 Fine, so let's continue our discussion. Let me start with a two or three minute review of what we talked about last, in the last lecture yesterday. You'll remember that yesterday, motivated by two or three strands of motivation, the first being an attempt to try to find an analytic approximation, or, you know, to find some useful analytic approximation to highly nonlinear behavior of black holes. The second being to make sense of this, to make precise this feeling that has been expressed over the last 50 years or so, that there is a sense in which the physics of black holes reduces to the physics of a membrane that lives at or near the event horizon of the black hole. And the third being to see to what extent one can, in any context, do something like ADS-CFT for Schwarzschild black holes. Okay, the second and third are more or less the same motivation, okay? We decided to study the dynamics of black holes in a large number of dimensions, okay? We followed Emperor Suzuki and Tanabe to explore the metric of a black hole in a large number of dimensions and realized that this metric was characterized by two lens scales. There was R naught, the size of the black hole, and R naught by D, the thickness of the black hole. And the important thing was that in the large D limit, these two lens scales were very, there's a hierarchy of scales between these two lens scales. The thickness is much smaller than the size. We then reviewed the computation of Emperor Suzuki and Tanabe of the quasi-normal modes of these black holes. And we saw intuitively how you might expect that while most modes, most quasi-normal modes for this black hole should have frequencies of order one over the thickness, which is D by R naught, it was possible, depending on the details of the problem, that some modes would have lower frequencies of order one over R naught. We saw from our intuitive motivation that at any given angular momentum, you should expect only a finite number of modes to be light, whereas you should expect an infinite number of modes, perhaps zero to be light, whereas you should expect an infinite number of modes to be heavy. The question, the interesting question was, did these light modes exist? Emperor Suzuki and Tanabe did explicit computations, found the spectrum of these light modes, and you'll remember we quoted their frequencies for scalar spherical harmonics, the frequencies where R naught times omega was plus minus square root of L minus one plus I times L minus one. And for the vector spherical harmonics, R naught times omega was I times L minus one. The sign of our frequency, we were looking at modes that went like e to the power i omega t, so positive imaginary parts corresponded to decay. We may contact with these formulas, perhaps in this lecture, once again. So we, given the fact that there were these light modes separated by a huge mass gap from an infinite number of very heavy modes, we decided to try to see if we could find an effective non-linear theory of the light modes. As we emphasized in the last lecture, this happens very commonly in physics. For instance, in string theory. In string theory, we have a few massless particles separated by a gap from very many massive particles. If you were just looking at the spectrum of the theory, you would see linearized massless particles, linearized massive particles. Now, because of the gap, you could ask the question, what is the effective non-linear theory of the light modes? That question, of course, has been asked in many ways in string theory and has a beautiful answer, Einstein's equations. So the effective theory of the non-linear theory of the light modes, Einstein's equations, perhaps in type two super-gravity, type two string theory, souped up to give you the equation of super-gravity. But because of the gap, it was consistent to talk of a non-linear theory of these light modes, ignoring the heavy modes. If you ask, what is the effective non-linear theory of the light, the massless modes of string theory, and in the first excited state, there would be no answer to that question. Because non-linear effects would bang to the first excited states to produce the second excited state, okay? So that's not a consistent question in itself. But there is a consistent question about, what is the effective non-linear theory of the light modes of string theory, the massless modes of string theory, because of the mass gap? In a very similar way, we're asking the question, what is the effective non-linear theory of the light modes of the light quasi-normal modes of the large D black hole, okay? That's the question I'm going to try to address, at least in part of this lecture. Now, at the end of the last lecture, I told you that it will be useful in addressing this question to keep in mind an analogy, namely, a similar question that has been asked and successfully answered in the past. And this question, the answer to this question goes by the name of the fluid gravity map of ADSEFT. At the end of the last lecture, I reviewed that, but I think I went too fast for you to really understand. So I'll spend another 10 minutes this lecture, just going through that review more slowly, just so that you keep the analogy carefully in your mind, who are for it. You remember that when we would discuss the analogy, we switched gears to considering Einstein's equations with a negative cosmological constant rather than the vacuum Einstein equations. This is just for the analogy. Nowhere else in the talk will there be a negative cosmological constant. I don't want you to be confused about that. Just for the analogy we did there. And we switched gears looking at four-dimensional gravity. We can do it in arbitrary dimensions, actually, but I'm emphasizing we're doing four-dimensional, five-dimensional gravity, five-dimensional gravity, just so that you see that for what I'm saying here, large D is not important, okay? And we discussed that while ADS space was one of the solutions to the equations, we discussed that there was also another interesting solution, which was this black braid, which had the dual field theory interpretation of the thermal equilibrium, okay? Now, something I didn't tell you about in the last class because we