 Okay, my name is Shiraz Minwala, I work at the Tata Institute of Fundamental Research in Mumbai, okay, and it's really nice to be back in ICTP. Thank you for the invitation and so on. My lectures are titled Black Hole Dynamics at Large D and I'm going to be giving four lectures. The first one that's today's lecture will focus on the motivation for the project, introduce it and give you a sort of analogy or remind you of a similar situation to what I'm going to do to the real interest of my lectures that has been worked out in the past. The remaining lectures will be more technical. In the second and third lecture, we will actually derive the effective equations of Black Hole Dynamics at Large D and try to explain how they can be recast into the form of stress tensor conservation and current conservation on the membrane and try to understand the emission of gravitational and Maxwell radiation. Okay, fine, let's start. So there are three, I thought, in order to motivate my lectures, well, there are two basically different strands of motivation that meet in the analysis that we're going to be discussing. And the first of these motivations is an extremely physical, quite remarkably an experimental motivation and it has to do with this famous event that you've all heard about detected by LIGO. So let me give you a five-minute reminder of what LIGO recently saw, okay? As all of you know, LIGO detected some gravity waves and that's great but more interesting for me is not just that they detected gravity waves but what the source of those gravity waves was. The source of those gravity waves, at least if we believe LIGO's analysis, appears to have been an extremely dramatic event. The event was the collision of two black holes. Each black hole was about 30 solar masses in size in mass and they collided with each other. So in order to understand, in order to get just a rough feel of this event, I'm going to take you through some essentially dimensional analysis just to show you the orders of magnitude of the energy intensities radiated in such an event, okay? So as we said, we have two black holes, each of about 30 times the solar mass, okay? And then the collision LIGO tells us that approximately three solar masses of energy were radiated. How long would we expect such a collision to take, okay? If you know nothing else about the problem, you might estimate the time scale essentially by causality. See each, as some of you may remember, a solar mass black hole has Schwarzschild radius of order three kilometers. The Schwarzschild radius of a black hole scales like its mass. So a 30 solar mass black hole has a radius of about 90 kilometers, let's call it a hundred kilometers, okay? So we've got one black hole with a diameter of 200 kilometers, another black hole with diameter of 200 kilometers and they're coming together to form a composite black hole with some diameter not too far from 400 kilometers, I mean roughly. Clearly this process can't happen faster than the time it takes for lights to go from one end to the other of this process. And let's use that as an estimate for the time scale of the problem, okay? So the estimate for the time scale of the problem is 400 kilometers, but time is not measured in kilometers so we need the speed of light, C is not set to one for LIGO, okay? So that's three into ten to the power five kilometers per second. So that's approximately, you know, I'm getting very rough on this magnitude, ten to the power minus three seconds. So this event, this LIGO event happened in approximately ten to the power minus three seconds. Now this estimate might be a little wrong, ten to the minus three might be ten to the power minus two, that's not too wrong. Now in ten to the power minus three seconds, this event radiated 30 solar masses of energy, sorry, three solar masses of energy. Now in order to get a sense of the scales, let's compare that to what happens in the sum. The sun's radiating quite a lot of energy, but through its lifetime it radiates approximately one percent of its rest mass, okay? Now what's the lifetime of the sun? The lifetime of the sun is about ten billion years, so that's ten to the power ten years, but each year is about ten to the power seven seconds. So it's approximately ten to the power seventeen seconds. This guy happened in ten to the power minus three. So the ratio of times, how much faster did this LIGO event happen? It happened about ten to the power twenty times faster than the lifetime of the sun. How much energy did it radiate? Well three solar masses compared to one percent, so about three hundred times as much energy as the sun will through its entire lifetime, okay? So that's three into ten to the power twenty two, roughly, times the intensity. The intensity of energy radiation is about three to the power ten to the power twenty two times that of the energy radiated by the sun. Now in order to get a sense of what that scale means, let's remember that there are about ten to the ten billion, so ten to the power ten stars in our galaxy, and about ten billion visible galaxies in the universe. So using the same very rough estimate that tells us we have about ten to the power twenty. We have about ten to the power twenty stars in the visible galaxy, okay? And so, this thing here is radiating according to our estimate, at three hundred times the combined intensity of all stars in the visible galaxy, visible universe, sorry, visible universe. So it was quite an event, okay? It's a magnificent event. You've got these two black holes coming, push, and enormous amounts of energy are radiated in very short time. Now, this event, this LIGO event, it's not the only dramatic event we've seen in astrophysics. For instance, events like supernova explosions are quite dramatic. However, for a theorist, there's something much nicer about the LIGO event than a supernova explosion, and that's this. A supernova explosion is an essentially inherently dirty process, by which I mean to model this process, it needs a lot of very difficult physics, which you probably will never know exactly. Equations of state of matter, of nuclear matter, very high energy densities, details of the composition of the star, and so on. On the other hand, to a very good approximation, the event described by LIGO is governed by what, in my opinion, is the most beautiful equation ever written down by human beings. Namely, the equation R mu nu equals 0, Einstein's equation in a vacuum. So from a theoretical point of view, the LIGO event is an extremely clean event. In