 Today's class, we're going to, you remember we started our discussion, okay, today's class is going to be a brief class, but unfortunately it's week 15 and I have to go somewhere, okay, but anyway, we're going to six o'clock. And the other thing is, next week, all of next week, I apologize for this, there won't be no class. For this reason, I'm going to propose we have three classes of you for the next few weeks, okay. So before we do anything else, let's fix some of the time we have left for the class. So part of Monday's and Thursday's, I need one more slot. Regular basis, can you suggest one? Is this slot okay? Tuesday. Tuesday? Tuesday. Tuesday? Tuesday. Tuesday or Wednesday? Tuesday or Wednesday? Tuesday. I'll just check which of the days, if I need the classes. Okay, Tuesday is fine. So shall we fix that? Monday's, Tuesday's and Thursday's. At least four weeks. Yes, but this is four weeks left. If you don't mind, we just take three classes. Okay, so we'll fix our Tuesdays to 30. Same class. Monday, Tuesday, Thursday. Okay, so we've begun a discussion of cosmology. We discussed that our universe, we're going to make a model of our universe according to the following metric. And we talked about the idea that the metric in the three-dimensional space, the metric in the three-dimensional space was going to be special. Today I want to make that notion of something being special in the model. So, firstly, let me make a couple of definitions. The model has seen before that definition of isometry is a coordinate change, such that the metric in the new coordinates takes the same functional form as the metric in the new. Okay, so let's look at an example. X i is equal to r i j x j plus e to the metric here. This is an orthogonal rotation. We'll get r r cross goes, r r cross goes as one. So you get d x i prime d x i a mixed matrix. This is the full set of isometry isometries. Isometries are defined as finite code. But as usual in the study of symmetries, as you probably can probably see in the video, it's much easier to deal with infinitesimal things than the finite. So an infinitesimal isometry is an infinitesimal coordinate transformation. That does not change the metric. But we can write down a simple equation for what's in the form of an infinitesimal isometry. An infinitesimal isometry is patterned by a vector field that changes under the vector field. We know that del is by 3 mu nu. This equals del mu sin tan mu. So you know by the definition of an isometry, the metric should not change. Okay, so under an infinitesimal, you know, a vector field patterned by an infinitesimal isometry, if at all there is del mu sin tan mu plus del mu sin tan mu, isn't it? A vector field that satisfies this equation is called a killing field. And parametrize is infinitesimal isometry. We work with infinitesimals because those are convenient. Well, of course, infinitesimals can be explanations. You can make the coordinate change smaller than another to generate the finite coordinate change. It's just you know lack of information. No. If they are the general matrix, there's no isometry. There are no coordinate transformations that leave its initial form unchanged. And no killing vectors. There's only special space values that admit killing vectors or admit isometry. Okay. This is the isometry. Are these the words that could be defined all over the history? This is our definition. Yes. They generalize these with a partial isometry. What happens in the rest of it? You know, if it may be useful for you, if you're only interested in physics, that's it. But this is not our definition. Okay. Now. I solute the equation looking for solutions of theta. You could think of it that way. If you want to know whether the space time has any isometry, you can look at this equation and check to see if there are any questions. That in general, there are no solutions. Or are there multiple solutions? Could have multiple solutions. But if you arbitrate, I hope there are no solutions. Okay. Why am I start talking about isometrities and the killing vectors? I start talking about that because I now want to tell you more precisely, I mean, by the statement that the space time is homogeneous and etc. Okay. So, let's consider a little batch around the ideal point. What does it intuitively mean that space is isotropic at least about that point? It means that all that if you set up a little local coordinate axis, it doesn't matter what, how you orient it. What does it doesn't matter? I mean, it means that rotation of your loading local coordinate axis can be generated by symmetry. What are the symmetries of our theories? The symmetries of our theories are the isometrics. The field in the theory is the metric. The metric doesn't change. Okay. So, the space is isotropic. It's said to me, isotropic is what I need to find. If all rotations of the local frame at that point are generated by the symmetries. How many parameters are there? Suppose we're in d-spatial dimensions. d-spatial dimensions. Since I don't want this cosmologically possible, context and problems, the spatial matrix. Okay. Suppose we're in d-spatial dimensions. If the a10 is even one, if the space is isotropic of only one point, there must be a certain minimum number of k. What's the number? p, p-1 by q, because that is the number of rotations. So, if the space is isotropic around a given point, then we don't