 So that gives you these many, and then there will be values of the value of the technology that's set. OK, now, we then quoted a film which came that completely symmetric spaces, a space that is both homogenous and isotropic, or has maximum number k, in which it's called a previous n-frame. Completely symmetric spaces that essentially work. That is, they are classified by the value of the energy scale. Now, given a space with a particular positive value of the energy scale, it's easier to produce a space with any other positive value of the energy scale by just changing the overall value. But to take the metric and multiply it by 0. OK, so that's essentially one space with positive energy scale. That's one space with zero energy field, and one space with negative energy. That's essentially three of the scales. There are three symmetric spaces that you can get. And once we know that they link, it's very easy to classify these spaces. And I just remind you that you have constructions. And I remind you of our construction. The guy with zero energy scale has one basis, that space. That space has p in the p plus 1 by 2, killing that plus, because it has the b minus 1 foundation. And it has the translations. It's the point coming up. The space with positive curvature is the scale. And in the previous class, we talked about this in a special case, p equals 3 minutes, totally general. This space, x1 squared plus p plus 1 squared, behind the sphere, how many symmetries do we have? How many symmetries in the sphere? OK, symmetries by dimensions. So all the rotations in p plus 1 by dimensions. And that is p plus 1, in a p plus 1 minus 1 by 2. So p plus 1 in a p by 2. Completely true in every dimension. The sphere gives you the maximum risk. But we talked about the base of parametricism, the coordinate pattern in the sphere, which we actually have. OK, and the space with negative curvature is also easy. So the space with negative curvature every day goes to look at the space minus x0 squared plus x1 squared plus x2 squared this is equaling to minus x0 squared. And part of this as a manifold, in many ways, are one component. And if you remember, we used these equations to derive the matrix of these maximally symmetric spaces. And I'll just quickly do that for you again in the form that is when these two are simply related. So the form we're going to take is to let x1 and x2 parametrize the coordinates on a p-dimensional sphere of radius r. So the matrix is going to be, so that this tells you that x1, so what it means to suppose x1 squared plus x2 squared let me use uniform notation, let me call it x. So I always solve for this, here for x, that's what we have. In each case, x1 squared plus x2 squared plus xp squared is equal to r squared. So we've got, we use photomorphism to this p-1 dimension space and matrix by the angle wherever they happen to be of a p-1 dimension sphere. And this radius. Then suppose we look at ds squared, which was plus and minus, and let me call this also p plus 1. It is uniform, it's the negative direction. Plus minus dxp plus 1 squared plus dx1 squared p squared. Then we plug in this part in here. And what do we get? Because we saw last class we get ds squared is equal to dr squared plus r squared d omega p minus 1 squared. When d omega p minus 1 is metric on a unit, on a unit p minus 1 dimension sphere. Remember, Synecchia forbids us from having a cross a dr v in the angle there because there's no vector field. That has the full rotational symmetry in this way. And that we get just dr squared with equal and what we are arguing currently factor 1 we are arguing really, we tell it for 3, 4, 3, 4. We argue this way. And this is the part that comes from this part. Because it comes from a part that is equal to here. But this was very simple because x p plus 1 was simply equal to, so let me write it, plus x p plus 1 squared. But x p plus 1 was simply equal to 0 squared. Now, so there's an r squared here. So we get either a squared and then plus minus r squared. That's what minus is. Minus reverse of the sphere and plus reverse to Euclidean this this, this other thing, this negative connection case. You see that we get the other side here? Because you put an r squared here and you take it to this side. That's what minus is. Okay, so now substituting it again, we got the metric with the s squared equal to all we have to do is plug this in here. So we get E, we have r v r the whole thing square by a squared minus plus r squared. You see, when you differentiate this, you get a factor of 2 you get a factor of plus minus up here with squared, so that's it. Okay, so you, okay? And then plus the r squared. Plus r squared d, okay, minus 1 squared. And then we add this up. You get and now there's also plus minus here because this came with a box. So we get plus minus r squared minus plus r squared plus a squared d, okay, r squared by a squared minus plus r squared plus r squared d, okay, so this one. And so that is equal to a squared over a squared minus plus r squared square plus others, where d omega p minus 1 is squared. In the special case, if we were particularly interested in dp is 3, this dp omega p minus 1 was sine square theta d phi squared plus d theta. OK. So these are actually the metric space. 100 s from here, the flat space, of course, the flat space, the metric is d x squared question the comments about this. OK. So now this is the starting point for our discussion of the first one, which, as we said, when the zero of the space is completely isopropagated, which is simple. And then, as we said, there are two natural forms. There are two reasonably natural forms that we should write in the lecture. There's a full space signature. One of the things that we've done before was ds squared is equal to dg squared minus, let's give it the name, d of t squared. And then, let's say we take this metric with radius equal to 1. So a of t is the radius of the space. And then, d space 3, this is either the metric of the sphere or the flat space, or if we need it, with radius equal to 1. There is another form. Now, by really finding time, there's another form in which it's sometimes convenient to do this. It's the same metric. And that's this one. It's equal to a of t of t squared minus 3 to the first and the second form, where I do is to do a coordinate change. So let's see what coordinate change will be. You see, suppose I go to a new coordinate, so t is equal to t of t. How will this metric change? Well, of course, this thing will just become a of t of t. But this guy changes in a slightly more complicated way. It changes as d t by d eta. So the thing behind d eta squared is d t by d eta squared. And in this metric, what we demanded is that the coefficient of this and this is the same thing. So now to make the variable change, what we have to do is to solve the equation that this is equal to a of t squared. These are the differential equations that we can solve to go from one coordinate