the sense that the fundamental equation governing it is known without parameters, and yet it describes this completely magnificent physics. Okay, so now let me tell you, now that we're also impressed by LIGO, okay, or by the event that LIGO detected, let's spend two or three more minutes on what LIGO actually saw, and how do they conclude from what they saw, that they were saying two black holes collided. Okay, so I won't go into any experimental details largely because I don't know them, but there's this curve with some strain, something like that. You see something like this, and then you see something like this, and then some. Okay, all of you have seen this plot, which describes the gravitational radiation coming as a function of time. And there are basically three distinct regions in this plot. So this is early, middle, late. Okay, the early region is believed to have been invented by the last inspirals of the two black holes, you know, the two black holes orbiting each other. As they were orbiting, they were inspiring into each other, okay? And when we're not too near this big region here, this spiral can approximately be described by systematic corrections to Newton's laws. What's called the post-Newtonian approximation. This is a very clean process, you understand what's happening analytically. And this process leads to a large part of LIGO's analysis. This final process is described by what are called quasi-normal modes. Through these lectures, I will explain what quasi-normal modes are, we'll get great familiarity with them. Roughly speaking, for now, they just linearized fluctuations of Einstein's equations around the black hole background. These also are completely well understood and controlled theoretically. However, there's this middle region which is really dramatic. It's the region where the bang actually happens, okay? How did LIGO analyze what went on here? What they did was to fit the results to numerical simulations, full numerical simulations of general relativity, okay? So there's some, you know, analytic stuff here, analytic stuff here, numerical stuff here. The LIGO analysis used a very large number of results from simulations of order 10,000. 10,000 simulations of general relativity. And that sounds very impressive, but I want to alert you to the fact that this is less impressive than it sounds, okay? And that's for the following reason. When we've got two black holes colliding, when we've got two black holes colliding, although the equations have no parameters in them, the initial conditions have a bunch of parameters. So let's pause now for two minutes to count the number of parameters, okay? So we've got these two black holes, they're coming in, they're coming in or they're orbiting in some plane. Let's suppose that the plane, the normal to that plane is the z axis. Yes, that's a convention, no parameters so far. Now the first black hole has a mass and an angular momentum. So I'm drawing the z axis here. The angular momentum makes some angle with the z axis, okay? So we need to specify Lz, Lx and the mass of the first black hole. So that's three parameters. Now the second black hole comes in. So I've chosen my x axis to be along the projected direction of the second moment, you understand? That's why it's three not four. Now the second black hole comes in and it has some Lz, Lx, but now x has a meaning because it's the projected direction of this. So it also has some Ly and it has its own mass. So that's seven parameters. And then plus there are three other parameters or two depending on how you count, these three other parameters are, you see now we fixed all axes. Z, x and y all have absolute meaning. With respect to these axes, the orbit of the two black holes has an eccentricity. So there's a direction, okay? And there's the amount of eccentricity. There's also the actual radius, but that's a function of time. So let's forget that, okay? So the eccentricity in magnitude and in direction, okay? So eccentric, sorry, we could call it ex and ey if you want. So that's two more. So the total number of parameters and a fair counting of the problem is nine parameters, okay? So we've got this beautiful situation where we've got a bunch of equations, no parameters, however the initial conditions of nine parameters. Suppose you want to fill out a grid, you've got, imagine a nine dimensional grid on which you put initial conditions, okay? And you want to discretize this nine dimensional grid. You want to discretize it, let's say, for simplicity, evenly in each dimension, okay? Evenly in each axis. How many points can you put per axis if you have in your hands 10,000 simulations? Well, that's the answer to the question, what, to the, sorry. Nine to the power what is equal to 10,000? And if you open your calculator on your computer, you'll find that this is approximately 4.2. What is about 4.2? Yes? It is unfair because it is. It is unfair, okay, I'm just, you might want to make nine, eight. It won't change very much. I mean, I'm just giving you rough orders of magnitude. I agree with you, they're cleverer than what I'm making out to be, okay? But just, all my numbers, four could become six, okay? If you take my nine seriously, and you ask, suppose I want to discretize each axis with 10 points. The number of simulations you would need is 3 billion, okay? These simulations, these numerical simulations, were very impressive, hard to do, okay? And I don't really know very much about it, but it seems to me that getting 3 billion simulations is a very tough job. Okay, so let me repeat, what have we seen? We've got this fantastic event, this fantastic analysis that uses analytics, analytics, numerics. Numerics has to be done by initial condition, and sort of tough to do. So if you want to improve, well, I don't know. Perhaps you might be motivated to ask, is there somewhere in it to analyze the actual bank? That is not just blind numerical simulation of general relativity. Can you get some analytic input into this, okay? My lectures will be an attempt to provide some analytics to understand this bank, from one point of view, that will be one way of looking at these lectures. I don't yet know how well it works for actual things like black hole collision, so let me say that. Clearly from the beginning, we'll make an attempt which is ongoing, and we'll see how that goes, okay? And let me immediately tell you what the attempt will be. The attempt will be the following. The attempt will be to generalize Einstein's equations, to give it a parameter, and then try to