 We'll talk about black hole stability. Thank you very much. I hope the microphone is working and can hear me. So let me start by thanking all of these program organizers for putting this great program together with many excellent lectures, which I'm unfortunately not going to be able to match in excellence. And Paolo for the kind introduction. So it's been of course a great pleasure working with Paolo and Leonardo Sinettore. So actually I started out as a physics student when I was an undergraduate, but then eventually converged towards mathematics. So I really do math now. But I hope I can get across some of the ideas that are used in modern mathematical treatments of the underlying analytic problems in this lecture. So as they relate to black hole stability. So I put global and micro analysis aspects into the title actually, in part because I hope to spend a good amount of time on the actual underlying analysis. Although I know that in real life the way things go is I always run out of time. So we'll see. All right. So it's great that you already had a very nice introduction to a lot of topics in GR. So what I talk about is joint work with, just say right away, so it's joint with Peter Hintz with my PhD student a few years ago and partly with, I would probably say much less about this, Dittri Hefner. And so again this is about black hole stability. And so various notions have come up, in particular in Bob Boyd's excellent lectures. So this is purely classical GR that I talk about. So the setting is you have a manifold M and the metric G. So this is Lorentzian. And I'm sorry but I'm going to use this signature convention that it's mostly minus. So I'm going to use this signature convention. It doesn't really matter. So I make it four-dimensional right now. It doesn't really matter, but sometimes I might write on a formula that should slightly change in general dimensions and I don't want to think about it. And this mostly doesn't matter either except again some signs come up sometimes. Okay. So we have four-dimension manifold. We are hoping to get some sort of Lorentzian metric on it. And you're hoping that this will solve Einstein's equation and it's not going to vacuum. So Einstein's equation looks like the richer curvature of G. So I'm going to put in a cosmological constant plus lambda G equals zero. So it's the cosmological constant which is fixed. So it's given. And so what you're supposed to do is to find G. And it already has come up in some of the lectures that the kind of equation this is is some sort of, at least once you start thinking about it the right way and do some things to it, there's some kind of a wave equation in this Lorentzian signature. So this involves up to second derivatives of G if you write out the curvature tensor. And so this is some sort of a partial differential equation. And the kind of PDE it is, if you think of right, so after some work, which I will get into, this is a wave equation. So it's not honestly a wave equation, but after some work it is. So the issue is exactly the, so maybe I just say the issue right away, so the issue, this has already come up as well, but the issue is the edificium morphism invariance or gauge invariance or gauge issues. So it's not honestly a wave equation. You have to do some gauge fixing to really make it into a wave equation. But anyway, pretending it is for now, you might want to think about the initial value problem which would be that you are given a hypersurface sigma, so co-dimension one, so it's three-dimensional in this case. So I mean the equation which is so for g, so if you think about having g, the picture would be that here's my m, whatever my m is, this is my manifold, and here's my sigma, and it's going to be a space like hypersurface. So sigma is a hypersurface, and you are given two things. One, a Riemannian metric on sigma, let's call it gamma, and two, a symmetric two-tensor, a symmetric two-tensor again on sigma, so let's call this k, and then right what you want to do is solve, so this is Einstein's equation, vacuum, but actually we're EE, and then solve EE with these initial conditions appropriately interpreted with the following extra conditions, so with conditions, so these initial conditions, that the metric g one, so it's this, g pulls back, so restricts, in other words, it's a tangent space, so pulls back to sigma as, so the pullback of g to sigma is minus gamma, so if you just restrict it to sigma, it's minus gamma, and the minus sign is because of the signature convention. The only place for the minus sign is annoying, but anyway, there it is, and two, that the second fundamental form, the extrinsic curvature form of sigma in m, in mg is k, so that's the initial value problem. Okay, and then I think it came up in the Christian lecture that you cannot expect to do this in general because if you had the solution of the problem, there would be some relationship between this gamma and k, so we need, so certainly we need that gamma and k satisfy constraint equations, otherwise no hope, solving the equation, the initial value problem. Okay, so that's the general setup in gr, and then the, so in general, this is of course really hard to solve and talk about, so something that's easier, and so this is what you do when this is a mathematician, if you have a very hard problem, then you simplify it, so what's easier, so very hard in general, so what you want to do is simplify, which is to say start with a solution G zero of Einstein's equation and then take the corresponding initial data, initial data, so this would be that's what I'm gamma zero and k zero on some sigma, okay, and then ask that if you perturb this initial data, while of course satisfying the constraint equations, perturb gamma zero, k zero, so what does perturb mean, this means that change them by something small and of course you have to have a notion of how you measure small and so this needs a, needs a scale, so needs a scale, some sort of a norm mathematically, so to say gamma k, so gamma is supposed to be close to gamma zero and k is supposed to close to k zero and ask if Einstein's equation has a global solution on this manifold M, so M is just this background manifold that came from this initial metric with these initial conditions, say initial conditions gamma k on sigma, okay, so this would be the, or one is, so first of all you want to know whether there is a global solution and moreover, if it exists, is it closed, globally closed, close to G zero, so again you need close, you need some notion of measurement, it's globally close to G zero and moreover, does it maybe even tend, even tend to G zero as you go out to infinity, as one goes to infinity, so this would be the stability, these two would be a stability statement, so this is stability, so if so, you would say that the metric is stable as a solution of Einstein's equation, so you can also add Maxwell and all the usual things and then you can ask similar questions, but it usually doesn't really