 Thank you. OK, hear me? Move this up a little bit. Hello? There we go. OK, so before we start with black hole perturbation theory, let's go back to our little post-Newtonian picture we had, like fried egg diagram. And I'm just going to zoom into one of the eggs, OK? So if I zoom into one of the eggs, here's my black hole. I said, this is the horizon. Imagine this is some sort of horizon, marginal trap surface, something like that. And what I want to do is I want to, oh, and way over there on the other side of the board, there's another black hole, OK, really far away. And now I drop you, or me, right here, crazy hair, because I'm very close to a black hole. But I'm happy because I can make measurements. OK, so the question is, what will I measure if I just sit there? Clearly, I'm not just going to measure the gravitational field of this black hole because this black hole is in the presence of another black hole. So there's going to be a tidal perturbation that's being generated by the companion. Now, we cannot really study this, at least not easily, within post-Newtonian theory. Because post-Newtonian theory, when we're dealing with black holes, it typically uses a stress energy tensor as that of a point particle that's distributional and has all these issues. So typically, remember, we had this region around each of the black holes that you sort of need to excise from your manifold. This is a whirl tube, it's like a cylinder, if you want. Because if I try to use my post-Newtonian expressions here, I'm just too close to this black hole so my fields are not going to be weak anymore. So my expansions in G are not going to be valid. So you typically eliminate this, and then the question is, OK. But sometimes you sort of need the metric everywhere, right? You don't just need the metric in between the two black holes. My other black hole would have been over here, right? So sometimes you need the metric everywhere. If you're trying to prescribe initial data for a numerical relativity for a 3 plus 1 evolution, you can't just tell the numerical relativists, oh, well, you have initial data everywhere, except in a two-spheres surrounding each of the black holes where the rays of the two-spheres larger on the horizon. Because they're going to be like, well, what do I do with that? How do I evolve forward in here? Moreover, if you're studying, for example, accretion disks, you need to know what the metric is very close to the black hole. If you want to study how light bends around black holes and understand, say, the image that the event horizon telescope took, do you guys see that picture that they took with very long baseline interferometry? You need to understand this based on here. So what we do instead is a technique based on black hole preservation theory. So what we do is we start a new calculation, and we say, well, I want to calculate the metric here. So if the other object weren't present, if this black hole is isolated, then the metric here would just be the metric of Kerr or the metric of Schwarzschild, the metric of your isolated black hole. But if I'm in the presence of an external perturbation, I don't care what that is. It's something. I have in mind always my happy black hole over there, but it could be anything. It's some external universe. I can parameterize the external universe in terms of the remand tensor generated by that external universe. And I can do an expansion in that remand tensor, assuming that the external universe is producing a small perturbation of the gravitational field right here. That is, the gravitational field here is dominated by the black hole with an external perturbation that's changing slowly. This is the field of tidally perturbed black holes. And as far as I know, the person that pushed this forward to the death was Poisson. And you can look at how he's constructed tidally perturbed metrics of Schwarzschild. So you have a Schwarzschild black hole here with an external perturbation. And he carries out this perturbation to higher and higher order. So I'm not going to go into detail for the construction of this, because I don't have time. But if you're interested in knowing what exactly do we do in this Poissonian region, you'll go there. Essentially, what you say is g is equal to, I'm just going to sketch it here, because I just can't resist, g is equal to g of your black hole. So say Schwarzschild plus some h perturbation. And this h perturbation is going to be expressed in terms of remand tensors and derivatives of remand tensors and second derivatives of remand tensors and so on and so forth of the external universe. And of course, if you just do that and you put it in here with the right index contractions, for example, see the metric is a rank 2 tensor. The remand is a rank 4 tensor. If you're an intimate contract, the remand tensor will have two vectors to have the number of indices work out. And so if you are over here, you're assuming that you're very close to the black hole. So you can assume you can contract this thing with two powers of the distance from the center to this object. And if you do that, this becomes a rank 2 object. And then it begins to look a lot like a tidal field, like a tidal force. Remand tensor, remember, is mass divided by separation cubed. You multiply by x square becomes dimensionless. And so on and so forth. The problem is, if you plug that into here, now at least it has the right