do perturbation theory in that parameter, okay? And the generalization we'll use is very simple. We won't change Einstein's equations because they're so beautiful. All we do is look at Einstein's equations instead of looking at them in four dimensions, we look at them in d dimensions. Take d to be large and work in the one by d approximation, okay? I want to emphasize that everything I do in these lectures will be purely classical, no quantum mechanics, okay? It's just meant to be a scheme for understanding classical physics and finite d, but in a one by d, one by d expansion, okay? The logic is very similar. The basic logic is very similar to you know, Toft's attempt to understand QCD, which has three colors, by doing a one by d, one by n expansion, okay? Generalize the problem, get a parameter, and then hope that the perturbation in that parameter is useful even though you might be quite far away, okay? So this is the first motivation for this problem. Any questions or comments for the problem I'm going to address? Any questions or comments about this? Great. Now, the second motivation goes as follows, and we will see as we go along, that these two motivations merge into each other, okay? The second motivation is as follows, it's been felt for a long time that there is some sense in which the dynamics of black holes can be captured by the physics of a membrane, okay? That lives on the event horizon or near the event horizon of the black holes, okay? This hope has been articulated in various forms, and roughly speaking goes by the name of the membrane paradigm. Roughly speaking goes by the name of the membrane paradigm of black hole physics, okay? And any particular articulation of this hope has been fall short of what I'm going to say. But the most interesting and useful version of the membrane paradigm would be the following. Is there a clear, well-defined sense in which a large class, a well-defined class of solutions of black hole dynamics is dual. Namely, is the same thing as the solutions to the equations of some membrane wiggling around in space, okay? If there was a nice duality between these two problems, instead of trying to solve the problem of black hole dynamics, you can get away by solving instead of problem of some sort of membrane wiggling around in space. That would be sort of interesting. This would be, in my opinion, the strongest form of a membrane paradigm. If these two systems were somehow just exactly in some well-defined parameter regime, just the same problem, okay? Please, please, correct. It means you find a metric as a solution to the Einstein equations that includes solutions that are non-trivial enough to include things like black hole equations, or extremely non-trivial solutions. Yes, oh yes, dynamical solutions of Einstein's equation, not boring static stationary stuff, okay, yeah, okay, great. So yes, I will not talk about black hole evaporation at all. There are various different things called a membrane paradigm. The thing formulated by Thorn and Damour and company were just analogies of the classical equations with membrane equations, okay? That's the membrane paradigm I have in mind, nothing quantum mechanical, okay? What we're going to show is that this hope is indeed realized in the purely classical sense we just discussed in the large delimit, okay? What we will show is that the equations of Einstein gravity in a well-defined regime that we'll describe as we go through these lectures for, in the presence of black hole, reduce to the equations of a membrane propagating in flat space, okay? The membrane will have certain number of degrees of freedom, an equal number of equations of motion, and Einstein gravity will uniquely, without any freedom, determine the degrees of freedom and the equations of this membrane systematically in one by the expansion, okay? This is the simplification, the quote-unquote simplification of Einstein's equations in the large delimit, okay? This is the quote-unquote simplification that people find for the Einstein's equations in the large delimit. The ability to replace the problem of gravity with a problem of a membrane moving around, that's where these lectures are going, okay? Any questions or comments about this? Said, we've got a black hole living in d dimensions, what? In the membrane paradigm as usually formulated, d was not important, okay? You could formulate the usual statement of the membrane paradigm in any d, okay? But the usual statements of the membrane paradigm were not, in my opinion, very useful. Basically, because they did not give you as many equations as variables. We'll see some of this as we go along. In what we're going to be doing, we will have, what in my opinion is a useful version of the membrane paradigm, okay? And in order to get that useful version, we will need a parameter. The parameter will be d, so it will work only in very large dimensions, in the one by the expansion, okay? Any questions or comments? For instance, nobody suggests, Kip Thorne is one of the inventors of the membrane paradigm, he's one of the pioneers of LIGO. He's not suggesting that he used his membrane paradigm to try to analyze LIGO, it's not useful enough for that. It's not good enough, it's some analogy really, it's not, yeah, okay? This will go beyond being an analogy, as you will see, okay? So these two motivations will come together, having a systematic expansion of non-linear gravitational physics in a new parameter, the parameter one by d, that expansion will make contact with what I feel is a useful formulation of the membrane paradigm of black hole physics. And that will, in fact, the expansion will basically be the membrane paradigm, in a manner we will see as we go along. Okay, any other questions or comments? So now let us start getting a little more technical. The subject of trying to use d, more precisely one by d, as an effective expansion parameter for black hole, for black hole dynamics, was first suggested by, the idea was first enunciated by Emperan, Suzuki, and Tanabe, henceforth called est. In a couple of, well they wrote, they've written maybe 10 or 12 papers on the subject, but the two that will be useful for me, that are immediately useful for me, I'll alert you to that first one. And I'm going to give you a quick review of what they discussed in these two papers, okay? So the first paper had many nice things, but the observation that I'm going to take away from it is a very simple one. But you will agree, I hope, very interesting. And the observation is this. Consider the