were so rushed is this, that before you start doing anything fancy, the first thing that you might do is to try to linearize Einstein's equations around the black braid. Okay? You linearize Einstein's equations around the black braid more precisely find the quasi-normal mode spectrum around the black braid. This historically was what was first done, it was done in the early 2000s. Okay? The quasi-normal mode problem in ADS is the problem of linearizing Einstein's equations subject to the following boundary conditions. Ingoing of the horizon, that's always what you do for quasi-normal modes for the horizon, but normalizable at infinity. And you remember we discussed why? The boundary conditions always should be that you're not smashing the black hole, you know? Doing something to it. In flat space, doing something to it is sending something in. In ADS also, doing something to it is sending something in. But the way you send something in in ADS is by turning on non-normalizable modes of the boundary. So you turn them off. Okay? This, the computation of the quasi-normal modes was what started this whole thing off by Soroka, Sohn, Coptun collaborators. You know, they computed these quasi-normal modes and very much like in our black hole problem, they found two sets of modes. The first class of modes, and there are infinitely many of these. Okay, so, sorry, I should back up. In our black hole problem, we first used symmetry to say all modes transform in spherical harmonics on the sphere. In the black brain, there's an easier symmetry than spherical symmetry. It's just translational invariance. Okay? So all your modes will be taken to be e to the power i omega t plus e to the power i kx. And then everything non-trivial is functions of the radial direction, this radial direction in ADS. It's all modes that assume to have this, this time and space dependence. Okay, this is k dot x. Is that clear? The problem of quasi-normal modes goes as follows. A, around the black hole, the problem was for every angular momentum, what is the set of integers? What is the set of discrete set of omegas you get for quasi-normal modes? Here the problem becomes very similar for every k, for every momentum in the spatial direction. What's the discrete set of quasi-normal modes that you get when you solve the problem? Okay? Once again here, there are two kinds of modes. There's an infinite number of heavy modes and a finite number at each k of light modes. The heavy modes had the property that omega sum function of k, but when you take k to zero, so you go to the long wavelength limit, okay? That function of k goes to some pure number times the temperature. That's the only thing it could go to in this problem because there's only one mass scale in the problem, namely the temperature, okay? And the important thing is that for the heavy modes, this number is non-zero. In fact, as you go to heavier and heavier modes, it grows. But all that we care about is that for the heavy modes, it's non-zero. However, there was some special modes which had the properties that the quasi-normal mode frequencies went to zero as k went to zero. You went to the long wavelength limit, these quasi-normal modes frequencies went to zero. Okay? And in fact, when you worked out the quasi-normal mode problem carefully, you found analogs to those formulas. You found omega was plus minus k by square root three plus i times a number times k squared by t. So two modes like this. And also omega, some other numbers, a number prime times i k squared by t. Actually, it's a degeneracy, but there are two modes like this. Okay? So you found four, you found four light modes. What do I mean by light? By light, all I mean is that as k went to zero, omega went to zero. These are zero modes. These have been normal modes rather than quasi-normal modes. Had there been no imaginary part? These would have been genuine light excitations which you might have tried to find an effective action for. Okay? But in this problem, things are dissipative. That's what makes these problems. And also that problem, the one we will come back to most of this lecture, different from integrating out the massive mode and finding an action, more difficult as well as more interesting. Okay? Now, so what we see here is that we have these black brains and we have a set of light modes. These modes are much lighter than the heavy modes if we go to the long wavelength limit because then k is much, much smaller than t. So in that limit, in the long wavelength limit, you might ask, reasonably ask the question, what is the effective nonlinear theory of these light modes? The light modes exist. So they should exist some effective nonlinear. Okay, that was the question. That was the analogy. Now, the four modes are omega is plus minus k by square root two, that's two. And these modes, but the same frequency come with a degeneracy of two. These turn out include dynamics to be the sound modes and the shear modes. And the degeneracy is just that there are two directions in which to shear. The two transverse directions to the wave. This is in four dimensions. If we were doing fluid dynamics in D dimension, that would be a D minus two degeneracy of that. Exactly, it's just the collective coordinates, the goldstone modes. Exactly. So in order to try to understand that, as I reviewed in the last class, what we did was to first take this black brain and rewrite it in a different set of coordinates, these Eddington-Finklstein-Ingoing coordinates and also looked at the exact solution of the black brains with a boost. So this metric here with U constant and T constant was an exact solution of Einstein's equations with a negative cosmological constant. In fact, large diffume-orphism equivalent to the original brain, done nothing. But then we looked at metrics in which we took the same form of the metric, but allowed U and T to vary slowly. The idea being that surely since the solution is, this thing is a solution when U and T are constant, if it varies very slowly, it should be near to a solution. Now, I wanna emphasize one thing. It was very important we did this in the right coordinates. And that's gonna have an analog in the problem that we'll solve in five minutes, or