number of indices, it won't satisfy the Einstein equations. So what you need to do is you need to multiply each of these things with some functions, f1, f2, and so on. And then you have to, which you don't know. These are unknown functions that have the property of going to 0 as you approach the horizon. Because as you get closer and closer to the horizon, then these perturbations should vanish. You'll be completely dominated by the black hole. And you go and you solve the Einstein equations, or you insert this into the Einstein equations, you get differential equations for these f's. And then you solve those differential equations for those f's with the right boundary conditions. And you have a metric for a tidally perturbed black hole. That's in a neighborhood around each black hole. Of course, I cannot take this metric and evaluate it here in between the two objects. Because now my expansion, that this distance is small, breaks down. But I don't have to use that metric in between. Because in between the two objects, I can use a post-Newtonian metric. So now all I have to do is I need to match that asymptotic metric, that post-Newtonian metric, to this black hole perturbation theory metric. So you do asymptotic matching or match asymptotic expansions. And ta-da, you have a metric that's valid in the entire domain. And that was my thesis, like 15 years ago. But the back in Einstein equations. Yes. So if you insert this into the Einstein tensor, you set it to 0. And then you'll get the different components of the Einstein tensor. And that defines differential equations for these f's. You also need an angular basis to this expansion. Typically, if your background black hole is Schwarzschild, you use tensors for the harmonics. I'm going to talk about that in a second. So this works all very good, except when that black hole, because this black hole, a little bit later in time, it's going to spiral in. These things are going to be moving around. It's going to be losing gravitational waves. So it's going to get closer. This one is going to get closer over here. Then a little bit later, it's going to be over here. And then that one's going to be over here. And eventually, they're going to sort of coalesce. So eventually, I'm going to have some sort of thing that merged. And what do I do now? I cannot use post-Newtonian theory, because I just be silly. I'm totally, absolutely wrong. I cannot really use this type of structure, because the objects that we're going around each other have already merged. So the remnant here doesn't have an external universe that's perturbing it. It got perturbed because of the merger. And now I want to understand the forward evolution of this. So what do I do? So this is what I'm going to talk about today. So I call this topic four. I call this black hole perturbation theory. And it's a little bit misleading, because when I talked about self-force, that's also black hole perturbation theory. And when I talk about metric of tidally perturbed black holes, that's also, by definition, black hole perturbation theory. But what I mean by black hole perturbation theory now is understanding the behavior of the peanut after the object has merged. Let's begin. So here are some good references for you to look over, because unavoidingly, when you're trying to make an omelet, you're going to break some x, which means I'm going to make mistakes on the board. And sometimes it's so to check that you're paying attention, but most of the time it's not. It just happens. So the good references that I've been going to be following are Birdie, Cardoso, and Starry Nets, so Birdie et al. This was an app J185. It's one of them, another one that. So that's like a review paper. It's like many, many, many, many, many pages long, but it's actually quite clear. Then, of course, you can go to Tukolsky's paper, which is not bad, but very, very mathematical. Oh, sorry. This one is app J185. This is not app J185. This is a review paper. So this is Birdie, Cardoso, Doso, and Starry Nets. It's a review paper. I forgot the journal, and I didn't write it down, but you can ask me later if you want. It's called Quasernormal Modes of Black Holes and Black Brains. Then Tukolsky's original paper was an app J in. This is Birdie. This is Tukolsky. Two, ah, two calls, deformable to that name. So this was in 1973. There's also Cocotas. He wrote a nice review as well with Schmidt. This is a Living Reviews in Relativity. And then there's Brian Nollert from Ireland, I think. CQG16 from 1998. So depending on which paper you look at, you'll get a slightly different take on this same topic. The first one, the third one, and the fourth one are more like review papers. The second one is just the classic Tukolsky paper that separates the equations. So with that said, let's begin with a simple example. The physicist's favorite example, Scalar Field. People have been asking me about like ADS and whatnot in this summer school. So let's consider a Schwarzschild ADS black hole because it doesn't really matter. Questions are going to be the same. So here's my Lagrangian. So you have some complex scalar field, because why not? And it has the standard kinetic term. You can add a mass term if you want. Not that it matters. For what I'm going to do, I'm actually going to drop this term because I don't like Klein-Gordon equations. They're bad. And then let's look at a d-dimensional. So this