system to another. So we'll see in the examples. It's easy to go between coordinates. These are two different ones, each of which has their own elements. In the first one, t is the proper time. We will very soon see, I'll write down in 10 minutes, the Christoffels symbols in the space. And you will very soon see that in this space, lines of constant position on the sphere or the universe in the space, the flat space, are geodesics in this space. So if you were an observer floating around in your rocket shipwork, you were a galaxy floating around in this space, if you started out at constant position x, and nothing was property to the no-forces of people, but the no-forces that you cancelled, then you would see that. So observers, like gangsters in this space, are labeled by constant positions, constant spatial positions. We'll see this. Constant spatial positions in this universe. And then t is simply the proper time, but time measured on a watch by one of these, by any of these systems. So t is a very nice physical object. It has not so nice. Basically, it's sometimes called conformity. And the reason behind it is that this space here is conformally optimal, this overall factor, multiplication of a metric by an overall factor is a maximum conformity of 1. So this space, up to this overall factor, is just dt squared plus the metric of the sphere or the metric of the flat space. The metric of the atlases is completely linear atlases. So it's time and space indirect product. Up to this conformity factor, it's time and space, not the metric of the atlases. This is for some purposes, this is outside. So t is the proper time for the flawless. It has what's called the conformity time. Both will be useful in many certain cases. Yes, sir? So what do you mean by 0, 0, 0, 0, 0, 0, 0, 0, 0, 0? So one set of juries in these six we've talked about. They have constant values of space. Other sets of juries, we'll have some of these. Because of this, on the eta square, you see that isn't the a of eta outside here. So we can have the juries in this space, and not the same as the juries in this space over here. So one set doesn't simplify things too much. Fine. Now is to try to solve an instant equation to determine, so we almost completely got the form of the vector. But not quite this. Please. Sir, since eta is not a right parameter, the equation of the geodesic equation would it be not either of the two equations? They, you see, for the particular geodesic space equals constant. We continue to be a geodesic. It's true that, you see, just being at a constant location in space, we continue to solve the equations. And how you parameterize, how you move along the geodesic, it's true that the time along the geodesic is not eta. So the lambda of the geodesic will be the bad function of eta, that we just wrote down in this equation. But just the geometrical part in space, which is always sitting at space equals constant. So that continues to be a geodesic. Yes? Because having a geodesic is not a geodesic, if that geodesic is not there, this dv square minus dv space will do that. Right. In general, that will not solve the geodesic equation. You see, suppose we set no matter, for instance, a retail curve, this thing to be dr, this sphere, then that will not solve an instant equation. Because the sphere is covered, so you will get something on the left-hand side of an instant equation, nothing on the right-hand side. What are we doing at the moment by making this assumption of symmetry? After having made the assumption of symmetry, we've written down the most general form of symmetry. You're not allowed to do more than that. You're always allowed to impose a symmetry. Because, well, OK, no, not always. You're usually allowed to impose a symmetry. I think always, always. You're always allowed to impose a symmetry because we've got some solutions instead of equations that allow for a symmetry. Then there's always, at least I'm not sure if it's always true. It's usually true that there are solutions to that symmetry. You may find that there are no solutions, in which case it's always true. I think it's always true. I would have to think about it. I think it's always true. This is a solution to that symmetry. OK? But what I mean is that, by, by making an ansatz that is dictated by enforcing a surface symmetry, you are not killing solubility of equations. But if you just make some random ansatz for the metric, that may not be what the right solution will be. It may not solve the equation of motion. OK? So all we have done is to write down the most general metric consistent with the symmetry ansatz that we wrote. And we're not allowed to make anything. We can't say, well, what if we die as a made-of-team of sanity? We could say that with reasonable measure on the right-hand side, it will not be sanity. And that's the dynamics. It's how the universe will expand as a response to the matter. It's not the audience. So you can set the GTT space of the symmetry. This was, it was a consequence of symmetry. It is a consequence of symmetry. Even there is a consequence of symmetry. You see, because what we want is a metric that has all these killing vectors in it. It has all the B in the plane-blast mark in the case of three-dimensions, all the 10-dimensional killing symmetries of the spatial vector. And once again, there is no, let's say, a vector field there is no vector field in the three-spheres that has all the rotational invariance of the three-spheres. So suppose you had a DTD space term. As far as the three-spheres were concerned, that would be a vector field. So we have one index. And if you wanted it to be, have, enjoy all the symmetries of the three-spheres, then you would have to be a vector field that was rotationally invariant and full SO4 symmetry of the three-spheres. But there's no such thing. There's a similar theorem for killing the three-spheres. Certainly it's an obvious theorem for flat surface. Because you would have to have a vector field that a vector field somewhere over, you rotate it and it rotates. Is that clear? So the fact that there is no vector field in any of these places that enjoys the full rotational symmetry, the full killing symmetry of these places, ensuring that it's not consistent with these killing symmetries, there is no DTB space term. So that is not an assumption. It follows from our ansatzus symmetry, from our demandus symmetry. But now, you might have thought there were two functions at the time. There's one behind this and one behind this. But what are the capabilities of my quadrant? So is it all DTB x times r z to be the product of the vector field? That's right. Suppose we've got some coordinates deep-down alpha on our station. So the DTB alpha term is of the form DT deep-down alpha and some A alpha which should be a function of line and the line. Now as far as the spatial coordinates are concerned, this is the one. In space, one from vector fields, it's a number in the