metric of a Schwarzschild black hole. In d dimensions, okay? So we have one minus. This is a metric familiar to all of you. The metric has a sphere, a two dimensional sphere of radius r. And in Schwarzschild coordinates, there's two parts to the metric. There's this dr squared by this factor that goes to zero at the event horizon. dt squared times this factor that goes to zero at the event horizon. This function here, f of r is equal to 1 minus r0 by r to the power d minus 3. Sometimes called the in blackening factor. And measures how far you are from flat space. When f is one, this metric is flat space metric. When f deviates from one, you're moving away from flat space. At least if you do it in a way that's not coincidence. Now the observation that Emperor Suzuki and Tanabe made was the following. Let us look at, let us consider r, any number greater than r0, okay? So that r0 by r is less than one. Keep the ratio r0 by r fixed and then take d to infinity. What happens to this in blackening factor? Well, what happens to this in blackening factor is very clear. You've got a number less than one to the power infinity. And that's zero, okay? So in this limit, f of r is one minus the zero. So it's just f of r is equal to one. What have we just learned? We've learned that if we keep r, any number larger than r0, keep that ratio fixed. r minus r0 fixed as I take d to infinity. The in blackening factors that are flat space. You just don't see that there's a black hole at all, okay? Why is this happening? It's essentially because Newton's law falls off so fast in a large number of dimensions. Okay, now this experiment didn't allow us to see that there was a black hole. Instead of another experiment we can do that allows us to see that there is a black hole by staying outside the black hole. When you're inside, you're stuck so you notice it, okay? But when you're outside, can you stay outside and see it? Of course there is. In order to do that, let's suppose r is equal to r0 in 1 plus r divided by d minus 3, okay? And as I take d to infinity, instead of keeping little r fixed, I'll keep capital r fixed, okay? What does that mean? That means what I'm doing is as I take d to infinity, I'm scaling towards r0 so that I'm going to a distance 1 by d, 1 by capital D. D is large, so d and d minus 3 are the same, okay? 1 by d off near to the event horizon, okay? Now let's see what happens to this in blackening factor. We get 1 minus r0 by r0, so that's good. 1 plus r to the power d minus 3 to the power d minus 3. And now d minus 3 is going to infinity, but this you may recognize, this limit. This you may recognize as being equal to 1 minus e to the power minus r, okay? It's a famous limit, 1 plus 1 by n to the power n is equal to e, when n goes to infinity. So what we see here is that if we scale towards the event horizon, so that we come within a distance 1 by d of the event horizon, as d is taken to infinity, then we start seeing that there's a black line, then we start seeing the event horizon, okay? Let me word this conclusion in slightly different language. The wording of this conclusion in slightly different language might go as follows. In d equals 4, you know, some reasonable number of dimensions, there's essentially one lens scale associated with the short-shell black hole, which is its size. But in a large number of dimensions, there are two effective lens scales. One remains its size, that's r0. But there's a second lens scale, which is its thickness. The distance beyond the event horizon that you have to go, before you start not being able to see that there was a black hole at all. And that thickness is of order r0 divided by d, okay? So two lens scales, size, thickness, thickness much smaller than size. If we work in units as we were, where r0 gets fixed, the thickness is r0 divided by d, okay? Great, this is already an interesting observation and was made in this paper. However, in an observation like this, you're never really sure whether it's telling you anything unless you do a calculation. And Emperor Suzuki and Tannabe proceeded in stages, but most completely in this paper, to do an interesting calculation. And I'm now gonna tell you about the calculation. What they did was to compute the spectrum of quasi-normal modes of the Schwarzschild black hole in large d. Okay, now some of you don't know what a quasi-normal mode is. So I'll give you a two minute introduction to that. The first thing to say about quasi-normal modes is that they're linearized solutions of Einstein's equations linearized around the black hole. However, this by itself does not specify the modes. Because these linearized equations are linear for second-order differential equations. And in order to get particular solutions, you need two boundary conditions. So quasi-normal modes are the linearized solutions subject to the following two boundary conditions. A, that modes in some sense that can be made more precise in infalling coordinates. But anyway, it's common to say that the modes are infalling at the event horizon and outgoing at infinity. We have two boundary conditions. We put one at the horizon, the second one at infinity. Things go in to the event horizon. That's a very reasonable boundary conditions. That's the thing we usually do with luck holes. The thing to focus on is that the modes are also required to be outgoing at infinity. Now, why are we interested in solutions of linearized Einstein's equations with such strange boundary conditions? Why the outgoing at infinity? You see, what quasi-normal modes are the answer to is the following question. Suppose you take a black hole and you thump it. For instance, by throwing a huge rock into it, okay? After you thump it, the black hole will wiggle around. And then, slowly, it settles down into the state that it wants to settle down into, okay? The question that we're interested in is, once it's almost settled down, the settling down should be a linear problem. And is there a basis of modes in which we can describe the settling down? Notice we're not interested in the thump. We don't care about the thump. We want to know after any thump, we stop thumping it, okay? Can you find a basis of modes in which it settles down? Now, thumping, as we've said, is sending something in. Sending something in is allowing ingoing modes from infinity. But because we're interested in what happens after the thump, we don't allow those ingoing modes. That's the reason that our boundary conditions are that modes at infinity are purely outgoing. You see, you may say, we