is going to be in d-dimensions. So d spacetime dimensions. So that's what the d stands for, d minus 2 divided by 4, d minus 1. Gamma is just some constant. And then there's going to be the Ricci scalar times phi dagger phi. So that's a coupling between my scalar charge, my scalar field, and which I'm going to call it the curvature. Just consider that. Let's see what happens. So if we consider that, look at, oh, and we're going to consider these situations. We're going to consider this Lagrangian. So what we're going to do is we'll understand the motion of this scalar field. I'm going to derive the equations of motion in a second. But I want to consider evolution of a scalar field of phi, d-dimensional, Schwarzschild, AdS, Blackhole. So I'm guessing that you have seen the line element. It's like the way I like to write it is minus f dt square plus f to the minus 1 dr squared plus r squared times the line element of the d minus 2 sphere. Need a bigger pen. So f here, bigger f, is 1 plus r squared divided by l squared plus some, or sorry, minus some r naught divided by r to the d minus 3. So l here is the ADM radius. r naught would be the mass in four dimensions. Gamma here is a coupling constant. Of course, constant. Thank you. This is Pewter's fault because he keeps on talking about ADM and it's in my head now. So the mass of this object, just in case you're wondering, is d minus 2 divided by 16 pi. The area of a d minus 2 sphere times this r naught to the d minus 3. And the area of the d minus 2 sphere is 2 pi to the d minus 1 divided by 2 divided by the gamma function evaluated at d minus 1 over 2. Now that it really matters, but those are all the sort of physical quantities. So this is the background. So I'm not going to consider the back reaction of the scalar field on the background. We're going to do just standard field theory where we just consider fields propagating on some background. Yes. That's to ensure that my equations look nice. That is irrelevant for what I'm talking about. So the equations of motion, you take that Lagrangian. You get box acting on phi equal d minus 2 divided by 4 d minus 1 times gamma r phi. Very simple. And I've set the mass to 0 because, as I said earlier, don't like Klein-Gordon equations. And of course, if you were to consider the back reaction of the field to the metric, then you would have to add to the Lagrangian, the Einstein-Hilbert Lagrangian. And there would be a variation that you have to do with respect to the metric that would give you the Einstein tensor with a cosmological constant term plus or equal to the stress energy tensor of that scalar field. I'm not going to worry about that part. So very good. So if I now take this and I evaluate it on this fixed background, background phi going to be equal to d times d minus 2 divided by 4. Now r is equal to just the cosmological constant term. So it's 1 over l squared and then gamma. And I have my phi term here. That's why I said, sure. So what we're going to now do is we're going to make an ansatz for the scalar field. So we're going to decompose this into plane waves. So we're going to do a sum on l and m of e to the i omega t for some constant omega times some psi of r divided by r to the d minus 2 divided by 2 times some ylm theta and phi and possibly other angles. So this is supposed to be the d-dimensional generalization of spherical harmonics, which is constructed such that it satisfies the angular sector of the Laplacian. And so if you do that and you plug these ansatz into here, magic happens. And this equation turns into f squared times 2 derivatives of psi plus ff prime times psi prime plus omega square minus v times psi equals 0, where v is equal to some potential that looks like this. d minus 2 divided by 4 d minus 4 r squared f plus 2 f prime over r r plus d gamma over l squared. So you're all physicists. And while I look for a chalk that I like, because I don't like any of these chalks, they're all very small, think about what that equation is. There we go. What's that equation? Physicists should recognize it like that. The Schrodinger equation. Good. Yeah, very good. It's the Schrodinger equation. And the way to see that is the Schrodinger equation is to go to a different coordinate system. So let me define the tortoise coordinate to be a star which satisfies the equation dr star by dr equals 1 over f. That's my new coordinate. And you can write it as an integral if you want. If you do that, this equation becomes d2 psi dr star squared plus omega squared minus v acting on psi is equal to 0. Kaboom. That equation I can solve. I could solve that equation a long time ago. OK, so it's a Schrodinger equation. So somehow what we're finding is that if you take a scalar field and you consider the evolution of the scalar field in a black hole background, then the evolution equation of that scalar field satisfies the radial part of the eigenfunctions here, if you want, satisfy a Schrodinger-like equation. And I didn't have to do it in ADS or in d dimensions. I'm just showing off. I mean, you can do it in d equal 4. And you can do it in asymptotically flat boundary conditions to take L2 infinity. Yeah, yeah, so the b here was just to indicate that this is the box associated with this background metric which I wrote down here. Well, so this was, if you just do it as a gen, if you don't consider a fixed background and you just allow the field to back react