spatial metric. So the question is that if this space had all the killing symmetries of yaw under that spatial manifold then there is one form here that had to be an invariant under the symmetry generator by those killing symmetries. But there's no such one. There's no such one form field. A alpha of the alpha, this one. That's it. There are some spaces where there exists an invariant one. Sir, there exists one. Exactly. So a circle here is a circle. Something like this. So I should say for B, greater than or equal to for this space being two-spheres away, then something I'm familiar with in the case of two-spheres from the fact that there are no vector spherical harmonics with anything. Scholar spherical harmonics start at anything. At some point in some course you study vector spherical harmonics. And you might remember that Jackson used to be Jackson. So the vector spherical harmonics are all these stuff. It starts at anything. Anything that, you know, helps that shape our proteins. Vector spherical harmonics always appear in multiples. It's a very easy thing to do. It's easy to understand both intuitively but it's true. So now we've got the form of matrix. Does it imply that the matrix is absolute? Yes, it would imply that the stress tensor should have related properties because otherwise it wouldn't be consistent with the matrix. Yes, but what about the matrix? Matter field. Distribution of matter is different. No, I mean you're giving the same stress tensor. Okay, I've known this. I've only seen the stress tensor. You know if you've got red stuff here and blue stuff here but they have the same energy. But then evolution can break this effect. No, no, no. Oh, you mean, but this affects the equations of motion of the matter. Yes. That would be right. So anything that either in stress equation C or the equation of motion of the matter Oc would have to respect this. It would be different. But if you've got red dust here and blue dust here both of all, you know, red galaxies and blue galaxies would be on the same. Whatever that means. With the same mass. Anything that is blind is not seen in the equations. Right. Okay, so now let's move on. Let's completely determine the form of our metric. Because we still need to know what this is. So so now let's start writing down how we're going to determine that. That has to be determined by solving that situation. Okay, so we take this form. Sorry, as usual we take this form of the metric and we start computing all the things about it. All the things one should compute and start computing crystals. Okay, and according to the policy we decided on a few classes ago we're not going to actually compute crystals which are the class. Okay, so I'm just going to go with the values. So okay, now for this, I mean in this form you have to choose one or choose equals to this. Actually the equations are comparably complicated. My word works with this form. Which is an irritating thing because it makes it harder to comprehend the form of this. But anyway, we're working with this form of the metric and we get I'm just going to go back to this. I'll write one of these just so that we can see all of them. We'll use ij as the basis. And that's cool. There will be a lot of time. And the kama ijk is dama ijk is purely spatial things which will be calculated which will never be needed. Because anything that we already know the answer to the curvature is coming from purely spatial dama ijk cases. It's that form with product of ancient dancers. Okay, and the first thing about this that I want you to notice is that now let's look at the GDC equation. Okay. I want first to justify that where is it? I want first to justify that the statement I made that lines of constant spatial locations are GDC equations. Okay. Let's remember the GDC equation. The GDC equation was d2 by d 1,000 from potential GDC of excited plus then we had ij by ij in the 26th hour of the form x and this type of music or personal music. Ah, ah. Which is called GDC. No, I'm using our GDC. I'm using our dimensions. It's Greek is all. That's right. Okay. Okay, we claim that x i equals constant and x 0 is equal to some function x 0 of power. We don't specify what the function is, do we? Obviously that is. You should just get it properly. Is the solution to the GDC equation. So where is this part? The x i equals constant self equation. So now if x i equals constant as a self equation then d2 by d tau square of x i should be 0. Okay. So that means that this right hand side should be 0 when we put an i in theta. But self consistently the only thing on the right hand side is d x 0 by d tau because d x i by d tau is 0. So the question of whether gamma i is 0, 0 is actually not true. But gamma i is 0, 0 is actually not true. So from the fact that gamma i is 0, 0 is actually true if true that constant spatial locations are due to this equation. Okay. So the Galax studies in topology will be a constant, labeled by constant locations in space. Okay. But where any which has a time max of the level of space in there? No, I don't think that's a waste. It was very important here. Let's write down the formula for G 0, 0, i. I mean all we need is the gamma 0, 0, 0. That's that's that's it. So that's gamma 0, 0, i is is equal to G i mu by 2 times gamma 0 G 0, i is G 0, 0 basis in which there's nothing of the action. So mu is a purely spatial index. So let's say that this guy is vanishes. But this guy is vanishes. If G 0, 0 depends on where you are in space then this guy is not vanishing. And it won't be true. An easy way to say Yeah, exactly. That's what the Schwarzschild metric is that has this property that counts over in space. But yet if you just put things about three-per-point space in another solution then what's the potential? So okay. So much for these crystallizations. Now the crystallization will be used to compute coverage components. By the way, so now this thing we did was a check that gamma is using to actually vanishes. So now if you use to compute the coverage components I'm going to write just write down what you need to do. So you get R 0 0 5 8 What can you say with R 0? What can you say about the answer it must be 0? Yeah, you would have to work through I'm thinking that way but the function of symmetry you see if a metric has a certain symmetry then the coverage would continue to have that symmetry. Now 0 i is an extra field under purely spatial limitations. It should respect all these killing symmetries of that. That makes it R is 0 R is 0 okay. So what remains R i g it is constant Now there are two contributions from R i g there's the part that you get with only the gamma i t keys and there's the part that comes from including the 0 components okay. Now the part that came from gamma i t keys we already know okay. Let's remember we had written that sometimes we told that R alpha beta gamma delta was equal to g alpha gamma g beta delta minus g alpha delta g beta gamma gamma delta gamma gamma gamma gamma