we'll start looking at in five minutes. And the reason it was very important that we worked in the right coordinates, one of the reasons is as follows, goes as follows. You see, black holes are regular solutions at the event horizon. But you wouldn't know if you looked at that. If you looked at a black hole, it looks irregular at the event horizon. Einstein thought Schwarzschild's solutions had a singularity of the event horizon. Einstein was no idiot. It looks like it has a singularity, but it doesn't, you have to work to see that it doesn't. But it's a slightly dangerous game, because something that looks like it has a singularity, but doesn't, might turn out to have a true singularity if you change something a little bit. And what we're interested in is changing the solution a little bit. Go on, wanna work in coordinates where a small change may turn a coordinate singularity into a genuine singularity. On the other hand, these Eddington-Winkelstein type coordinates, okay? A manifestly non-singular at the horizon. And therefore, remain non-singular if you make some small change. There are more and more motivations, but since this is not an electric and fluid gravity, I won't discuss those motivations in great detail. But what was very important was that we did this in the right coordinates, okay? Trying to implement the slowly varying thing in different coordinates that are related by singular coordinate transform can give you different starting points, okay? And then, as I reviewed in the last lecture, what we did was that then we could show that this was the first term in a systematic expansion of solutions of Einstein's equations. You have to correct this. But it was possible to correct it to solve Einstein's equations, not just a leading order in wavelength in units of temperature, sorry, of derivatives in units of temperature. But first, for some leading order, provided the starting point was not completely, was not completely arbitrary, but obeyed some equations of motion. Equations of motion were the equations of conservation of a stress tensor. And the stress tensor was made up of this velocity and temperature field in the following fashion. Leading order was t to the four, four u mu u nu plus eta mu nu. Plus the first sub leading order, there was a t-cube term plus a shear tensor term. I won't even define what shear is because fluid gravity is not the main point of this lecture. But okay, and blah, blah, blah, blah, blah. And this whole procedure's consistent only if this equation of motion is obeyed. What was important as we emphasized in the last class was that there were as many equations of motions as variables, okay? And so this procedure established a one-to-one map between two apparently different dynamically problems. First problem, solving Einstein's equations in ADS space in spaces with event horizons in the wrong wavelength limit. Second problem, solving the equations of hydrodynamics. These, the conservation of stress tensor by the stress tensor's function of velocities and temperatures. It's a fancy way of saying the equations of hydrodynamics, Navier-Stokes equations, relativistic generalization of Navier-Stokes equations. Okay, and if you could solve one problem, you could solve the other. And you might think that this problem is a bit simpler than the other problem because it has no gravity, it works in a fixed background. It's one lower dimension than the other problem. Now, you might ask, did this hydrodynamic problem that we looked at, did it, you know, what is the connection between that and the light quasi-normal modes that we started this discussion with? So the connection is simply this. You take these equations of hydrodynamics and it's one obvious solution. The solution is u is equal to 1, 0, 0, 0, fluid's addressed, and t is equal to constant, equilibrium. That's the solution that corresponds to the black ray. When you take these equations, just hydrodynamic equations, and linearize around the solution to find the spectrum of small fluctuations. And what do you get when you do that? You get this, which is the same as this. So there's non-linear theory. Is the non-linear theory of the light modes? And we see that from the fact that when we take this non-linear theory, put it around the solution of interest and linearize, we recover the lightness, okay? So we, by this means, we have checked that we found the right non-linear theory. It's hydrodynamics, which reproduces these linearized fluctuations as sound and sheer oscillations. But it's much more than the linearized theory. It describes all kinds of things. People have now, by now, this is an old story, and people have looked at it in great detail. For instance, people have shown that you recover turbulent solutions in this limit. This is a huge story with fluid gravity. It's not our concern at the moment, except it's an analogy. This ends my review of the fluid gravity correspondence as an analogy to what we're gonna be doing. Any questions or comments before we proceed to the actual problem at it. They're linear, yes. So the quasi-normal modes, it's exactly, the analogy is exactly like what we said about your string theory. Suppose you're doing vertex operator calculations. You have vertex operators for massless gravitons, and the vertex operator tells you that the gravitons obey the equation del squared h mu nu is equal to zero plus some gauge conditions. Those are the equations of linearized gravity. All they do is give you small fluctuations. Little boring gravity waves. But the full theory is Einstein's equations of general relativity, which has much more. So that's an indication that there exists a nonlinear theory. But it doesn't determine the nonlinear. Is this clear? Are there questions or comments? Excellent. So now we're gonna go to the problem at hand. We're looking at black holes in large D, and we wanna find an effective nonlinear theory of the light modes. The more the effective theory that will allow you to understand what happens after something's thumped. The theory settles down, but in a full nonlinear fashion. That's the question. That's the problem. We've said this. Okay, now let's recall the steps here. The first step was to move to the right