on the metric, then that one would have to be also enhanced for an equation for the metric tensor. This is what I was saying in words. I didn't write it down. OK, so this looks cool. This is how a wave scatters off H-R-Shoot block. So it's a scalar wave. But the metric is like a scalar field or a bunch of scalar fields. So you sort of suspect that if you do the calculation right and you consider what happens to the vibrations of spacetime after my peanut forms, they should satisfy some sort of Schrodinger type equation. Intuitively, you would want that to happen. And indeed, that's exactly what happens. So now let me derive for you how that comes about. Now, I'm going to do this calculation on a Schwarzschild background because I want the lecture to take less than Hubble time. But you can definitely do this on a Kerr background. If you do it on a Kerr background, then you get something called the Tukolsky equation, which tells you how the perturbations or how psi 4, which is one of the Neumann-Penro scalars, what equation the psi 4 quantity satisfies in a decoupled way in the sense that how this psi 4 decouples from the other Neumann-Penro scalars and the other metric degrees of freedom. I'm not going to go into any of that. Instead, I'm going to talk about Schwarzschild. That's much easier. So Schwarzschild perturbation theory. So we begin by decomposing the metric into your background metric, Schwarzschild, plus a metric perturbation, H mu nu. Moreover, it's convenient to decompose H mu nu into two terms. Well, first of all, to decompose it into sums over lm's. And I'm going to tell you what those sums, why I've done this. So there are some quantity H mu nu lm with an a here. And then there's a quantity H mu nu lm with a p here. So what I am doing is I am decomposing the metric perturbation into a piece that's polar and a piece that's axial. And by what I mean by that is that if I act the party operator, meaning like I flip my triad, so I take x to minus x, y to minus y, z to minus c, I act this on H, lm, polar, or any quantity that's polar. Let me write it as quantity a, lm, that's polar. Then I'm going to get minus 1 to the l. Time my quantity back. Whereas if I act the party operator on a quantity a, lm, that's axial, then I'm going to get minus 1 to the l plus 1 of that axial quantity. That's how I define the axial and the polar parts of any quantity, OK? So I can always do that. And now what I need to do is I need to do something similar to what I did over here. I need to decompose my field in spherical harmonics. The little problem is that this thing here is not a scalar. It's a tensor. And so you cannot use just regular scalar spherical harmonics. But there's a generalization to tensorial quantities, which allows me then to, so there's a generalization, generalization spherical harmonics, called tensors spherical harmonics. So your scalar one is still ylm. Oh, and I mean four dimensions now, by the way. And I mean asymptotically flat spacing, simply because dear god, OK? So there's a scalar one. There's a vector one. So the vector quantity is yalm. There's two of them. And there's salm. And this yalm, this is just the yalms that you know and love. But you take a covariant derivative of the scalar spherical harmonics with respect to the metric on the two sphere. So that's what the column means. So definitions here. Column means covariant derivative with respect to omega AB. Omega AB means the metric on the two sphere. So like 1, 0, 0 sine square. So I can definitely compute or define Christoffel symbols associated with this metric. And from that, I can construct a covariant derivative just like I do in GR. And then once I have that, I just take this covariant derivative on the y. But of course, this symbol a, so a, b, c, dot, dot, dot, these are going to span only the angular sector. So theta and phi, because otherwise you cannot. Otherwise this doesn't make sense. So S a lm is defined as omega AB times yb lm. And so this quantity happens to be axial. This quantity happens to be polar. And these things are called vector spherical harmonics. And then there's a generalization for spatial tensors. So there's three quantities yab, which is ylm times the metric of the two sphere. And this quantity is polar. Then there is zab, which is the ylm with two covariant derivatives plus an extra term that you need to add in order for the equations to be satisfied, which is also polar. And then there's a last term, Sab, which is the covariant derivative of the vector spherical harmonic, column B, symmetrized. And this quantity is axial. So I define these quantities. Yeah, how is that? How is it racer lowered? So you define an inverse to your omega AB such that omega AB times the inverse of omega AB gives you the chronicle. No, so if you raise, one of these quantities are going to be factors of sine square. Let me continue, and I'll tell you about it afterwards. So I always am not going to finish. So just as a note, tensor spherical harmonics, and I'm using this word to imply scalar vector and the spatial tensor ones here, satisfy orthogonality conditions that I'm not going to write down, but they're similar to the scalar ones. So things like, for example, integral on ylm, yl prime d omega is like delta of ll prime mm prime, things like this. Except that, of course, now you have indices, so you have to be a little bit more careful, but there are similar