may, perhaps we're not interested in modes going out, but we can't control that. Because as a black hole wiggles, it might radiate, okay? So we have to allow outgoing modes. That's not something in our hand. But we can specify that nothing's coming in. So that's the reason for this boundary condition, okay? So boundary condition at the horizon, boundary condition at infinity, okay? So that's what quasi-normal modes are. Any questions or comments about this? Why quasi? It's because the boundary conditions are not Hermitian. And so these modes, you see, now we've got a second order differential equation with two boundary conditions. So we're going to get particular frequencies. And these frequencies will have imaginary parts, as we will discuss. That's the reason they call quasi. Because they're not quite normal modes, yeah, okay? So now that we understand what these quasi-normal modes are, let's think for one or two more minutes about how if you were so inclined, you would solve this problem, okay? What you would do is to take all of Einstein's fields around the black hole, expand them in some convenient basis that utilizes symmetry as much as possible. Clearly that basis, because our solution is symmetrical, is a basis of spherical harmonics. Einstein's equations have modes that have no indices, one index or two index along the sphere directions. So your expansion will be in basis of scalar, vector, and what I'm going to call tensor spherical harmonics, just two symmetric indices. The spherical harmonics. Once you focus on a particular symmetry sector, each of these modes, because you're event horizon, you've got infinity, can wiggle a certain number of times between event horizon and infinity, okay? It's like a particle in a box problem, okay? So our modes will be labeled by L, M, with sector, L, M, and then N. Which is roughly the number of nodes between 0 and infinity, okay? So in each angular momentum sector, you have an infinite number of modes. Now, let's look at this problem at large d. At large d, as we've said, this black hole has a thickness. We've got a boundary condition at infinity, but you see, once we've gone past a region r0 by d, we're just in flat space. So flat space can just be propagated, you know, you can bring the boundary condition from infinity down to r0 by d, just by propagation in flat space. So the effective boundary condition for this quasi-normal mode problem can be imposed at the horizon. And so at one at r0, and the other one at r0 plus r0 by d. So do you see that the problem of solving for quasi-normal mode is like the problem of solving a wave equation in a shell. A shell of radius r0, but thickness r0 by d, okay? Once you've seen this, it's very clear that modes that wiggle in this direction will have omega of order n, which is number of wiggles, divided by r0 by d, so r0 times d. As we discussed, if you look through the details of the boundary condition, omega will have both real and imaginary parts. We'll come to that in a moment, okay? However, depending on details, it may be that the boundary conditions allow modes that have no wiggles in this direction. If so, they will wiggle only in the sphere direction. Such modes, what should you expect their frequencies to be? Clearly you should expect their frequencies to be of order l, because that's l is the number of wiggles, okay? Divided by d, sorry, it divided by r0, so that's the length scale over which they wiggle, okay? So just from this picture of a black hole with this thickness, okay? You might suspect that with luck, you need luck because the boundary conditions may forbid modes that have no nodes. May require half a node, for instance, okay? Then all modes are heavy. But with luck, there are two kinds of modes in this problem. The light modes, those that have no wiggles in the small thickness direction. And all others, the heavy modes. Emperor Suzuki and Tanabe did a computation and this is exactly what they found, okay? They found that while most modes are extremely heavy with omega of order d by r0, both real and imaginary parts are very large, okay? Okay, any given l, okay? In the scalar sector, and for later use, I'm gonna record the quasi-normal mode frequencies. The quasi-normal mode frequencies turn out to be i into l minus 1, plus minus square root of l minus 1 in the two scalar sectors where my time evolution is e to the power i omega t, okay? So this corresponds to dk other than special that we have come to that mode. And in the vector sector, this was just i into l minus 1. The scalar two modes, the two is this plus minus, vector 1, tensor 0, no light modes, okay? And a whole slew of heavy modes. Which will be of no interest in this talk, so I'll just ignore them, okay? So what they found is that the spectrum of quasi-normal modes, the spectrum of quasi-normal modes of this problem has a few light modes separated by a mass gap. The mass gap is afforded d, okay? From an infinite number of very heavy modes, okay? Moreover, the situation is even better than it sounds in the following sense. All of the very heavy modes don't just oscillate very fast. They also decay very fast. So what have we learned? What we've learned is that if you pump a black hole in large d, and you wait till it reaches linear, you thump in a linearized way, okay? Then the problem might be complicated to start with for a time of order 1 by d, but then the heavy modes decay away. And you're left with a much simpler problem, a problem of only light modes. It's simpler because there's infinitely fewer degrees of freedom in the light modes than there are in the heavy modes, okay? Something else that Emperor Suzuki and Tanabe found when you look at their paper in detail is that these light modes have support purely within the thickness of this membrane. That is, they decay away like e to the power minus r, was that coordinate that we discussed before. As you move away from the event horizon, very much like the black hole solution itself decays away like e to the power minus r, okay? So there is a sense in which these light modes live entirely in the membrane region. So this thickness, I'm gonna call through these lectures, the membrane region. The light modes live entirely in the membrane region, okay? Now, whenever you have a situation in physics, where you've got a few light modes separated by a parametrically large mass gap from into the number of heavy modes, you should hope that there is an effective theory of the light modes obtained by quote unquote integrating out the heavy modes. This effective