know what number of k is in this. Next question. What does it mean for the space to be homogenous? What means for the space to be homogenous is that in the local fashion and by extension everywhere. There is, okay, so I should speak more precisely about what I said about the isometrics. But there is a killing vector that vanishes at the point. So, what are the isometrics? Isometries are, you know, everything to rotate but not for the point to move. So, we would like killing vectors that vanish at that point, but whose derivatives, you know, in the neighborhood of that point, generate all rotations. And so, the example of this is rotations in that space. Okay. So, of course, we had the origin of that space is rotation in that, but the killing vectors that generate these rotations are x mu and mu minus x. They vanish at the point, but they rotate things in the neighborhood. But this is what I mean by a space being isotropic at a given point. And if it's isotropic at that point, then it has at least p into p minus 1 by 2 killing vectors. Well, I mean by homogeneity. Well, I mean by homogeneity is that there is a killing vector that takes you from a given, from any point to any other point. Infinite, infinite, infinite. They would have that by extension of that space. Okay. So, there is a coordinate change. So, it takes one point, then another. The tropic can be defined about a point. Homogeneity is a, if a manifold is homogenous, it has at least p killing vectors. So, that's the dimension of the matter for me. I'm going to quote some results for you without just proving them. But these results are very nicely explained in Weinberg's book on the chapter of Symmetric Spaces. I'm not going to go through the proofs basically because we're running out of time. A killing vector field need not vanish at a point, at least. But if a killing vector is an isometry, then it does. That's part of a definition. So, an isometry is like a local rotation. So, the translational killing vector doesn't vanish. First result. Okay. Okay. More than p into p plus 1 by 2 killing vectors. And the way you prove this result is by showing that suppose you can kill vectors, they come in. Then, from using the killing equation, you can prove that del sigma del rho of zeta mu is equal to minus r lambda. This is a general identity. You can prove the identity. Just using actually. I will try to prove. I will try to prove. I will prove for you. But it's a general identity very nicely explained in Weinberg's book. Okay. Why is this important to me? This is important because, look, what we've done with this equation is express two derivatives of the killing vector in terms of the killing vector. So, how do we take this equation to differentiate? So, we'll get an expression which is called three derivatives of the killing vector in terms of derivatives of the derivative and derivatives of the killing vector. But we've already expressed two derivatives of the killing vector in terms of the killing vector. So, what this means is that once you know that the killing vector itself and its first derivative, then you know what the sum of it is. That's the big kind of the killing vector field is contained entirely in the value of the killing vector and the value of its first derivative at a point. At one point, that determines the whole vector field. You can construct the whole field by telling us is this clear? Okay, how are we going to do a few more lines? So, this is, you can show this in a simple idea till you get its consequences. Okay? So, that tells you that the number of possible killing vectors let's look at a point. The number of possible killing vectors is entirely determined by the number of pieces of data in the vector field at that point. And it's derivatives. Moro's derivatives are all free because they represent this equation. So, now can you somebody help me do the counting? How many pieces of data are there because the value of the derivative at a point in this equation? How many pieces of data are derivative of a field at a point? Wait, which question? Suppose we didn't impose this condition, how many pieces of data are derivative of that at a point? Peace. Peace, quiet. Now, with this condition, how many? P and 2? Now, there's something in this constant, something by 2, what's this? You're moving the symmetric part. You're counting all the answers in vector field. That's P minus 1 vector. This is the number of pieces of data at the derivative point. There's also data is the value of the killing vector. How many pieces of data are in the value of the killing vector at the point? P. This is the maximum number of killing vectors killing vector fields are a matter for data because these are the independent pieces of data which completely reconstruct the killing vector field. The maximum number of killing vector fields you can have in a manifold is P into P plus 1 by 2. Now, let's look at this P into P plus 1 by 2. It happens to be the number that you get by adding P into P minus 1 by 2, local rotations. And P, the local translations. So, from just this counting numerology, it might suggest to you that, and it's true, that a space type or a space that is both isotropic and homogenous. See, it's isotropic of a one point and homogenous that is isotropic of every point. The homogenous is the vector that goes one every point to every other. So, that space type that is