orthogonality relations. And what is nice about this is that these angular tensor spherical harmonics satisfy the angular sector of the Einstein equations when the Einstein equations are evaluated on a spherically symmetric background. So that suggests that I should use for my ansatz for the axial thing a matrix. I'm going to split it into two sectors. I'm going to split this into the tr sector and into the theta phi sector. And in the theta phi sector here, I'm going to use the axial. Did I label this backwards? Maybe I labeled this backwards. It doesn't really matter. So in this box, I'm going to use some halm times sa lm. And here I'm going to use some hlm times sa b lm. And the idea here is that this ha and this h are functions that depend on t and r that I haven't yet determined. And the capital index a here, just like the small ones here, span theta and phi, the capital a, b, c, dot, dot, span the tr part of the space. So for example, ht theta of this quantity is going to be equal to, so I take this box here and I set for the capital a I set t and for the little a I set theta. So I'm going to get ht lm, which is a function of t and r, times s theta lm function of theta and phi. So that's what the notation means. Sorry, I didn't invent it. So I'm going to do that for one of the terms. And for the other term, I'm going to do something similar, but I'm going to use the other terms. hlm mu nu p. I'm going to write as ha b of t and r, lm here, times the ylms. OK, maybe Nico should choose a different part of the board to write this. How about I write it right here? So h mu nu polar lm, I'm going to write it here, this, that. OK, and so over here I'm going to use some functions ha b lm of t and r, times the ylms. Over here I'm going to use some other functions, p a lm, times y a lm. And over here I'm going to use r square times k lm, this again of t and r, of t and r, times y a b lm, plus r square g lm of t and r, times z a b lm. OK, so for example here, if I wanted the t theta part of this quantity, then I would go here and I would put for a t and for the other a theta. So I would get p t lm of t and r, and then y theta lm of theta. Hope the notation is somewhat clear. It's this thing. So this is, so h sub t here is the t component of this two-dimensional vector. OK, and this thing here, so this thing here so it's a theta component of this two-dimensional vector. Capital h, here, this is a scalar, this is a function of t and r. Yeah, so I don't need different scalars here for the different components. It's a bit of historical reasons why people did it this way. Remember, people were doing things without computers, everything was being done by hand, and they just chose this particular parameterization of the metric perturbation, and it has stuck with us, so. This is not the only way you could do this calculation. This is what we're going to do. OK, so now comes the remarkable part. And by that, I mean like, I never thought about it. Surely, you can come up with something better. Probably because Nikon made a mistake. Also, it doesn't really matter, right? You can absorb it into the h. It's just a matter of like how many terms you want to pick and keep it. But you just sort of, you know that that thing is going to scale as r squared. So if you want h just to go to 1 in flat space, then that's why you factor this part. Anyway, so then you take this, you plug it into here, and then you take this, and you plug it into here, and then you take this in threshold coordinates, and you plug it into the Einstein equations, and then you suffer a lot because you have to calculate the Einstein tensor. But because these angular functions are constructed such that the angular sector of the Einstein equations on a spherical background separate, you know that the Einstein tensor will also be proportional, different components of the Einstein tensor will also be proportional to these quantities. So if you do the calculation, you can verify this. And indeed, if you look at the TR part of the Einstein tensor, obviously to 0th order in the perturbation, the Einstein tensor is identically 0 because Schwarzschild, the Schwarzschild metric annihilates the Einstein tensor operator. So this is to first order in h. So you get some quantity gLm times yLm's here, for example. So this thing here is a 2 by 2 matrix, symmetric matrix, with derivatives. All of the functions is h, a, lm. So this is with time and r derivatives only of the h, a, the other h, the h, a, b, in principle, all of the metric functions that you're trying to solve for. So there's some complicated mess with t and r derivatives of that. But what's nice about it is that the Einstein tensor separates like this. The angular part can all be absorbed into this yLm. And then you just have some complicated differential system for the TR sector. Similarly, the off-diagonal piece of the Einstein tensor can be written as calligraphic ga, again, some other complicated mess, times yLm, plus some other complicated mess, times sLm. And finally, if you look at the spatial spatial part of the Einstein tensor, you get some gLm times yBLm, plus some hLm, calligraphic hLm, times zABLm, plus some calligraphic iLm times sABLm. Apologize for the notation again, but this calligraphic hLm is not the same as this hLm. This one's calligraphic and that one's not. This gLm without indices is not the same as gLm with indices, with one index, which