gamma gamma now it follows from here that if we take this object and we contract okay. So we contract let's say alpha with gamma Okay. So we contract alpha with gamma then when we get here we get is equal to now we stick to three dimensions So, we get plus minus 1 by a square into 3 g beta delta minus g beta delta 3 comma 2 is equal to plus minus 2 by a square. So, 2 by 2 square into delta h. So, there is a term that is just 2 by a square. This is the part that you get pretending that there is only space, but there is a whole set of time. Where? A square or A square. No, I raise 1 x. So, but there is also a contribution from the fact that you got this, you know, that you got this a square, that a square is a function of time or all the other stuff. So, when you get that out, okay, now I am going to follow that out. It should say first work out things for the case of positive. First work out things for the case of the sphere and then we finally write down all the answers. So, for the case of positive for the curvature, you get 2 a square into 4, okay, and then in relation you get that a naught square plus a times r i j has to be proportionate to delta i j. That again was basically a consequence of symmetry. You know, there is no other symmetric density, okay, and then this is the coefficient of that. This is the value of r i j and this is the value of r 0 0. Okay? Now, with these in hand, we are all set to interact with equations. Okay, but before we start writing down x 10 equations, we need to know what comes from the right-hand side of x 10 equations. What is the matter that fills the universe? Now, of course, you know, the matter that fills the universe is a very complicated stuff. It is made up of the quantum field theory that makes up a standard model of physics made up of walks and electrons and neutrons and protons and you know, all that stuff. Okay? And the actual thing on the right-hand side is that comes from the very complicated actual quantum state of the stuff. You know, you want to actually solve that your day. Okay? And then we'll come to the right-side action. But what we're trying to do is take a course break. And there is this wonderful thing about matter that, you know, almost about the stress density of any reasonable quantum matter that comes from it. And that at longer distances, it can be, the stress density can be approximated by that of the fluid. But we're just going to have a 500-tutorial influence. It's made up of some collection of stuff. Okay? The great thing about fluid is that the important thing about fluid is that it is assumed locally to be in some sort of thermo-liquid. Okay? So suppose you have some collection of stuff. You look at a little local region of that stuff. That little local region is assumed to be thermo-liquid. And the parameters of this thermo-liquid are functions of space. So in the simplest case that we've been dealing with, thermo-liquid, what? So let's look at, for instance, a gas reporter. These quotas may be interacting with each other through interactions induced by integrating out these two quotas. They interact with each other. And they, through these interactions, they interact with each other. Now what are the parameters of thermo-liquid? We're not going to look for a gas reporter. A gas reporter is a parameter of thermo-liquid. That's simply the temperature of the gas reporter. Okay? There's no particle number. It's a particle number that's not observed for it. It would be for less than electrons. Okay? In which case the parameters of thermo-liquid would be temperature and can be connection due to that. But in this case it's just temperature. Now, in general for thermo-liquid, there is some frame. So this thermo-liquid stuff could be, and how fast you're moving could be different. Just like the temperature of your stuff can be different and different equations. How fast you're moving can be different equations. So in addition to the parameters of thermo-liquid, there's also velocity field. Those are the variables. The variables of your thermo-liquid have some temperature as a function of x and d. And the potentials, for instance, as a function of x and d. Also, the velocity field of this equilibrium of stuff is a number. And the other parameters of thermo-liquid are the variables. The variables of charge are the variables. Okay? These are the variables of the stuff that is approximately in thermo-liquid. Approximately in thermo-liquid. But the parameters are different, yeah. Temperatures are different. You see, if equilibration happens, because things collide with each other on some sort of length scale, on the length scale, the mean-free path, the temperature, the fractional change of the temperature is substantial on the length scale of one mean-free path. Then it's basically meaningless to talk about equilibration. Because then the very notion of the temperature is passing. So, what you require for this description to make sense is that the fractional change of the temperature on the length scale of a mean-free path is small. Okay? So, you require, for instance, that d d by d x divided by x is a fractional change in 2 l m f 3 is my fractional change. And d m f b is the mean-free path. This is the condition for thermodynamics to make a good description of what we discussed. And this condition is satisfied. It's reasonable to consider the parameters of equilibration as being the basic dynamical future. This is a huge simplification of dynamics, because instead of thinking of all the complicated things, it builds a freedom of microscope recklessness. We need only what I want to temperature and I can make a prediction of the velocity. The mean-free path will be a motion. So, whether this is true or not, it will depend on the particular configuration. But when this condition is satisfied, where does it come from? Well, it's from the fact that what you're trying to think of is stuff equilibrated. But, you know, equilibration happens because of collisions that happen over a length scale. The parameter, if you're equilibrated, you're equilibrated at a given temperature. If, on the length scale of a mean-free path, the temperature is not well defined, meaning changes in fractional difference, you're clearly not equilibrated. So, the whole thing is not a system. Again, so flow dynamics are actually, okay, so these are the variables. The equations of flow dynamics are simply the statement of the conservation of the stress density and other charges due to which, due to these type of conditions. Let's do some counting. How many variables do we have? Well, how many variables are there in the game? A temperature? Let's look at the simplest case of four times the stress density. So, it's temperature and chemical protection. So, there are four dimensions. There are four variables. The three variables are the velocity and one for the temperature. How many equations is the conservation of the stress density? Oh, same number of equations