isotropic of one point and homogenous manifold has the maximum number of killing vectors, possibly, which is P into P plus 1 by 2. And then there's the name of what it's called, it's maximally symmetric space. Such a space is also isotropic of every point. Another bit of algebra you can do to show, again, nicely given by the worksbook, is that if a space is isotropic about every point, then it must also be homogenous. So, all these are the same thing. So, when I talk about maximally symmetric space, there are many characterizations of it, all of which I wrote. A, that it has P into P plus 1 by 2 killing vectors. B, that it is isotropic of every one point and homogenous. See, that is isotropic of every point. All of these three completely say the same thing and define maximally symmetric spaces or homogenous and isotropic spaces. So, all that I've said now, I've now explained the algebra that I could have in a total of 15 minutes. Then the result in that way, it takes a little more time. It's a unique landscape for maximally symmetric space. See, suppose you've got some symmetric space, you can of course take it at this scale, take the metric and multiply it and then overall do the problem. That will also be maximally symmetric. That's a trivial question. Clearly, if symmetric spaces are, you've got one symmetric space, you can generate one binary because I have symmetric spaces by scale. Now, the scale will change the coverage. It's something that is strongly covered because it's small, but completely covered because it's big. So, symmetric spaces can't be completely unique because there's the scale coverage. It's like for the scale. Big, small, small. However, you might wonder why the symmetric spaces are unique up to this scale. The answer is yes. So, more precisely, given the signature of the space time, everything at the top of the space, but everything at the space time, given the signature of the space time and the value of its Richie scale, Richie coverage, which is a measure of how big it is, the scale of the space, the size. By the way, can somebody tell me if I scale the metric, G goes to lambda, G. How is the Richie scale? It's different. G goes to lambda, G. That same distance goes to lambda. The radius goes to it. So, G is a radius square. Let me argue. The coverage of, the coverage of has terms that go like gamma, gamma. Remember, gamma was like G first, DG. So, gamma and coverage also, things like gamma. So, gamma and radius. So, our new loop scales like gamma because it's gamma, that's gamma. And this is independent of the overall scale of the metric. You take G to lambda, G. Our new loop distance won't change. But, our new loop is equal to G first. Our new loop. So, our scales like G first. So, you multiply metric by lambda, the Richie scale is equal to what? Big space and small coverage. It's interesting, right? Okay. So, So, So, just get back. Yeah. So, the uniqueness there. The uniqueness there is that, given the value of the Richie scale, that's just setting the scale. And given the signature of space back, there's a unique, there is a unique, completely symmetric. Okay. This takes a little more proving because it's also nicely proven. All of this information. Okay. It is very easy to construct the metric. All maximum spaces. And I'm not going to, but actually the metric space that we constructed of already last class was history. How did you know there was maximum space? Well, it's intuitively obvious that it's homogeneous and isotropic. But, can you count the number of killing vectors in S3? What is the set of coordinate changes that leaves the metric? S3 can change. Think of S3 being better. Rotations. Four dimensional rotations. Four dimensional rotations. How many four dimensional rotations are there? Six. Six. What is three into three plus one by two? Six. So, S3 has the number of isometrics in S3. It's six. Therefore, it's a maximally symmetric space. So, we're searching for a space of homogeneous space time. So, what we need to do is to search for spaces of space times as many isometrics as we can have. So, now the example S3 has given us has shown us how to think. So, but firstly, all equals zero clearly the unique manifold is flat space. Flat space has been to be plus one isometric, obviously. Speed of P max one by two rotations plus P translations. So, flat space is maximally symmetric. The sphere of maximum is symmetric. Now, let me try to find other examples of metaphors that are maximally symmetric. Now, this matter this is close to our output this is sort of what I said. Suppose I you can do this matter. Suppose I look at the manifold minus x zero squared plus x one squared plus x two squared is equal to what you suggested last time minus x one is equal to minus x two. Let's think of this as embedded in our one column. One time like language. In like ordinary processes. As a manifold, the manifold is left embedded by all the range transformation. Because the expression is the range. So, every range transformation match once in this manifold plot the points on the same manifold. You don't leave the cycles out of this by the range. So, now, where is the number of ison entries of this manifold? Six. Because there are six in range. Now, let's think more about this manifold. What's the signature of this manifold? It's something that we have to understand. Firstly, what is the dimension of this