is not the same as gLm with two indices. But all of this quantity is this one, and this one, and this one. Well, they're all not. OK, let me erase this. So this one is not a 2 by 2 matrix. Obviously, it's a two dimensional vector. So it's this one, but both the components of these quantities are all some expression that depends on only time and radial derivatives of these metric functions. And similarly, g and h and i here are scalars. They're not vectors or matrices. And they're, again, some function of derivatives in t and derivatives of r of these metric functions. So that's what's nice of using this angular basis to expand this metric perturbation. I mean, it's constructed such that this will happen, such that you can separate the tr sector from the theta phi sector. And I'm sure it is related to SO3 and other mathematics that you can probably tell me more about. What I care about is that that separation looked really cool. So there's two miracles. You see, one miracle is that I managed to separate the tr sector from the theta phi sector. That's not, well, I mean, maybe it's obvious, but I mean, it's not trivial. But this is still a couple system of equations because each of these horrendous quantities that depend on the derivatives of tr, t and r of these quantities, they contain the derivatives of all these quantities mixed together. And it's a mess to solve. You still have, in principle, 10 equations for 10 unknowns, which are these metric functions that are over here. And you can impose a gauge if you want. Not that you have to. So I'm not going to. But there's a gauge called the Regi-Wheeler gauge that people sometimes speak, where you essentially, see, you have more than you don't need to have all of these degrees of freedom present. At the end of the day, you know that you have, for the metric tensor, 10 degrees of freedom. Well, we have 10 degrees of freedom. 4 can be eliminated through our coordinate. The pendulum can be eliminated via coordinate transformations. 4 are constrained by your constraint equations. And only 2 are really free or propagating. So you can impose a gauge to eliminate, say, 4 of these quantities. And the Regi-Wheeler gauge is one of these choices. But we're not going to do that. Instead, what we're going to do is we're going to note that since I'm in vacuum, each of these things has to vanish. Each component of the n-centers has to vanish. Moreover, for example, since these components have to vanish, that means since the YLMs can't vanish, because that just would be silly, then each component of this differential operator has to vanish. Similarly, since these components have to vanish, then this sum has to vanish. But remember that YA is orthogonal to SA, in the sense that if I integrate them, I get 0. That means that this has to vanish independently of this. And the same with this. This has to vanish independently of this, which has to vanish independently of that. Which means each of these things have to be 0, which means I can linearly combine them, if I want, because 0 plus 0 is still 0. So if you take F, so I'm going to define F just to be here. F is equal to 1 minus 2m over r for Schwarzschild in Schwarzschild coordinates. So if I take F divided by r and I multiply it by 2 over r times 1 minus 3m over r times this calligraphic ILM plus 2f times the calligraphic h, the r component, lm minus f times the radial derivative of ILM, then you would agree that because each of these three terms depends on these quantities, each of which has to be 0 independently, this whole thing has to be 0. And you do the math, and ta-da, you get minus dt squared plus dr tortoise squared minus a potential acting on a function that I'm going to call psi rw lm has to be 0. And I have to tell you what these things are. So this is vrw, vrw lm is equal to f over r squared l times l plus 1 minus 6m over r. And psi lm rw is minus f over r times the r component of the hlm minus the radial derivative of the hlm function divided by 2 plus the hlm function divided by r. So what have I done here? How did I get this equation? Magic. So what I did was I said, look, each of these things have to be 0. And I want to decouple the system of equations to find an evolution equation for the two propagating degrees of freedom in my theory. There's only two degrees of freedom that are not constrained, the plus and cross gravitational waves, effectively. And what I have found is that one way to decouple these equations for one of these degrees of freedom is through this combination. I'm not claiming this is the only way to decouple the equations. It just happens to be one of the ways to decouple the equations without imposing any gauge. And so by combining things like that, all of the gauge degrees of freedom cancel, exactly. And what you're left with is a wave equation with a potential, like Schrodinger equation, acting on some combination of the metric functions. It is a linear combination of the components of the instant sensor. A very clever combination with very clever, like if you put here a 4, then things on the couple, for example. And so this thing here is called the Regi Wheeler. Italians make correct my spelling here. Tulio, right? Tulio Regi. Regi. Sorry. And Wheeler equation. This thing is called the Regi Wheeler potential. It's completely specified by the background and by the LM mode of your spacetime. This thing is called the Regi Wheeler