theory in our context simply just a classical theory. We're only interested in classical physics, okay? But it's going to be a non-linear theory, okay? We want to go beyond this linearized approximation of quasi-mormon modes to look at the full non-linear problem of the light modes. So, more precisely, what we're going to try to do in these lectures is to determine the effective non-linear dynamics of the light modes. Now, what might you suspect? You might suspect the following thing. These light modes live entirely in the membrane region, okay? So, this already might lead you to suspect the following, that the non-linear theory will somehow be the theory of a membrane with some degrees of freedom living on the membrane, okay? And that the reduction in number of degrees of freedom is just the fact that this membrane has one less dimension than the bulk, okay? This is, all these statements will be true, and this is what we will find, okay? So, what we are going to try to do is to take the observation made by Emperor Azuzuki and Tanabe of the existence of this mass gap at the linearized level and use that to motivate the construction of an effective non-linear theory of black hole dynamics. That effective non-linear theory will break down over time and distance scales one by D, but it will be perfectly valid over time and distance scales of order one, okay? And notice that because the heavy modes are all decaying, you shouldn't have to tune conditions in order to have this effective theory takeover as a description of dynamics, because if you do something very violent to the black hole, perhaps all modes are excited, but then you wait long in times, in units of one by D, the heavy modes all decay away, and you're left with the light modes. However, you may be left with a very non-linear configuration of these light modes. That's when our theory takes over, okay? So, this is the theory we're going to try to find in these lectures. The effective non-linear theory of the light modes, light meaning anything that varies on time scales one rather one by D, okay? The last thing to say about this is that as we will see, there is a sort of precise sense in which the modes we're looking at are decoupled from the outside. More precisely, we will see that the coupling between these modes we keep and the modes at infinity will be of order one over D to the party, okay? And so there is a sense in which what we're doing is a classical version of the ADS-CFT correspondence for Schwarzschild cycles. You know, Mildesign now is a regional argument for the ADS-CFT correspondence, took a decoupling limit of a near-horizon region of, let's say, the D3 brain, stuff happening here, didn't talk to stuff outside there, and a second description of that told you that that decoupled theory was in fact N equals 4 Yang-Mills theory, and that led to ADS-CFT. You could of course ask, are there interesting decoupling limits of Schwarzschild black holes? And in a finite number of dimensions, the answer is just no, there are not. There's nothing you can do here that doesn't talk to there quite a lot, okay? However, as I'm going to explain to you, if you're an infinite number of dimensions, a large number of dimensions, you can approximately achieve that. Decoupling is violated only at this order. So another way of thinking of what we're talking about is as a classical and approximate, because at any finite D, it's approximate. Totally really infinite D, where we don't want to go, right? Okay, this is a classical version of the ADS-CFT correspondence for Schwarzschild black holes. And just to say, what are the two sides? On one side, there's gravity. That's like gravity in ADS-5 cross S5. On the other side, there is this membrane theory, just non-gravitational theory. That's like the field theory, or more precisely like hydrodynamic approximation of the field theory. As we will now describe. Okay, any other questions or comments about this before we proceed? Please, why the coupling between the light? It's not really the coupling between the light and heavy modes. It's the coupling between light modes here and light modes far away. You see, we've so far talked about the modes just situated around the black hole. But of course, this is a theory of gravity. So you can have gravity waves, which can have arbitrary wavelength, and so it can be light outside the black hole. So the question we're asking is if we are excited to light dynamics here, energy conservation would allow it to radiate light gravitons. Does it happen? Okay, and the answer is that it happens, but only at this order. Is this clear? It's very much like the same question for D3-branes. Okay, you've got light modes in the near-horizon geometry, and you have light modes outside, but it's because of the tunneling barrier that they decoupled. Something very similar is going to happen. Is this clear? Question. We have all the boundary conditions locally, and every bit of the horizon is that things are going in-going. So if you have two horizons, or seven of them, and each of these horizons, you need to set a boundary condition, and it's always in-going, okay? Because what we're going to do is to cut out the inside of the event horizon as you will see as we go along, okay? So wherever you've got, you've cut out bits of space, you need to put a boundary condition. So everywhere you need to set the set of entries, okay? At least for the quasi-normal mode problem, this is what we're going to do as you will see. Any other questions or comments? Yes. There's also quantum mechanics. So quantum mechanics we're just not going to consider, okay? H bar is zero in these lectures. At least for the first motivation, the motivation for trying to get an approximation scheme for LIGO. We know that real-world quantum mechanics is completely irrelevant, so what are we going to, what? You can think of this as a way of generating an approximation scheme for solving the equations R mu nu equals zero, classical equations, okay? In the presence of lack of some, okay? Imagine just setting H bar to zero. H bar to zero, okay? Yes, okay. Can you study brain solutions? Now we're going to be in flat space in everything we do, and brains in flat space are always unstable, okay? Unless you've got some charge, at least just to play a pure energy brain in flat space is unstable because of the Gregory LaFlamme instability, okay? Now you can try to do that and try to understand the dynamics of their instability, and that's an interesting problem, but that's not where we're going