coordinates just to study the equilibrium solution. So we're gonna imitate that. The first step is to view the basic Schwarzschild black hole solution in the right coordinates. Now what are the right coordinates? Well the right coordinates are coordinates that will turn out to be convenient for what you wanna do. So what will these right coordinates be? Almost the same as the coordinates we use for fluid gravity, but with a little twist in the tail. So let me show you these coordinates. Let's start with the Schwarzschild metric in Eddington Finkelstein coordinates. I'll write it down, and then I will ask you if you, let me write it down first. Two dv dr minus dv squared, one minus r naught by r to the power d minus three plus d omega d minus two squared r squared. Now this is not the same as the metric I wrote down at the beginning of the last class, but exactly the same manipulation that went, that took us from this metric to this metric with u being one zero zero zero. We'll give us that. It's exactly the same manipulation. Namely, the r here is the same as the Schwarzschild r, but the v here is T Schwarzschild was equal to v minus integral one by f of this one minus r naught. It's exactly the same manipulation, so I won't go through it again. Is this okay? Anyone confused about this? In fact, this is where Eddington and Finkelstein invented their coordinates. Look at Schwarzschild, let's not look at black brains and ADS. Okay, great. So you might think, and this is what we thought for a while, that we should look at the problem in Eddington-Finkelstein coordinates, but it's somehow very inconvenient. And the inconvenient thing is that it's tied to a particular frame. What proves more useful for us is to do a second coordinate change, starting from Eddington-Finkelstein coordinates to what I call Kerr-Schild coordinates, okay? And the thing about Eddington-Finkelstein that proves inconvenient is the following. When you set r naught to zero, this space reduces to flat space. Visually obvious that the space is flat space. To get the space back into form, a form of flat space that is completely manifest, namely dx mu dx mu, you have to do a further coordinate transformation. The idea of Kerr and Schild was, well, you see, these coordinates are nice because they're good at the horizon, but they're not so natural for infinity, okay? Kerr and Schild, okay, on the other hand, the Schwarzschild coordinates were nice at infinity because they were manifestly flat-spaced there, but not so good at the horizon. So Kerr and Schild thought, you know, had the idea of inventing a coordinate system that was good both at infinity and at the horizon. Let's see how you do that. We did, to go from the Schwarzschild coordinate to the Eddington-Finkelstein coordinate, we did the coordinate change, ts is equal to, I wrote that somewhere. I erased it, I made it, okay? ts is equal to v minus dr by one minus r naught by r to the power d minus three. Now, what does this become at infinity? At infinity, this is ts is equal to v minus r. To get infinity back to what it was in Schwarzschild coordinates, now let's make the coordinate change. v is equal to t plus r. So that, if you plug that into here, you'll find that at infinity, t is the same as ts. But because all of the singularity business comes from this guy going to zero, and since this is a non-singular coordinate change at r equals r s, this coordinate will remain non-singular at the event horizon. So now you've got a coordinate, by doing this, you have a coordinate that is manifestly flat at infinity, but continues to be non-singular at the event horizon. Okay? It's a convenient coordinate system and one that I will employ extensively through these lectures. You will see why it's so convenient in two minutes. Let's just implement it first. So all I have to do is to take this and plug it in here. Okay? So let's do that. We've got two in a dt plus dr dr minus dv squared in a one minus r naught by r, sorry, minus dt plus dr, the whole thing squared, to one minus r naught by r to the power d minus three. Okay? So let's write this as two dt plus dr into dr minus dt plus dr, the whole thing squared, that's the one part, plus dt plus dr, the whole thing squared, into r naught by r to the power d minus three. Now, here we just simplify the dr squared term which was minus, sorry, this is dr. This was minus dr squared, but there's a two dr squared here so that just becomes dr squared. We get dr squared. The dt dr term, that's two dt dr, minus two dt dr, goes away. And we're left with minus dt squared, plus of course, the part I've not written which is r squared, omega d minus two squared. This part is simply the metric of flat space. This, you could rewrite as dx mu, dx mu, okay? But there's an addition, and the addition, can people see down here? Probably not, right, let me ask. And the addition, so the final metric is d r squared is equal to, oh, maybe I should, it's better to write in the following way. G mu nu is equal to eta mu nu plus oh mu, oh nu by rho to the power d minus three, where, let me write down a bunch of definitions. Rho first is just r naught by r, r by r, r by r naught, r by r, r by r naught, oh mu, n mu is d mu rho by square root d rho squared, where d rho, it's norm, this square root d rho squared, is taken in the metric eta mu nu, okay? So this is, to be totally clear, d mu rho by square root of d mu rho, d nu rho eta mu nu, okay? And oh is equal to n mu minus u mu, okay? Why is this the same as this? Sorry, why is this the same as in explicit coordinates, this is, you know, it's, this is the same as the statement of the metric d r squared is equal to dx mu, dx nu eta mu nu, plus plus dt plus dr, the whole thing squared over r by r naught to the power d minus three, okay? And u mu, in this particular case, was simply minus one, zero, zero, sorry, I wrote that, I was a bit disorganized while writing that, maybe you can take a look at what I've written for a half a minute, and tell me if it's clear, okay? Now the final form of the metric is flat space plus something, this something manifestly decays at infinity, and is naturally written in terms of a function defined in flat space and a one-form field defined in