manifold? Three dimensions. Now, what is the signature? Public use. You see, we want to know what is this good set of coordinates of this manifold. You can see it by doing equations. You see it by by the following. This equation can be read as x zero squared is equal to e squared plus x one squared plus two squared plus x two squared. So, at every x one, x one, x three, we can solve for x zero. We can solve for x one, x three. So, we can think of the space being parameterized by x one, x two, x three. So, the x one, x one, x two, x three are spatial dimensions. A spatial. Perhaps this gives you some proof. The reason for believing that that this thing will be the signature of the three. I'm going to show it. By the way, this is not the direction. Suppose you try to solve for e x one squared. If you solve for x one squared, you will get is equal to x zero squared minus x squared minus x two squared minus x three squared. And then there will be some values of x two and x three which are large enough. So, that you can solve for x one squared. So, while x one, x two, x three are good coordinates of the mathematical neighborhood. If you try to choose some other set, that would not be good. Again, I guess I'll place an R three, R four. In the last section, we will place a plus plus plus because then you expect that it will always be positive. Any distance in the distance? Because I don't think any distance on R four is R one comma three. Suppose I take this hypersurface in one comma three and look at this. Because it is solving it in R five. It is solving this equation in R five. In R one comma three. We will see exactly what calculation we need to do to check this. But it's not true that every hypersurface will be of the signature. Suppose we think the hypersurface x three equals zero, that would be two plus one direction because this which has signature minus plus minus. So it depends on the hypersurface. You know what I'm saying is very intuitive. There's time and there's space. If you said t equals zero for instance, you've got a spatial hypersurface. If you said x equals zero, you've got a hypersurface that includes that. This is some funny combination of the two. The solutions in alpha x one, x zero, x one, x two, x three. Besides that, the sequence is a bit an R four. Firstly, we now work in R one comma three thing. It's based on the metric of the embedding space there. It's minus, it's dT squared minus x one squared minus x two squared. Okay? Yes. Now I've written down a particular equation for hypersurface. There's a question. Is this hypersurface of signature plus plus plus? Or is this hypersurface of signature minus plus plus? Depending on the equation in question, the answer will be different. So it's not an empty question. What is the signature? Signature means, are you dealing with a space or a space time? Now we'll take the R four to the half more place over the R What? If there is a minus signature, then it's a space time. Yes? They won't all be x zero equal to constant. No, it's just what is the signature of the metric? Yes, that's what I asked. It's not some temporal equation. It's not some x zero equal to constant. So it's not completely space time. Are you making up this? Others? It's just, it will be some space in which there is some action. And are all directions in which you go to space directions or some of the time directions? Physically, is there some measure of when signature is not in place? It's not constant. How do I find out what the signature is? Okay, so let's do it just a little bit here. You do it by computing the metric of the sectors. Basically, you just take it there. But there are many ways. But let's just do it by computing the metric. Okay, in this particular case. So now, in this case, the coordinate system that we use for the sphere is now, the first coordinate system we use because in this case, it does not have a so-called symmetry. Basically, I have a problem, but there is a 2C. I was supposed to write x1 is equal to r sine theta cos theta. x2 is equal to r sine theta. x3 is equal to r sine theta cos theta. And x4 is equal to a squared plus r squared. This is very natural coordinate system. Let's compute what the metric is. So the original metric was dT squared minus dx1 squared plus dx2 squared plus dx3 squared dx0 squared and it's pluggy. So, I'm just following the calculation we did last class. From here, we get r squared d omega 2 squared. Last we also get dr squared dr squared exactly as we got first time. So from here, we're going to get plus dr divided by squared root of a squared plus r squared r combine these two terms together we get minus r squared d omega 2 squared minus a squared dr squared over a squared. So this was minus but let's see what's the positive and positive and if physically we see a ego on this and always have a positive then it should be a positive it's simpler than that, start on a point move in every direction let's see if it's positive or positive so in some sense by introducing nonpositive definite matrix We are introducing some additional structure on the hypo-surface. Well, you know what we are doing is looking at a hypo-surface in an embedded space which was not possible. This is not possible. We are pretty much by virtue that we have this additional structure on the hypo-surface. Additional possibility. Possibility, namely the signature of the