to go. We're going to work with stable configurations, so yeah. Though as you will see, what we're going to get is a local theory of horizons, and then you can stitch up that local theory into more or less any configuration you want, but if you stitch it up in a weird configuration, you'll get unstable situations. But the real answer to your question is we can consider whatever we want. We're going to get local dynamics in this thing, and then, yeah, yes. Good, ADS backgrounds. This is something that MPRAN collaborators have considered for static and stationary black holes, and that some people were, or maybe are, working on. Bidisha I know was working on this at some point. We don't have any clear results on that. Should not be difficult to generalize what I'm saying to ADS and this at the backgrounds. Should be very simple, but I've not done it. Okay, yes. What is the role of Newton's constant? Has no role for a very good reason. The equations of motion R mu nu is equal to zero. Newton's constant does not appear in the equations of motion, okay? That's a matter of how you define your R naught. You can choose to scale the parameters in your problem so that there's no new constant. You see, this is an important question, let me say a little more about it. Einstein's equations in vacuum appear to have a dimensionless coupling, because we write down the Lagrangian. It's one by g times square root gr, let's look, let's, nonetheless, the classical Einstein equations have a scale, a scale in variance in them, okay? And that follows because although square root gr is not left invariant under scale transformations, it scales homogeneously under scale transformations. For quantum mechanics, scale transformations don't leave the action invariance, so for quantum mechanics, it's not scale invariant, but classically, at least there's a way of viewing the problem where classically there's a scale invariance. So it plays no role at all. So really, the equations we're solving are equations without a parameter, apart from d. R mu nu equals zero. Other questions, go ahead. Yes, you'll see as we go, we use it very crucially and you will see as we go along. Okay, now I have maybe how many minutes left for this lecture, but 15, five, 10 minutes, 10 minutes, okay. In the remaining 10 minutes, what I'm going to tell you about is an analogy, is an analogy, I'm gonna remind you of an earlier example of this sort that has greatly, that greatly guided our thinking for this problem. And this earlier example goes by the name of the fluid gravity correspondence and was a subject of lectures I gave here four or five years ago, I can't remember exactly when. Okay, I'm gonna give you a 10 minute reminder of that subject because keeping that in mind will be helpful to understanding everything that we do for over the next few lectures. Okay, so this is 10 minutes to remind you of an earlier story of an effective theory of event horizons that has worked out very nicely and has proved to be very useful. Okay, this is called the fluid gravity correspondence and the setting for this subject is the equation r mu nu minus r g mu nu by two is equal to minus d into d minus one by two g mu nu. Okay, so the setting for this other story was a slightly different equation than the one we're studying. It is, in fact, Einstein's equations with a negative cosmological constant where we are in d plus one space time dimensions. So I just wanna tell you that what I'm gonna do again over the next five or 10 minutes will have no logical link to what we're gonna do later. It's reminding you of an earlier story that works in a way that's similar to what we're going to work out, okay? Our problem is in flat space, not in ADS as no cosmological constant. However, in my analogy, there's Einstein's equations with a cosmological constant. This was studied in great detail because of ADSEFT. Okay, this is a consistent truncation of type II B theory and ADS5 cross S5 as well as many other gravity theories. Okay, and we'll study to understand that context. But I'm gonna tell you very briefly what was understood there and just as an analogy to keep in mind. Okay, here, this equation has black brain solution. Somebody was asking me about black brains. This equation has very nice black brain solutions, which I'm gonna write down for you. Okay, this d into d minus one by two is chosen. Ah, there's no, oh, there is. Yeah, this d into d minus one by two was chosen so as to ensure that ADS space with unit radius is a solution to these equations. I normalize my cosmological constant so as to make ADS space with unit radius solution to my equations. So this equation of course has very famous solutions, ADS space, which in Poincaré coordinates can be written as dr squared by r squared minus plus r squared dx mu dx mu where the metric contracting these is eta, the field theory metric, the boundary field metric. Yeah, that's ADS space. However, there's another almost equally famous solution of these equations, which is the black brain solution in ADS space. Okay, and that solution is dr squared over r squared into one minus pi t by r to the four. I'm sticking by the way now to d equals four, just for concreteness. It's easy to repeat in any dimension. Minus dt squared in a one minus pi t by r to the four. Okay, plus r, sorry, r squared, plus r squared dx i dx i, okay. This is a black brain solution, which Schwarzschild radius, this all is to the four, sorry, with the Schwarzschild radius r naught is equal to pi t. t has some constant, but of course I've called it t because it turns out to be the Hawking-Beckonstein temperature of the black, of this black brain, okay. So there are these nice black brain solutions in this problem, okay. And the black brain has the interpretation in the boundary field theory as being due to thermal equilibrium. Take the field theory, put it at temperature t. This is the gravity solution that governs that field. Now, you see, you might ask the following question. You might ask, what is the solution due to thermal equilibrium, but not in the rest frame that you are sitting in, but in the boosted rest frame. What is the dual to thermal equilibrated stuff moving with a constant velocity u? And the answer to that question is obtained just by boosting the solution. But before boosting this, I'm gonna rewrite it in another set of coordinates, okay, and you will see why in two minutes I choose these other quantities. These other set of coordinates are called any ongoing Eddington-Finkelstein coordinates. And you get these