flat space. The one-form field is u mu, the function is rho, rho of course gives rise to another one-form field, namely it's normal vector, n mu, okay? This is a fancy way of writing the black hole metric in curtshield coordinates. Questions or comments about this, is this clear? How is it related to curtshield time, yes? So what we had was that T s was equal to V minus integral d r by r, what was it? Yeah, just substitute back, yes? So it's like a tortoise coordinate with r removed, yes? So the tortoise coordinate reduces to T minus r at infinity, remove that part, but you let the stuff happen at the horizon that desingularizes the horizon, okay? Great, so it's a very clever coordinate, but for us, firstly it's non-singular at the horizon. Secondly, what's really important for us is that it's naturally written in terms of mathematical structures that are naturally defined in flat space. Now you remember what we were after. What we were after was to demonstrate that solutions of black hole dynamics are one-to-one correspondence with the motion of a membrane that lives in flat space, okay? For this purpose, it's very nice to deal with metrics defined in terms of auxiliary quantities that naturally live in flat space, not in the metric itself, because in the dual theory, we're interested in quantities that live in flat space, not in this preformed metric itself. The dual description, namely membrane, has no metric. Just lives in flat space. This is a complication we didn't have to deal with in fluid gravity, because in fluid gravity, all the structures lived at the boundary and the boundary had a natural metric. Here, even though the metric in the bulk has changed, we're looking at dynamics in the bulk, we want to deal with effective dynamics of something living in the undeformed flat space. And the Kerr-Schild form of the metric very naturally regards metrics as constructed out of one forms and functions living in flat space. So that's very convenient for us. Excellent. Now, the next thing we did for fluid gravity was to take our solution and boost. So we'll do the same thing here. We take the solution and boost it. But having written it in this form, the boosting is simply totally trivial. Instead of U mu being minus one, zero, zero, zero, you replace U mu by any constant vector that squares to minus one. Okay, so boosting just replaces this line by the equation U squared is equal to minus one, U is equal to constant. This is an as good solution of Einstein's equation, the exact solution. What does it represent? It represents a black hole, it's not sitting there, but it's hurtling somewhere at velocity U mu. Clear, trivial, nothing interesting. What's the next thing we did here? After boosting, we decided to make the boost parameters functions of space and time. And the temperature functions of space and time. Let's imitate the procedure. Okay, so suppose we take this metric here and in that metric, just take U mu to be U mu of X. We demanded here that the functions of space and time were functions that were varying on very long length scales compared to the temperature. In this problem, we do not have the luxury to vary on very long scales compared to the analog of the temperature, namely R naught, because we live on a sphere and there are no length scales larger than R naught. However, we do have the ability to vary on very long length scales compared to the membrane thickness, R naught by D. This is the key point. The fact that there were two length scales in this problem allows us to be on long length scales with respect to the smaller length scale while still fitting inside the sphere. We will demand that this variation, the length scale of variation, let's call the length scale of variation, is much, much greater than one by D. One by D, let's say. In particular, when we actually implement our analysis, we will assume that there's no D in this problem. This varies on length scale one, not as some inverse power of D. We also here in this problem allow the temperature to vary as a function of where we were. What's the analog of that? Roughly what we wanna do is to allow R naught to vary as a function of where we are, but there's something else that had no analog in the black-brained problem. And that's this. You see, we've chosen a black hole situated somewhere. There's no particular reason why our back hole should stay situated at the same point. Move around. So the analog of allowing the temperature to vary, the right analog for that just turns out to be very simple. It turns out to be the following. Take rho, this function rho, to be an arbitrary function of x. Take the function rho as I've written it, to be an arbitrary function of x. By x I mean x and t, all coordinates of Minkowski's space. Okay. Take rho to be an arbitrary function of x. Now, as you will see in two minutes, there will be restriction on u and rho to get the program started on. But for now, we just let u and rho to be arbitrary functions of x. The justification is that it works. Hang on for five minutes, ask that question. A, yeah, hang on for five minutes. In this problem, what got us started was the almost obvious statement that this quantity, while not an exact solution of Einstein's equations, solved Einstein's equations to lowest order in the derivative expansion. Because at lowest order in the derivative expansion, you take no derivatives of t and no derivatives of u. If you don't take derivatives of t and u, you don't know the difference between whether this configuration is varying or not. And the non-varying solution solved Einstein's equations. Once you had that, you could roll the machine. Maybe just a brief interlude here about something genuine. You know, what we are doing both in fluid gravity and in this problem that I'm looking at now is perturbation theory. What we've done is identify a small parameter to try to systematically construct a class of solutions in perturbation theory in this parameter. And though you might not understand this when you do your undergraduate courses, perturbation theory is an art rather than a science. And the art goes as follows. You wanna find some exact solution that lies somewhere here. You've got a parameter in which to perturb. The