is there. Okay? However, you know, the equation of the conservation of stress density is an equation for the stress density. The variables are very temperature-intensive. So, this is not a dynamical system unless you have a way of expressing the stress density in terms of temperature density. The key ingredient in little dynamics is what's called constitutive equation that expresses the stress density as a function of thermodynamic fields. Now, we can understand this concept due to relations. You know, what we make of this, because what? So, let's see. What we want to do is to express this stress density as a function of these thermodynamic fields. Let's first, you see, in building the notion of, in the very notion of thermodynamic fields is the idea that things are locally originated. So, things are very small compared to, are very slowly compared to some of the roots here. So, it's sufficient, in fact, it's stupid to ask for more, to specify this stress density as a function of all the variables in an expansion and derivative system. The term that you want to specify is the term with zero. So, the term with zero derivatives. Now, the term with zero derivatives is very easy to specify, because if the term has no derivatives, then it's the same for an equilibrium field. You can't tell the difference between whether the flow is an equilibrium field. So, the first thing that we do is to ask, what is the stress density look like in equilibrium? So, team you whether or not, if there are no derivatives, you can have union or union, and you can have G meter. You only need things that have to write. Let's find the right names for the coefficients. And now, let's get our sine convention. Now, sine convention is G 0 0 is positive. And G ij is negative. So, let's look at the ij components of these. I mean, the components that are orthogonal to, yeah, they are orthogonal. So, I'm going to write something there and then you can do it. So, I'm claiming that at 0 of all the derivatives, the stress tensor of a fluid must take the form, T v of u is equal to epsilon times u. So, I'm saying that this is force to us. Firstly, that it has to be proportional to this and this. That's force to us, because there's nothing else to the right of x. That has two derivatives. We're looking at the terms with zero derivatives. Yeah, they could be forms later on. Energy density of p is the pressure. These are just names. But they are important names because they connected due to the equation of thermodynamics. Now, why is this the case? Well, let's take the special case in which u was 1, 0, 0, 0. Everything was fluid suppressed. Then how did the stress tensor take the form? In the 0, 0 component epsilon plus p minus p. And the one-month component, this would be p v p, which is how we want the stress tensor to come. Pressure is that force with the ii component of stress tensor and the 0, 0 component of the form. And now we can go to a for a fluid in equilibrium. You can go to any other, this is always true. You always make the boost to go to the coordinate system in which we have 1, 0, 0, 0. So, this is the case. So, this is what the stress tensor of a fluid is. Well, epsilon is the energy density of p is the pressure. Now, I would gain the meaning by writing this form. Yes, we have. We gain the fact that we that instead of, you know, 90, I think there are two functions of temperature just by symmetry analysis. But they are not because once, for a fluid, once you know epsilon, you also know p from the equation of state. Suppose you have this free energy as function of temperature. From the function of energy, you derive both the energy density as well as the pressure. Pressure is like minus the energy, and the energy density is the appropriate energy. These are not independent things. From how we dynamics, in the particular case, for instance, of gas, where there is only one filter of charge, there is only one unknown function of energy, which is for instance the pressure. Or instead of the temperature, you can treat the energy density as well. And then the pressure is determined as some function of the energy density. From the equation of state, the particular fluid has some equation of state. You need to know the equation of state. That equation of state, which is just determined by the intrinsic properties of the fluid, not where you put the fluid. Determine the pressure of function of energy. So, in the particular case of a photon gas and so on, you want the temperature of charge, there is basically, you know, just a single, there is just one single here. Then, namely, the temperature of the function of the space of the energy density. Now, higher orders in the derivative extension introduce more terms, the stress density. But it turns out that, and as we will see later on, there are all these terms are actually negligible in a thousand, a thousand times. So, these terms are basically utterly negligible for most of the cosmology. So, we are going to forget about that. We just write this form of the stress density. And forget about all the other terms. The stress density is all the stress density of the coefficient. And then what is the equation of motion of fluid dynamics? The equations of motion are simply the statement of the conservation of the stress density. Okay? Now, this is changing at every point that how do you assume it to be isopropagyl. So, it will have to have the p itself, it will have to have the symmetry at each point. Meaning, you see p is a function of both time and space. It will have to have the same value, suppose we are on a scale, and every x. But it will be allowed to be a function of time. So, in our situation, p will be completely off. That's the only way it could happen to us. That way, it should have the same stress density as the symmetry of the crystals. Is this clear? Because this is the unique scale of fluid. The unique scale of fluid that has the symmetry of the sphere as the constant. So, it doesn't develop very well. Now, we know up to the equation of state, which we are going to leave unspecified right to the end. Because there are many interesting questions of that equation of state. Of the effective equation of state that is filling the universe. After that question, we know the form of this stress density. Let's say a word about this equation of state. You see, suppose we have an Einstein's equations, and the only matter was effectively the galaxies. Then this matter would be effectively like dust. The galaxies aren't constantly colliding with each other. So, there's no pressure. So, it's the galaxies. It's sometimes called a cold matter. So, the galaxies, and you know, interstellar hybridism and so on. Effectively behaves like dust. And the equation of state is really easy. For those galaxies, you plan in this form of the equation of state. The stress density. On the other hand, as you all know, there's this cosmic microwave matter. There's gas of photons that's filling all