coordinates. Basically, one way of motivating these coordinates is that if you don't like metrics blowing up on you, okay. You might try to make a variable change that kills that block, okay. So the variable change that you might like to make is r is equal to r, but t is equal to v minus integral one by one minus pi t of r squared one minus pi t r to the four. Now, why do you wanna make this variable change? You see, what good thing happens with the variable change? Let's take d of this formula. We get dt is equal to dv minus one over r squared into one minus pi t by r to the four. I keep missing out this. Pi t r to the four times dr. Now you see that this guy, this dt square, when you square it, will have a factor of this quantity square. There was one factor of this, so we get one guy in the denominator. There's a minus sign, so that kills this. That's the point of making this variable change. That this unpleasant blowing up on you goes away with the variable change. Moreover, there is a term that is dv dr. What does that become? There's one factor here, but there's one factor in the denominator here. There's a factor of two because there are two terms you can get from here. The minus sign cancel this minus sign. So the metric becomes ds squared is equal to two dv dr minus dv squared. The term that was just dv squared from here. In a one minus pi t by r to the power four plus r squared dx i squared. Once you have this metric, we're gonna boost it. Boosting it means just take this black hole and make it move with velocity U mu instead of velocity one zero zero zero. Velocity is always relativistic velocities. So at rest is one zero zero zero zero. So this guy corresponds to the velocity U mu is equal to one zero zero zero or U mu is equal to minus one zero zero zero. Notice then that the metric ds squared is equal to minus two U mu dx mu dt, sorry dr plus minus, minus U mu U mu dx mu dx nu into, I'll be done in two minutes. In a one minus pi t by r to the power four plus r squared p mu nu. dx mu dx nu where p mu nu is the project on to space. Is the project on to space. Is equal to U mu U nu plus eta mu nu. That this metric reduces to this metric for the particular case U is equal to minus one zero zero zero zero U low as minus one zero zero zero. But since by boosting by performing a Lorentz transformation of the coordinates x mu, you can take minus one zero zero zero to any vector of any time like vector that squares to minus one. And since boosting is just a coordinate transformation, map solutions to solutions. This guy is now a solution for any constant U mu which is time like and such that U squared is equal to minus one. So far I've done nothing. I've just given you in sort of nice language the metric of a boosted black brain. The idea of the main idea of the fluid gravity correspondence was the following. Suppose we look at the metric that in which U mu is made a function of x. It's not a constant anymore. Continues to satisfy this condition. But this function varies as you move around. Also this T which was a constant is now a function of x. Roughly speaking, this might sound to you like it describes a fluid with varying velocity and varying temperature. Now, the idea was let's say when we take the variation scale of this problem to be large in units of the temperature, okay? Then this configuration or just configuration doesn't solve any equations. When you, however, it does solve the Einstein equations when U is a constant. Therefore, when U's derivatives are very small, it approximately solves Einstein's equations, okay? So you might use it as the starting point of a systematic perturbative construction of solutions to Einstein's equations in an expansion in the length of variation of U and T. And if you attempt that procedure, you find that you can indeed systematically correct these solutions to give you solutions of Einstein's equations in an expansion in derivatives of U and T, but not for every starting point values of U and T. Only for those starting point values that obey the following equation of motion. D mu T mu nu equals zero, where T mu nu is some number, which is irrelevant to us, T to the four times U mu U nu plus eta mu nu, or plus systematic corrections to this in derivatives. The first term is T, some number times T cubed times sigma mu nu where there's a shear tensor and goes on. Okay? What I wanted to tell you was that, you see, we're trying to understand in this context the effective theory of almost constant horizons. Horizons that are almost black and green, right, but very, very slowly. We started with a configuration that looked like it generates these effectively slowly bearing horizons, and then tried to promote this configuration, not a solution to any equations, a genuine solution of the equations. And we found we could do it, but not, it wasn't, anything goes. We could do it only if the starting point velocity and temperature fields obeyed an equation of motion. The equation of motion was a conservation of an effective stress tensor, which in ADS-CFT was the boundary stress tensor, but Einstein's equation gave us an expression for the stress tensor in terms of temperatures and velocities. Now, there are three velocity fields. You might think it's four, but it obeys the equation u squared equals minus one, so that's three, and one temperature field, and four equations of conservation of stress tensor. So there are as many variables as equations in this problem. So this procedure when carried through tells you that there is one-to-one correspondence between slowly varying horizon solutions of general relativity and equations of an effective theory of hydrodynamics. Okay? Because the equations of hydrodynamics are conservation of the stress tensor, but the stress tensor expresses functions of velocities and temperatures. Okay? So in that context, there was an effective theory of black hole horizons. You could substitute solving Einstein's equations for instead solving an effective non-gravitational theory. The non-effective non-gravitational theory was the theory of the event horizon was the theory of the Navier-Stokes, the relativistic Navier-Stokes equations. Okay? So this is the paradigm we will keep in mind as we proceed. In large D, we will try to do something like this. We will try to substitute solving Einstein's equations for solving some effective equations. Just like here, they will turn out to be effective hydrodynamic type equations, though with many differences as we will see. Okay, thank you, I'll stop my lecture here.