function, or master function. And you can prove that this thing is gauge invariant. So you can pick a different gauge, you get the same thing. This is truly one of the combinations of the two propagating degrees. One of the combinations of metric functions that gives you a true propagating degree of freedom regardless of gauge. And if you look, it depends on big h and little h. So it depends on these functions. OK? So it's contracted completely from this part of the metric. So you would expect that there must be some other function that can be constructed entirely from this part of the metric. And indeed, you can do that and you find, do you want me to write the other linear combination, or do you want to just write down the equation? Choose your own adventure. Well, I was going. Yes, so there's another combination that I can write down that decouples for the other. So do you want me to write that down, or should I? OK, I'll write that down. OK, so this one's more difficult. You take 4 times f squared divided. By the way, this is not written down anywhere. I do not understand why. It's left as an exercise for the student, which I had to do as a student. And it sucked. But we have maple now, so you can do it in maple. I don't know what it is. Or maybe it is written. I just didn't know where it was. So 4f squared divided by some lambda, and I'm going to write down, the glm sub r of the field equations minus 2f divided by r times the hlm of the field equation, plus 2f divided by 2 lambda l times little lambda l plus 1 times, prepare yourselves, minus 2r squared f dr of calligraphic mlm plus 12m divided by lambda l f squared times calligraphic glm rr plus 4f over r, calligraphic glm, plus r over 2 lambda l, 4 lambda l lambda l minus 1 plus 12 2 lambda l minus 3 times m over r plus 84. That's correct, 84. m squared divided by r squared. And all of that times calligraphic mlm. Close that, OK? Yes. And now let me do some definitions over here. So little lambda l is just l plus 2 times l minus 1 divided by 2. Capital lambda l is equal to little lambda l plus 3m over r. And calligraphic mlm is equal to minus calligraphic glm tt divided by f plus f times calligraphic grr lm, OK? So what I've done here is I've constructed some mlm that is just a combination of gtt, of gtt and grr with factors of little f. That's what this m is. Then lambda sub l, the little lambda sub l, that's just a constant that depends on the l number of your harmonic. Capital lambda l is like little lambda l, but it also has a 3m over r dependence in it. And then you just put this thing together. Oh. And that's also equal to 0, right? It's also equal to 0 because it's a linear combination of 0s. So that in my life also equals 0. So if you do that, well, you have to decouple the equation. So you know that you've constructed some sort of differential equation for some combination of these quantities that combination is gauging variant. So one way to do it is to find what other combination can you make of these quantities that would also be gauging variant. So you can do that. Once you have that combination, you can rewrite the different components of the Einstein tensor in terms of those combinations. The equations will simplify a little bit. And then you have to try to figure out, how do I combine different components of the Einstein tensor to the coupled equations? How exactly did Sir William and Griff do this? Or Regi and Wheeler? I mean, the Regi-Wheeler one is easy compared to that one. I have no idea. They didn't even have computers back then. So what's remarkable, absolutely remarkable, is that if you do this, what you find is that, so I'm going to take that last equation over there, it becomes, so let's call this equation 1, so 1 becomes minus dt square plus, you guessed it, d2 r star square, so sorry, d2 r star minus some potential times some other function, zlm. This must be 0. And it turns out that this potential, cm lm, is equal to r, sorry, f divided by r square lambda square l times 2 lambda l square 1 plus lambda l plus 3m over r, which you recognize as capital lambda l. I'm just writing it out. Plus 18 m squared over r square lambda l plus m over r. And this psi lm, zm, is equal to r over 1 plus lambda l, the klm plus 1 plus lambda l glm plus f over capital lambda l fhrlm minus r times the radial derivative of the k function lm minus 2 over r 1 plus lambda l, the r derivative. So very similar, right? Very similar, very similar to this one. And so this was done by Cerulean-Moncriff. So this thing is called the Cerulean-Moncriff equation. This thing is called the Cerulean-Moncriff potential. This thing is called the Cerulean-Moncriff master function. And what you notice is the Cerulean-Moncriff master function is made of combinations of components that appear here. Furthermore, you notice that it satisfies the same type of wave equation as the Regi Wheeler function does, except that the potential has changed. Furthermore, you notice that the potential only depends on properties of the background and the l number of your mode. And if you say that psi lm, cm, or psi lm, or w goes as e to the i omega t, then you can convert these wave equations into a Schrodinger equation for your radial eigenfunctions, just like I did for the scalar field. So I have done this for you guys in an arbitrary gauge. If you put in a gauge, then things simplify like a lot. But I just wanted to give you the