the space. These photons are not fresh. They're moving around in the light. If you take a little box, they're constantly going in and out of the walls of this box. So, there's no pressure. And we know what the equation of state is about. Photon gases. Can I even tell? What's the equation of state? Regularity. P is equal to O by 3. Actually, follow students. E by 3. A stress of the stress density. For gas of photons. Must vanish. If I take the trace of this, I get 1. If I take the trace of this, I get 4. So, E plus B must be 4 times B. Therefore, 3 times B must be E. Therefore, B must be E by 3. The trace of the stress density for our gas of photons is repenisous. It's scaling that. So, it emerges in the blackboard. Yes, sir. You get from a detailed calculation. But, you could figure this out in general. No problem. Okay, great. That's it. So, this is one kind of equation of state. There's another kind of equation of state. Okay? There's a third kind of equation of state. Which is of great interest because it looks like it's there. And that's the equation of state for our cosmological constant. Let's talk about that. Look, these photons is normal. Me and you, partner. So, I get 4 pixels through... You know, the flute. Yeah. It's true. You see, you're asking how long the photons take. Well, there's no infinite path in some... Firstly, that's the whole basis. Because the photons interact with each other through loops of the... So, they're not exactly... But you're right that they interact with very long timescales. So, you know, the question we're asking is, if you could take some photons here, set them out a little bit, how long would they take to be cool? It's true that the timescales would be very long, but we're also talking about it. They're going to cause a lot of big accidents. It's a good question. How do these two good timescales compare? I suspect that, you see, if we were out of this regime, if the infinite path was effectively infinity, in this case, then we would have to worry about the other... You're saying they would never... They would never equilibrate. No, I think it must be the use of the photons. We're in the regime where equilibration happens much faster than in that kind of logic. So, the electrons in this... Yeah. Well, they would be interacting for all kinds of reasons. There would be this interstellar dust. The photons would interact with that. And then there would be loads of that dust. But others of equilibration would be through interstellar dust. You see, in a black body, you know, why do you get such effective equilibrium? Because you keep getting the walls in the black. It's that interaction that essentially happens. So the other stuff will be at some temperature and then it will equilibrate the matter. And with the galaxy, too. There will be various mechanisms for equilibration that will happen faster than cosmological matter. And even for the photons, yes. Okay, but there's one more effective form of matter. That's not really a form of matter. That's just the energy density of the vacuum itself. Okay. But that can be treated like an effective route. And it's convenient to do that so we can use one uniform set of formulas for all possible kinds of matter. And that's the gospel of logic. Okay, so let me just go back to this brief discussion of gospel of logic. You know, I had this constant dot 15 years ago. You might well say, like, I know you should say network. But there's no particular reason to... There's another particular term in Einstein's equations that looks like a zero for all intents and purposes. We can just forget about it. However, now this seems very good. This seems increasingly growing in the cosmological aspect. It's there and it's not zero. So we have, in a serious discussion of gravity, so let's start. We're trying to find the right conditions for sign. Where are you using the details of the equilibration process? That's not necessary. It's not necessary. All that you require is that equilibration happens on the length of time scale faster than the length of time scale of the radiation approach. Once that happens, there's one other place where we're in some sensitivity. You see, the next term in the theoretical expansion, let's say the viscocity term. Actually, with so much symmetry, most of these terms are zero. But anyway, in principle, there are other terms in the derivative function of the stress terms. Though the magnitude of those other terms depend on the details of the equilibration process. For instance, viscocity is proportionate. And the fact that we're keeping these terms but neglecting the others is effectively the statement that mean-free path is smaller than the length scale of the radiation approach. You see, there would be another term which is effectively mean-free path times derivative. Because we're assuming that the mean-free path is smaller compared to the mean-free path divided by length scale of radiation. So that is an expansion, therefore, it's a good expansion. We assume that this equilibration becomes an expansion. What is an expansion in the mean-free path? What is an expansion in the mean-free path? That's it. But actually, the kind of symmetry that we have most terms that you could write down on the right hand or the other side would be easier. Because there's a bulk viscocity that's not the same. So there are terms that you could write down. That would be non-zero. And we know. Okay, fine. So where are we? Yeah, so about the cosmology we're talking about. So we don't look down by this kind of action. You know, when we start looking at this kind of action, we don't look down to square root g, that's all. Plus some number, which I will not bother with now. Okay? But we discuss by writing this down. That there's another term that one could have written down. You remember our logic for writing down this action. Whereas if you get dimensional analysis, all the other terms, and you assume that the dimensionful scale in the problem was due at this point, all the other terms were basically, in terms of four and six and a half. So the derivatives were completely negligible. However, there was a term that was zero. That seemed to dominate the understanding just by dimensional analysis. So if you assume that the coefficient of this, this has two derivatives. This has zero derivatives. If you assume the coefficient of this, it's given that the dimension of the analysis is made up by Newton's constant here. This would be far dominant. If we invalidate flat space as a solution, we will approximately find set of equations. And observationally, of course, that's not true. So lambda, if it's there, is very near to zero on the scale of the mass associated with Newton's constant. But such a term is allowed by its lectures about other branches. There seems to be no particular reason not to have it there. Now Einstein originally wrote down this equation about this term, and he famously introduced such a term in order to try to