art is in choosing the starting point of your solution to be nearby to you. If you start here and try to perturb there, won't work. What you have to do is to find the starting point of perturbation theory. And for that there are no rules. You have to use intuition, you have to guess. You have to keep trying till it works. What do I mean by starting point works? What I mean is that the starting point is near the solution that you want and solves the equation that leading order in your parameter. In non-trivial uses of perturbation theory, this is the interesting step. The rest in the end is mechanical. You know what to do. Solving some equations. Where physics, the interesting physics intuition comes in is in choosing your starting point. Okay? Here we had a nice choice of starting point and we wanna check whether this is an analogously good choice of starting point for our perturbation theory. Yeah, that's the question we're asking. So the way that question becomes precise is following. What we want to ask is does this configuration solve Einstein's equations at leading order in large d? Forget about some leading orders for now. We can use perturbation to correct the theory, to correct it up if it solves a leading order. We can correct it up to some leading order, but does it solve a leading order in large d? This is an acceptable solution at leading order in large d. Yeah, that's the question we're asking. So let's look at this solution for a minute. This configuration for a minute. Let's look at it. This, to answer this question, will take us a little time. To answer it properly, we'll take us a little time. But let's first look at the question, look at this metric and say what we can say about it on general rules. The first thing about this metric that I want you to note is that, oh, and something I didn't even tell you about here was that I will always choose my velocity field U mu such that U mu N mu is equal to zero. Okay, note that that was the case here. For instance, for the static black hole, U was in the 1-0-0's at chip yaw in the time direction, whereas N was in the radial r-adaptation, okay, okay. Oh, and there was one thing here, I'm sorry. When I boosted, I needed to tell you what r was. Once I boosted, it continues to be r by r naught where r squared is equal to p mu nu x mu x nu. p mu nu is the same as in fluid gravity, eta mu nu plus U mu nu. You understand, r is the spatial, is the magnitude of the spatial direction orthogonal to velocity. That's what it was before boosting. The boosting just, this is a boost, a boosted version of that statement, sorry. So the boosting involves replacing r by this along with replacing U. I'm sorry, I should have mentioned that before, sorry for that. Is this clear? Sorry, so to come back. So to come back, what we're doing now is to look, so this velocity vector was normal to N. We're gonna interpret the statement as follows. We will be particularly interested in the surface, rho is equal to one. For instance, in the case of the Schwarzschild black hole, rho equals one was the event horizon, because r is equal to r naught, okay. We'll be particularly interested in that surface and the surface rho is equal to one, we will define as the membrane world volume. This velocity field here is normal to the normal vector to rho. So in particular, normal to the membrane world volume. Now, this statement is simply the fact that the velocity field is a field that lives on the membrane, okay. It's an intrinsic vector field on the submanifold, namely the membrane world volume, okay. So, now let's go back to the analysis that we wanted to do. The analysis we wanted to do was the following. First thing I want you to note is that O mu is a null one form field. O dot O is equal to zero. Why is that? It's because N dot N is equal to one, U dot U is equal to minus one, and O dot U is equal to zero, pretty clear. Now, simply from the fact that O is another one form field, okay. This form of the metric, and now my analysis is for general U mu and rho. Not just the lack of a metric. If we wanted to invert the metric, we want a G mu nu, it's extremely easy to do, okay. I'm gonna give you the answer and then you'll see if you believe me. Where all raising is done by the metric, eta. Why is this the case? In order to check that this is the inverse of that, what we have to do is to take the inverse metric and dot it with an index of the metric and see if we get delta or not, okay. So eta dotted with eta, upper dot of the eta, lower gives delta, no problem, okay. But there are three other terms. One of the terms, the most difficult term is has an O dotted with an O, but O dotted with O is zero because O is another vector field, okay. So that leaves you with two remaining terms. One of these terms has an eta dotted with an O and the other term has an eta dotted with an O, but with a minus sign. So these just cancel each other. Because O mu is this null vector field, inverting the metric was very easy. It wouldn't have worked if this was an arbitrary vector field because you would have got contamination from the O square term, is this clear? Okay, so now the thing I'm gonna do is to determine what is the norm of the normal vector. What is the norm of the normal vector in the spacetime metric rather than in flat space? Remember that the normal vector was defined to have unit norm in flat space in the eta mu new metric. What is the norm of the normal vector in our actual metric, okay. That's very easy because when we dot through eta we get one because n was defined such that n dot n through eta is one. And when we dot n with O, we get one because O is n minus u and n dot n is one, u dot n is zero. So we get one minus one by rho to the power d minus three. Is this clear? N dot n in the metric that we're looking at is one minus one by rho to the power d minus three. The really nice thing about this formula is that this is equal to zero at rho equals one. Now what does that tell you? Rho equals one is a co-dimension one sub-manifold of the whole space. N is the normal vector to that that co-dimension one sub-manifold. What we've just seen is that the norm of the normal vector of that sub-manifold is zero. Okay, now manifolds, sub-manifolds, such