expressions in general. And this is cool, because if you solve those equations, what are you solving for? What are we solving for? So you remember what we are solving for? We are solving for the pinot. We have this pinot thingy that got produced after the merger of two black holes. And it's in pure vacuum. Now it's going to be a Schwarzschild black hole with a perturbation. So maybe let's take the pinot to not be so deformed so that I can treat it as a small deformation. But still, some sort of form black hole is going to vibrate. It's going to produce gravitational waves. This is what this is telling us, that the perturbations of the background satisfy wave type equations. And those wave type equations will radiate gravitational waves. So you expect that psi Cm and psi Rw should be related to h plus and h cross, the two metric perturbations. And indeed, you can show that h plus minus i h cross equals 1 over 2r sum on lm l plus 2 factorial l minus 2 factorial square rooted of psi lm cerulemon-criff plus i. OK, I'm going to introduce a new one. Psi lm Cpm times ylm minus 2 of theta and phi. So this thing here are spin minus 2 or spin weighted spherical harmonics. OK, so spin weighted spherical harmonics are functions of theta and phi that satisfy equations that look a lot like the equations that spherical harmonics satisfy, but they have in generalized by the addition of a spin parameter that's related to something I don't want to get into. So let's not talk about it. But it's just some function of theta and phi. And this psi Cpm, see, I told you that I decomposed these equations here to construct this regi-wheeler. But this is not the only way to do it. You can construct another master function if you want. So psi Cpm R over lambda l of hr, t minus ht, r plus 2 over r ht lm, for example. And this psi Cpm equation, if you combine the right components of the instant tensor, will still satisfy this equation with the same potential, but it will be acting on psi Cpm. So the equation is the same. It's just the metric degrees of room that you play with are slightly different. But bottom line is, if you go and you solve for those wave equations, you're learning something about how the spacetime is vibrating in order for the spinnet to relax to its final state, to its final configuration. And so price, and others have shown, that's what the pistons were priced, Cis, Cunningham, and Ms. Moncriff, again, show that the end state of this vibrating black hole will indeed be Schwarzschild, for example. And then the modes are going to be radiated away in quiescent normal modes. So these are the quiescent normal modes. So this is the expression you have for the quiescent normal part of the gravitational wave spectrum in vacuum. And from that, you can go and you can calculate fluxes if you want, and whatnot. So this ties back to the self-force problem. Because now, if instead of doing this, what I had was a black hole that's perturbed by a small particle, I can do the same thing. I can still say G is Schwarzschild plus a metric perturbation. I can go through the same procedure. And now, when I combine these things, I'm not going to say that's equal to 0. I'm going to have to say now that's equal to some function of the stress energy tensor. But that's the same equation with a source. So an Emory, if you want the gravitational waves emitted by the black hole due to an Emory to leading order in the dissipative part of the self-force, satisfies sourced wave equations like that that you can then solve numerically to find the gravitational waves emitted for a given geodesic, which you can then use this gravitational wave. You can take the time derivative of this, square them, and compute the flux carried away by gravitational waves. So now you have E dot, and you have L dot. And in principle, you can try to construct something like Q dot. And now you can correct your geodesic, the geodesic you used to solve this equation. This is one of the clutch methods. In fact, that's exactly what Scott Hughes proposed in the 2000s. Right. Good. So I've tried to cover a lot of material, because Paolo asked me to talk about black holes. And I know I probably didn't do the best job providing all of the details of the calculations that I presented. That's why I'm showing you these references. But I did that because I really, truly believe that as grad students, what you should try to do is learn as many new techniques as you can possibly find. There's a lot to physics and to gravitational waves and to geometry beyond just specifically post-Newtonian or just specifically, you know, quasi-normal modes. And you never know what techniques you're going to need in your research later on. And when you get older, like me and us, you'll find that the amount of time you have to learn new things, it shrinks exponentially with time. It's like a decaying function. So this is really your opportunity to learn new stuff. So just absorb as much as you can. Take these references that we have all been sending you and look them over. Try to learn the basics so that in the future, when you run into problems where you need to use things that are a little bit outside of your expertise, you know where to go, you know what to do. And you're a bit broader than just a pure specialist in that one specific calculation. So it's been my pleasure to teach you all. Thank you very much for staying awake. And we'll end there.