have static units, as a sequence equation. And then he famously withdrew this term in print, calling it his greatest lambda here. After Hubble discovered the universe, he had a question. But you know, once the science is not an issue of opinion either of great mail, because something is allowed by the equations, the safe one says to put it there when you start solving equations, and then compare with experiment. So let's say that we start with an Einstein equation based on such an action. Now, what are we going to get as an equation? Let's start there in this action. When we vary this action, we vary... So let's say s is equal to g, then g alpha theta r alpha theta r minus alpha theta. Now, let's vary the equation with respect to g alpha beta. When we vary with respect to g alpha beta, we get r alpha beta. When we get minus r, then what it is again? G alpha. So that was the total depth. These equations, if you write lambda this way, Einstein's equations are replaced by instead of r alpha beta minus half r g alpha beta plus lambda g alpha beta is equal to... This is the effect of modification of Einstein's equations. Let me introduce this new term. No, no. This is r alpha beta minus half r g alpha beta is equal to a by k g alpha beta lambda g alpha beta. We can absorb this part out into the metastress tensor. If we associate the stress tensor, t tilde alpha beta is equal to 1 by 16 by k g alpha beta 2 to the power of beta. So if you take t goes to t plus t tilde, then Einstein's equations take the usual form. So you can think of Einstein's equations being modified only by imagining that there is a metastress tensor of this form. Like a side write plus a metastress tensor of this form that permeates the universe. Again. Effectively given a stress tensor. Exactly. This is... It sounds strange? Yeah, it just... It's just the zero point energy. You know, in a quantum field theory, the vacuum in a quantum field theory can be thought of as a bunch of harmonic oscillators. The zero point energy of all these harmonic oscillators summed up could leave you with the rest of the zero point energy. So it's not terribly strange. The thing that's really strange in quantum field theory language is why it's this... Yeah, right. But you know, maybe in some meta sense you think it's strange. Then you have to explain it. It looks like it's there. Now, let's look at this stress tensor a little bit more. Okay. So this stress tensor, yeah, please. Well, if this harmonic oscillator supplies oscillators, so it's the energy density of everything. But you see, the point is that it's there even in empty space. It's just the zero point energy. And of the graph. Of everything. You know, when you've got nothing around you, you've got no photon you can see or gravity you can see. If there's still some rest of your energy density. Where would it go? What would it decide? In the vacuum. Yeah, what would it have to decide in some way? No, no, no. It's just the vacuum has a non-zero degree. And she gives an explanation for that. Like there's no constitutional explanation. I mean, you know, it's that... What is it? What is constitutional? But I don't know, Casey. You could ask for a harmonic oscillator. What is the explanation for... For a harmonic oscillator, it doesn't seem all right. There's an interaction. In fact, it's just the uncertainty. And it's the energy of that stuff. So it's the energy of the vibrations of everything. You see, let's say you've got a photon. Each mode of the photon is a harmonic oscillator. And so, if you accept the mode by mode to understand where that comes from, this is just summing over all such modes. But I have to say, that, you know, it had long been expected that this would be zero. And it's really experimental that's driving us to seriously consider that smart. There are many caveats here, some of which have been explored. Okay, man. You know, a harmonic oscillator, one can see delta p delta, it's using an experimenter. You cannot see the same thing in the fields, the uncertainty of fields. No. What do you want to see using that? What do you see in a harmonic oscillator? You can observe the particle. You can observe that. You can estimate, I was going to say, you can estimate zero point energy from the high center. So, you know, zero point energy exists because of the uncertainty of things. Zero point energy must exist. Must exist, yes. Same thing for fields. Using an experimenter. Using an experimenter. The thing is that even for a harmonic oscillator, suppose you change, suppose somebody came and said, look, I don't believe there's zero point energy. The harmonic oscillator, energy of a harmonic oscillator starts at zero and then you've got to write. It's next straight is h bar omega. Next straight is 2 h bar omega. Show me an experimenter. Can you convince me that this is not the case? There's nothing. Because only energy difference is at a measure. In a sense, there's a spreader. That's right, that's theory. I don't think you measure the spreader. You can measure the spreader. But suppose he says, show me an experiment. Directly the measure of the energy is now zero. Again, it's a experimenter except when gravity couples to energy. It doesn't come to the energy differences. It comes to energy. Because of that, the zero point energy is made sure. Due to the response of gravity. So it's a subtle thing. It's a subtle thing. The only way we're seeing it is of energy. Is in how gravity couples to the spreader. Yes. No, no, no. You see, it's proportional to gravity. The spreader stands as proportional to the metric. And eta. So let's say I have an enzyme free. And I have to measure the energy density. Yes. In other words, my G-alphabet also. Yes. Say you're in class 1. You measure the g0. The point is that it's not only energy. It's also pressure. It's not only g0. That's also Ti. Ita is the spreader of the beta. You boost. How can I say when I say it's eta-alphabet, I'm at the alpha beta side. What? Theta-alphabet is proportional to the metric. If you're metric, that point happens to be there. Then my beta-alphabet is also proportional. It's proportional. Yes. You're going to boost the chain and get the same answer. Accelerate the chain. I mean, this is the answer. The alpha beta is the answer. Whatever coordinates you use, this is the answer. Can you guess? Sir, do we have an absolute zero of energy in classical special relativity? For example, if we have a particle moving, then the expression for energy is E squared equals to P squared plus M squared. Now, if I were to redefine E by E equal to square root of P squared plus M squared, plus an iterative constant, then the four-vector then the combination E comma three-vector P will perhaps not form a four-vector. Yes, you're right. But what you can do, you see, that's because you want the what you're trying to do is to change the energy of a particle compared to it's not being there. That is in the end also an energy difference. Imagine