didn't work, it doesn't work because the remont tensor is a 4x4 matrix of 4x4 matrices, the energy moment tensor is a 4x4 matrix of numbers that do not match, cannot be proportional to each other.

Fortunately, there were some hard-working Italians who read Remon's paper in the 19th century and discovered all sorts of things that can be done with Remon's tensioner. Basically you can distill it so you can collapse like they say the remon tensor to make something called the Richie tensor after Professor Richie and the Richie tensor only has two indices and then you can collapse it again to get something called the curvature scaler so when Einstein are trying to think of what Can you put on the left side of your new equation for Gravity? It has several ingredients to play with.

It has the remon turnbuckle but also the Richie turnbuckle and also the warp climber. Now you will instantly guess that I want to take the Richie. tensor because it has two indices. I'm going to set it equal to the energy moment tensor which also has two indices. If you're guessing that, congratulations, you're as smart as Albert Einstein because he guessed it didn't work. It doesn't work because energy is not conserved and various other problematic things, but again, being Einstein, he worked harder and came up with this R mu new minus 12 r g muu = 8 Pig t mu new which is the correct generalization of the equation from Newton to the relativistic context and now we call this the Einstein equation, you saw it a few minutes ago, but now you understand it perfectly, but the point is that you don't see why, even if your understanding is only 99% and not 100 %, you see that it is not just sh off or atrocious right as these symbols have meaning these muses and news talk about directions in space-time they will take different values ​​0 1 2 3 r on the left and G on the left are different geometric quantities that le tell you about the curvature of SpaceTime t muu to the right tells you about the energy momentum G is Newton's gravity constant 8 and Pi that you've known before, so if you get this far and appreciate that you now have an equation, the next step Of course, it's like for Newton's equation is to solve the equation, what that means is that what you want to know is the metric in spacetime, so the metric is what you need to actually calculate the force on a rocket or a planet or whatever or a black hole, okay, the metric is implicit here because you use the metric to define the remont tensor and the remont tensor to define the Richie tensor and the curvature scaler, but it's a little bit indirect, in fact, it's kind of super indirect, so just so you know that I've written a component of the remont tensor in terms of the metric here is so in principle you could sayokay, I'm going to be brave, you know, you give your least favorite grad student the job of trying really hard to figure this out and Einstein.

He himself realized the level of complication, the amount of math you have to do to go from metric to remon and back and forth and solve his equation, and he said, "Nobody's going to do this." "I'm not doing this." I'm Einstein, so no one else is going to do this. Carl Short Shield went and did it. Short Shield is a fascinating story. He was an astronomer and mathematician by profession, but remember, look at the dates in 1917, there was a war. You may well remember World War I and Schwarz Shield because he was good at math was given the task of calculating ballistic trajectories for the Germans fighting the Russians on the Eastern Front and, but I don't know what the details were, but apparently there were a leave from Shor or something, not from the Navy, so it's not shely, but he had some time off from fighting on the front and was allowed to go to Berlin and spent his free time listening to Einstein's lectures on relativity, it's an election.

I can't argue with that and Einstein gives his equations and says no one is going to solve them. Shield comes back to the front and sits down and works them out and then how can you possibly do this right? You make a lot of guesses. a lot of assumptions, right, you make yourself say that doesn't sound very scientific. I'm going to use the German word, I'm going to call it an anot an onot it's an assumption and in this case what Schwarz Shield said is look, I don't care about the general arbitrary solution to anything. I care about the Earth moving around the Sun, so I'm going to ignore the Earth and Jupiter and all I'm going to do is take the Sun placed in the middle of my coordinate system.

I'm going to use spherical coordinates just like we talked about, instead of txy Z, it's t r thetaf as you can see in the equation up here and I'm going to assume that all off-diagonal elements are zero, etc., etc. me too. I'm going to assume that everything is static, nothing vibrates and then reduce the problem to something that you can actually solve and this expression here is the famous Schw Shield metric. This is the exact metric for a perfectly spherical empty space solution to Einstein's equation. a star or a planet or something and Einstein himself saw it and was instantly impressed, which is memorable because Einstein was handed a lot of papers and he usually wasn't instantly impressed.