hypo-surface. Yes. Is this here? Okay. So, this space has a name. Okay. This space is called Euclidean Antidecitus Space. Euclidean Adesis. Okay. It's a space as we mutilated last class, just like the sphere has a space of constant positive curvature. This is a space of constant negative curvature. Because, you know, all the formula is following. This is the curvature of the sphere. The rigidity of the sphere was 1 by 8 square. And all the formula is for the space which is lower than the rimless space by 2.6 square. To the curvature of the sphere, the rigidity of the sphere would be minus. So, this is Euclidean Adesis space, the space of constant negative curvature. It's a maximally symmetric space because it's a minimized space. It's 6. It's an R4 version of a hypo-surface. That's right. It's an R4 version of a hypo-surface. Yes. But you know what we do? Cleverly, it's a hyperbola rarely in R1, overall it's a spacelink. We do it cleverly so as to get such a large amount of symmetry. You wouldn't get such a large amount of symmetry for a hyperbola in R1. The linear manner holds a symmetric space, it's a completely symmetric space itself. Signatures with a space with a signature plus plus plus. That's completely classical by the energy scale. When we produce the manifold with every value in which it's set. So it's completely produced all symmetry, the Euclidean symmetric spaces. So they are the sphere of every radius, Euclidean area has a boundary, and it is the spacelink of every radius. And plus this. These are all three dimensions and the proof would be proof of every dimension. These are all Euclidean maximally symmetric spaces. Is this clear? Since we've all done this exercise, now there's a little exercise in which one could do. And one could ask the following question. Where would we have for minus x 0 square, minus x 1 square, minus x 2 square, minus x 3 square, minus x 0 square, minus x 2 square in R2R. Because this is a hyperbola, this is R2R. This also has six killing vectors. It's SO 2 comma 2. Same number as XO4. Okay? What is the signature of this space? What are the two possibilities? Could be one type of two-space or two types of one-space? Where is it? One type of two-space. Same kind of ideal. Just solve for x 0 square plus x 1 square. What are the time dimensions? What are the space dimensions? This space is what's called anti-discipline space. Okay? This space was called completely anti-discipline space. This space is called anti-iscipline. One of the spaces of other maximally symmetric spaces with signature one type of two-space. There's another maximally symmetric space with signature one type of two-space, which is called DC to space. That space is defined by the equation and is similar to this. We will come to the study of DC just when we start on the word space times with the conglotary constants. So I'll postpone the discussion of that. But just to see from this general construction, that it is very easy given a signature, given any signature, it's very easy to produce maximally symmetric spaces. Just by starting from such type of surfaces in RM comma H. Basically, this idea completely classifies, gives you a construction over all the symmetric spaces in a symmetric space. Anti-discipline space is of no normal importance in the real world. But it's very important in this kind of study of strip theory. You could, just mathematically though, you could produce spaces like this of signature M comma N. By starting with something that's signature M comma N plus one. Yes, sir? Sir, let's see if we can do some free theory on the strip theory. It's possible that boundary conditions put on one surface of one type of signature in the other side. Are you working on a hypersite or the whole space time? You're working on the whole space time. The whole space time is this hypersite or just R in problem? No, no, it's the... No sir, let's say I'm working on a solution of IH times. Sir, this is a different question. It's not separate, it's a related question mainly of particular IH times. Our initial conditions would be on some IH spaces. Yes. Which would be space time. Yes. I'm asking, can we solve, ideally, on one space time IH and other spaces time? You're asking, can you put initial conditions on hyperservices which have a time? It's going to be normal euphesies. It's going to be called an initial equation. Still at the mathematical level, you can ask. You see, the equation del squared equals here is solved. You want to know what is sufficient data for solving del squared. Alright, the answer is in this paper. It's focused on... Moving where? After it was conditions on the time like that. Yes. You see, if you have a bump, it's just a box. The wall of a box has a time lag. So it's certainly permissible to impose. It's permissible to impose. Absolutely. It's certainly permissible to impose. But I thought Harshan's question was, in our format set, could you, you know, what surfaces can you put initial data? Okay, whatever it is, let's postpone it after class. I'm going to stop the class and do it. Fine. Okay, great. So, now... Okay, I think maybe we should just actually stop. So, let's see. So, where have we got? Okay, understand our maximally symmetric spaces. These will be the building blocks we'll just focus on the gene. There's something I wanted to say. So, okay, so let's get the difficulty. Sorry. I'm done. I'm done. Okay.