the particle. Imagine it not being there. That's completely well defined. But, suppose you're trying to change the energy of the vacuum of space. There's nothing that can be distinguished there. You see, any time you could imagine an experiment in which you you've taken energy difference, that's always there. But there's always going to be one ambiguity. Because the energy of that there being nothing is simply the arbitrary. On top of that, you know, you start with some arbitrary zero and then you will look. But that's arbitrary zero will always be arbitrary. And in general, relatively, that arbitraryness is also a mystery. Because coverage of space time reacts to the energy density not doing some difference to some absolute notion of energy density. That sits on the right-hand side and the left-hand side will have either a positive or a negative sign. Now, the way we've written it I should have remembered t-alpha-beta is equal to g-alpha-beta times 16 by k. Now, there's the formula to use the following terminology. The lambda which gives rise to stress with positive pressure so that that's the thing that we call as acceleration theory. It's called a positive positive. Lambda, which gives rise to a stress tensor with negative pressure spools things because it's deceleration theory. It's called a negative component. Now, it's important to know that it's the pressure that determines whether this thing is positive. Because the pressure and the energy density have opposites. Okay? So the positive cosmologically constant comes with a negative energy density and the negative cosmologically constant comes with a positive energy density. Okay? So such a term is a possibility in the United States, equations. It's called a cosmologically constant. Einstein had to introduce this term because he realized that a positive cosmologically constant could lead to positive pressure. That could balance the flow of gravity and lead to a static universe. Okay? Something he didn't realize at that time was that the static universe is unstable. It wouldn't drain its flow. And since then you've seen that the universe is not static so that's not a good motivation for introducing this term. But the good motivation for introducing this term is that it's possible. It's there consistent with all the principles of our theory. So maybe it exists. Okay? So there are three effective forms of matter of stress tensor that we've discussed. The stress tensor that comes from the constant. The stress tensor that comes from dust. Dust. The stress tensor that comes from photons. Okay? And on the right hand side there are equations. You could imagine all three equations. And we'll allow the country of oxygen. Yes? I mean in the end it's a matter of words. So I would point that I'm not supposed to say that it's not something we would do. In the end it's a matter of words. So I would point that I'm not supposed to say that it's not something we would do. In the end it's a matter of words. What we are doing is solving an equation. It's so simple. So in some sense I mean the curvature. If there is an equation with the term constant times g alphabet. You might want to think of it as the as the energy of the vacuum. Because every with you it's basically a matter of words. But suppose vacuum did have an energy density. Then it would give rise to such a term. Okay? Basically what you want in the end is a theory of this term. You want your fundamental theory of particle physics to predict the right equation with the right value of time. Is there any interpretation about a constant by doing that? Well we don't even know for sure that there is such a term. Because what we know is that there is an acceleration of the universe. But there are many other phases because of that. No I don't think You see again energy of the vacuum is not important. What's important is the one point function of the vacuum. What this is is a source of the vacuum. So your theory whatever it is should predict that in the vacuum there is a one point function of the vacuum. So it's not quite the vacuum. And if you assume that okay? The equation of that is essentially the same as the calculation of the vacuum. When you take this one point function of k equals 0 and i equals 0 it's essentially the calculation of the vacuum. So the same calculation you do to calculate this thing is the calculation you do with the vacuum. But in the end you are completely right. Many people don't need to give these words to this term. What we want to do is to get the right equation for that. It seems like there is such a term in the correct equation of the universe. And the outstanding part of the question is how do you produce such a term? Starting from all of them. The rest is a bit of a fancy thing. Okay, we are almost out of time but let me just repeat it. So now we understand what kinds of terms we are going to be willing to put on the right hand side of hindsight. Now with this understanding in place we want to go ahead and solve Einstein's question. Firstly, this stuff has been abstracted. What value of UU and UU are we going to give? Are we going to give the matter fields? Well, it's pretty clear to me if these things have to have the symmetries of the problem they have to be addressed. Because otherwise there will be a vector field, namely the velocity vector field in special directions which would create the symmetries of the problem. Okay? So in our problem what we are going to do is to set UU 0 as the only non-zero value and then U0 squared will have to be equal to 1. So U0 so in our problem what we are going to do is we are going to do E plus B the coefficient of G0 0 is minus GII which is B. So what we said for U is irrelevant because this is not irrelevant. So what we are going to do is to set U0 U0 E is 0 0 equals Now at this term we have to be such that we want in our cycle we mentioned U U U G U U Is it 1 or how you just want to gloss because the time comes Okay, fine. So what we are going to have then is that U0 U0 will be equal to G0 0 then we will have G0 0 up over G1. Okay? So G0 0 will be equal to epsilon times G0 0 GII will be equal to B times minus B times GII I0 GIJs So I understood the equation of state relating epsilon and B So by the way, what is the equation of state for the equation of state is used for B. Do you have some understanding of the equation of state is? We are going to now plot this equation to understand the equations and compute what we did. We will have to do that in this class. So in the next class we find the equations of motion of cosmology equations that we determined both epsilon is a function of time as well as A is a function of time in cosmology and then we start trying to determine the coefficients. Okay, what have you decided? I mean, I said one extra class, right? Do you remember what day? What? Tuesday and was it the afternoon? What is the afternoon? Tuesday afternoon, okay? So Tuesday, it's a tomorrow? We tried. In fact, we found it.