He usually had a reason not to believe them. Shield solution. He was very, very happy. Now from here, you are now cooking with gas. Now you can calculate the movement of the Earth around the Sun. or more importantly, the movement of mercury. You can calculate gravitational lensing and gravitational redshift and a bunch of other predictions. One of the reasons Einstein became a celebrity so quickly is because his equation led to predictions that were tested and successful. Arthur Edington led an exp uh expedition to search for the gravitational lensing of light during a solar eclipse. It is believed that he was not perfectly rigorous with the data, but he got the answer that he knew he wanted to get Einstein was very happy, but I don't want to talk about it now, let me because we were running out of time, so let me talk about a fact that you can deduce from this equation.

I want to let you know that it's not just about you knowing some symbols here. you can get something out of them, so remember that we have interpretations or meanings for these different components of the metric tensor. G TT is the speed at which time flows for an observer compared to the time coordinate, etc. look at gtt, it's the top left left right corner 1 minus 2 gm/r r is the radial coordinate G is Newton's constant m is the mass of the thing and wait a minute when R is equal to 2 gm, this total amount is equal to zero, it's 1 minus one so time doesn't flow or something, what's going on at R is equal to 2 gm and then you look at the RR component, the next one diagonally down and it's if the TT component were zero, this is minus the reciprocal of zero? one over zero, which is infinite and that's bad, so as soon as you write this you're like this, something seems strange seems to be happening at R is equal to 2 gm, so what did people get right?

They said they calculated the value of R is equal to 2 gm. so you know the mass of the Sun, you know the radius of the Sun, what radius we're talking about here and what you quickly discover is that the relevant value of R, which is now called short shield radius for obvious reasons, is a lot. smaller than the actual radius of the Sun and this solution is only supposed to be valid outside the surface of the Sun, not deep inside, so people said and correctly, the fact that this metric seems to go crazy when R equals 2 gm is not physically relevant. for things like the sun, earth, etc. it only refers to places where the solution is not supposed to be valid anyway, some more imaginative astronomers said yes, but what if the thing collapses?

You can see a three-hour biopic about one of them if you go. to see Oppenheimer Oppenheimer and his student uh harand Snider didn't solve the static case, but they allowed a sphere of M to collapse and showed that it collapsed and went beyond the short radius of the Shield, so this could be physically relevant, what is it? you know what it is, it's a black hole here's an image of a black hole, not the black hole, you know what I mean, it's an image of light approaching a black hole, this is from the Event Horizon telescope, but here just To make it vivid, remember what is g0000 g gtt?

It is how fast time passes compared to the time coordinate. I'm plotting it now 1 - 2 gm/R so this is r in units of GM on the horizontal axis and it's g0 0 and then you see that in Ral 2 GM this number goes to zero, so at that place the rate of passage of experienced time is zero from the point of view of an external observer. We call that place the Event Horizon. It took decades of intelligent mathematical work. It wasn't like that until the '50s, people figured out how to just change the coordinates and there was nothing, there was no barrier that you hit in the Event Horizon, you could go through it if you wanted.

These coordinates are not very good there, but the coordinates are. just human inventions, they're nothing very profound and also there was work by people like Shandra Seikar and others who showed that these collapses can actually happen in the real world and they finally started saying in the '50s and '60s that maybe these holes black people really exist today. Of course, we take pictures of them, we hand out Nobel Prizes like candy to people who look at them in various ways, and the point of this is the last slide, which is that the reason we care so much about these equations is because they're smarter. . that we Albert Einstein was pretty smart he derived that equation that we show you here in the middle of the slide but he never knew about black holes he never fully understood gravitational waves he was always kind of uncomfortable with the idea of ​​the Big Bang the equations we just sat there smugly and said you'll eventually figure it out and we're getting to the same thing with Newton's equations or with Maxwell's equations or with Schrodinger's quantum mechanics equation we do some experiments locally here in our short, brief lives we take some leaps of imagination to sum up all this complex physical behavior in this concrete little poem and then we extend it to circumstances that we haven't even imagined seeing yet and a good equation will teach us something that will work, yes, the equation knew it all. together thank you