that the norm of their normal vector is everywhere zero have an n. They're called null manifolds, okay? And they're sort of interesting things, normal vectors. For instance, the manifolds that include the normal vector in their tangent space because the manifold is defined, the tangent space is defined as all the vectors that are normal to the normal vector, but n dot n is zero. So the normal vector lies in the tangent space. Okay, these are interesting manifolds and would be interesting just for mathematics, but we're not mathematicians since I'm not a mathematician, and I'm interested in them for physics. Where have we seen null manifolds before? Null manifolds occur whenever we study black hole physics because of the classic result that the event horizon of a black hole is a null manifold. Black hole event horizons are null manifolds. In fact, in a dynamical situation where a black hole is formed and then settles down asymptotically to let's say a Schwarzschild type black hole or a Kurt type black hole, often the most useful definition of the event horizon is the following. That the event horizon is the unique null manifold which agrees with the non-event horizon of the static equilibrium, the stationary equilibrium black hole at late times. What have we seen here? We've seen two things. We've seen that rho equals one was the event horizon of the Schwarzschild black hole. Because that was simply r equals r naught. Second, we've seen that in great generality rho equals one is a null manifold. So if this metric that I looked at has the property that it settles down at late times to a Schwarzschild black hole or as I will show you later to occur a black hole or anything stationary, okay? Then this null manifold has particular physical meaning. It's not any old null manifold. It is in fact the event horizon of a spacetime because then it would be the unique null manifold that agrees with the non-event horizon at late times. Is this clear? Now, as you know, just as in fluid gravity, this u and rho will not turn out to be arbitrary functions. They will have some equations of motion and the equations of motion will be dissipated. And so we'll plausibly guarantee that at late times everything settles down into a stationary configuration. In fact, probably you can show this, okay? So we will assume that at late times, that while at finite time, u and t are whatever they are, at late times it just settles down to equilibrium. And therefore the configuration rho equals one, the sub-manifold rho equals one has a beautiful physical interpretation. It's the event horizon of a spacetime, okay? So first thing we've learned about this metric. Metric has an event horizon. We can write down what the event horizon is. It's the manifold rho equals one. Beautiful statement. Usually very hard to find event horizons of spacetimes. Okay, here it's very easy. Okay, please. What do you mean? Yes, but then I changed it, right? I made this an arbitrary function of, for instance, had I extended it such that u dot n was not necessarily equals zero, then it wouldn't have been the event horizon. It wouldn't have been a null manifold. Certainly it was the event horizon of the Schwarzschild black hole, but now we're looking at a class of solutions that's more general than the Schwarzschild black hole, but for this full class it remains the event horizon. Okay, good. Other questions, comments? Okay, now, why is it important that this is the event horizon? This is important for us for the following reason. That rho greater than one is outside the event horizon. As you can see from the fact that when rho goes larger than one, you reduce to flat space. Rho less than one is inside, okay? We want to solve, but we want to solve Einstein's equations. Okay, we want to solve Einstein's equations, but we're not interested in solving Einstein's equations everywhere. We only want to solve Einstein's equations in order to predict the evolution of the outside, okay? What happens inside for rho less than one is not our concern, okay? That's what happens there is something that anyone who's suicidal enough to want to jump into a black hole would care about, but we not being such suicidal people don't care about, okay? We only want to see what a large deligo observer will observe when he's sitting outside the black hole. We don't care what happens inside the black hole, okay? Now, normally in physics you can't say I care about this region, so I'll ignore this region because what happens here affects what happens there after some time. But the great thing about event horizons is that they cause the boundaries. So what happens inside the event horizon just doesn't affect what happens outside at all. You know, this statement, if it's the first time you're hearing it, you might think that I'm pulling a fast one on you. Okay, certainly that's what I thought when somebody told me the statement first. But it's true, and in fact it's a crucial element, ingredient in the numerical relativity program. People who do numerical relativity stop their codes once they show they're inside the event horizon because they don't care, and it's very unfortunate that they didn't care because otherwise they would have to run their codes up to the singularity which would have trouble. Okay? So it's a remarkable thing about general relativity that there are these causal boundaries. So you don't have to, if you're interested in the outside, you don't have the, you can forget about the inside. So we're gonna forget about the inside. In fact, we can do better. We'll do a bit better. We'll solve a bit into the inside, but we don't care. What happens, happens for rho less than one. That's between God and the suicidal people, okay? Not our concern. Okay. So we're interested in rho greater than one. Rho equals one rho greater than one. No, this is genuine event horizon. Genuine event horizon, yes. That's what we're heading towards. You know, what we're going to try to do is to make sure that any little patch of this thing is a patch of some schwarzschild black hole. Okay? Which perhaps goes along the lines of what you're, if you're asking for more than that, I don't know what to say. But this, these are near the event horizons, as we will see, these are little patches of schwarzschild black holes. Hang on. Actually, you know this well, but, okay. Please. Yes. Yes, exactly. And in particular, that you assert that at large times, it becomes a stationary black hole. That was a crucial element, okay? Now, Atisha asks a good question. Normally, an event horizon is a very etiological thing. We could be sitting inside an event horizon, depending on a shell that will collapse in the future, we would not know it. How come in this situation, we know so well where the event horizon was? She's very similar question in fluid gravity, so let me take a minute to answer that question. Okay. In fluid gravity, the answer to that question is that, that event horizons are much less etiologically in the ADS space than in flat space. Why is that? The definition of an event horizon is this. It's a causal boundary that separates points from, separates those points that can reach the boundary of ADS from those that cannot. Okay. Now, unlike in flat space, you can reach the boundary of ADS in finite time. So suppose you know that a point from here can go whoosh and reach the boundary of ADS. Then you know you're outside the event horizon, no matter what is done here. You can send 20,000 shells in the future, won't affect that. You were outside the event horizon. Okay. And this happens over a time that's set by the radial position of where you are. And in fluid gravity, that's basically over a time set by the temperature. So fluid gravity, positions of event horizons are a causal, but only to the extent one over T, okay? One over T is a short lens scale in the problem. Everything's varying on lens scale, large compared. So essentially fluid gravity, the fluid variables locally determine the event horizon. That was crucial for the consistency of the whole fluid gravity picture because event horizons define entropy currents and the entropy current and fluid dynamics is a local function of the fluid variables. What's the situation here? Actually it's very similar. It's very similar because we are looking at solutions, because we're restricting attention to class of solutions. The class of solutions are such that outside a little region of around here, you reduce the flat space. That little region is only of thickness R naught by D. So normally in flat space, you have to get all the way to the boundary and that takes an infinite amount of times, okay? But in our class of solutions, in our class of solutions, once you get out of this little shell, it's flat. So you know you're gonna make it to the boundary, okay? The class of solutions we're looking at will not allow some strong gravity waves in addition. If you were doing that, you would have to do a different analysis, okay? But in this class of solutions, as you see, outside this little shell layer, space is just flat. So once you get out, you're home, okay? So this distance being one over D tells you that the location of the event horizon is essentially local in time over time scales one over D. Very similar to the graph. So now we know where the event horizon of our space sits. That's solved half of our problem because we don't have to worry about solving Einstein's equation for Rho less than one. We still have to worry about it for Rho greater than one. But what can we say about Rho order one greater than one? Okay. Suppose we look at Rho minus one equals order one, okay? Then this metric is exponentially close to flat space. I'm broken the wrong metric. This metric is exponentially close to flat space because we've got some unit norm one form divided by a function that's larger than one to the power D. That's extremely small. So distances such that Rho minus one is order one is flat space and flat space happens to be an excellent solution of Einstein's equation. Up to exponentially accuracy for Rho minus one of order one, our metric automatically solves Einstein's equations, okay? Where is it not automatic? So let's do a more serious estimate of that. Where it is not automatic is as follows. Let's write Rho as equal to one plus Rho minus one, okay? And then Rho minus one will be small. I'll just multiply and divide this by D for fun. Actually, I should really be doing this by D minus three but since D is large, D and D minus three are the same thing. Now we're interested in one over Rho to the power D minus three but I'll just write it as D because it's irritating to carry the minus three alone. Which is one over one plus D Rho minus one by D to the power D, which is approximately if this thing is not too large, E to the power minus D into Rho minus one. Imagine for instance that this has had fixed as D is taken to infinity, okay? So what this is telling us is that we have to, this stuff is non-zero provided Rho minus one is of order one by D. If Rho minus one is of order one by D, Rho minus one is alpha times one by D. Alpha is any number no matter how large that held fixed as D goes to infinity, the equation is not automatically solved. But for larger values of Rho minus one, the equation is just solved to very good accuracy in the largely extension. So what's the situation? The situation is that we've got this membrane surface. Rho equals one, okay? That's the membrane surface, Rho equals one. There is a little thickness around this membrane of order one by D where we have to worry about whether this metrics all Einstein's equations or not. Outside this region exponentially approaches flat space. Wonderful, solve Einstein's equation. Inside this region, all hell breaks loose because one over Rho to the power D minus three, which becomes very small when Rho is larger than one becomes very big when Rho is less than one. All hell breaks loose, but we don't care. Unicorns could be created there for all we care. It's causally separated from what we do. So in order to ensure that Einstein's equations are solved for the purposes of predicting the outside, all we need to do is to check whether they're solved in this little region. Is this clear? How much time do I have left? Minus one. Okay, okay, so then we'll wait till the next lecture, which is just in the afternoon, so that I can further analyze the question in this little next lecture. Thank you.