introduce too much extra complexity, so vacuum and no fields is just as good for this particular problem, okay, so all right, so this has a very long history and I should say a little bit about this, so in fact the first mathematical treatment of this, so full proof that something is stable, I mean okay, so what's the simplest solution of Einstein's equation, one might think it's Minkowski space, in some ways it is, but it's actually not so well behaved as an analytic problem, so in fact the first result was due to Hamlet Friedrich in the 80s who showed that the sitter is stable, so I should say that, so there are subtleties of exactly what you mean in various places here, like in this close two or tent two and so this can be different in the different statements I will describe, so I will state, I will describe in more detail the last thing that I want to talk about which is the black hole statement, but I won't give the precise stability notion for this first result I mentioned. So Friedrich in the 80s, I mean of course I should say that there was a lot of work in physics and math in this field before, but this is the first general no symmetry class, et cetera kind of statement and then what's mostly known in math is the famous Christodoulou Kleinermann work which is that, so this was the early 90s which is that Minkowski's space is stable and this Friedrich's paper is a normal paper of I don't know, let's just say in the order of 50 pages so I think it's more than 50 but not a whole lot more, this is a very big book, so in some sense Minkowski is, I mean at least by the page count Minkowski is a harder problem and I hope to say something about why these sort of things are hard. So I should say that the sitter, I'm not sure if the sitter came up in an earlier talk so let me just quickly say what the sitter is. So what you do, so this is the sitter saying you can be in n dimensions, doesn't matter, I mean the equation is slightly different looking but anyway, so what the sitter n is is just the one sheeted hyperboloid in Minkowski's space, so let me write this r1n, so Minkowski, a one higher dimension. So it's going to look like this, so this is the two dimensional, the sitter in three dimensional Minkowski, so it's a one sheeted hyperboloid and what Friedrich showed that in an appropriate sense if you perturb initial data for the sitter, and then hope to solve, in fact you get the global solution and it will, so if you put it a bit, it will be close to the sitter and in fact in an appropriate sense it will, so this requires effort to make precise, you can think of this decaying to the sitter. Okay, so, and then maybe just mention, this is a big book but there's a significant simplification due to Lydblad-Edronianski, so this was the early 2000s, so this is simplified, simplified this work of Christopher and Kleinman but getting weaker conclusions in the process. Okay, so, let me erase this. So what I talk about is the black hole statements, so I will talk about mostly the desitory version, so one thing I didn't say, okay so I wrote down the metric, but the point, I mean the sub-manifold of Minkowski, the metric is the pulled back metric and the point is that this has positive cosmological constants, so this is lambda positive and this is lambda is zero. That's right, that's right, so I will say a bit about this later on, so you are solving nonlinear wave equations, so it's an assembler of nonlinear PDE but nonlinear evolution equation, there's no reason, even for an ODE, there's no reason for solutions to exist globally, so again they can very easily blow up. Okay, so what I will talk about is, I'll talk about black hole metrics and there's two types corresponding to this lambda positive and lambda equal to zero, so the lambda positive, these are the Curd-Desitter metrics, or desitter curve, I don't know which way you want to call it, again lambda you think of as fixed and then, so these depend on two parameters, M and A, so this would be the black hole mass and this would be the angular momentum and then of course there's the lambda equal to zero version which is, so let me call this GMA, I'm just going to go up both of them GMA, so again it's the same thing, so I have an MA and they have exactly the same meaning but you think of this as a rotating black hole in space time with the positive cosmological constant, so in ds and you think of this as a rotating black hole in Minkowski, so lambda is equal to zero. Okay, so these are the real, the G zeroes in my previous notation, the initial things that you want to think about and then you ask exactly these questions, so you're going to perturb the initial data for these metrics a bit and ask if you get a global solution and what does it behave like, okay. Now, one thing I should point out that I think also came up in some ways is, so from a math perspective the first thing to think about is the underlying manifold on which you are trying to do something, so it's more important than something like a metric, so what's the underlying manifold? And so, let's talk about curde-sitter, so our manifold, so I'm not going to write down the metric but maybe I'll write down the special case when a equals zero, then the metric, so this is Schwarzschild-de-sitter, GM zero, then it looks like something like this, so the real linguist version is something like this, and I think it was called F yesterday in one of the talks, dt squared minus mu r inverse dr squared, but so remember I have the negative sign convention, mostly negative, so it's the standard round metric on the sphere, the two sphere, and this mu r, so it's the generalization of the formula for Schwarzschild, so Schwarzschild is the lambda equals zero, so formally it's exactly the same thing, just let lambda equals zero, so this is the Schwarzschild-de-sitter, and so one thing you see, right, is that there's this division by mu r, which can be an issue because this mu r can vanish, so the way this mu will look is that this is my r-axis, I'm going to plot mu, I'm going to plot mu in some places because I can't remember how it looks globally, but in that case, this will go to minus infinity as r goes to plus infinity, so you will have the largest zero, I'm going to call it r plus, I'm going to have a smaller, the next smaller is r minus, there's yet another one somewhere, I guess it's going to be, maybe not actually in the Schwarzschild-de-sitter, so this is the interesting part, and so when you really should think about r and r plus, otherwise you get to divide by mu, so real the manifold to start with, to start with you just have r, you have the real line in T cross r minus r plus in r cross the sphere as your manifold, and there's some other formula that works very similarly for the general a, so that's not an issue, I mean we didn't sum the limit anyway, so this is your manifold, but this r minus and r plus that coordinates singularities really, so the moral is that it's very easy to take a good object and make it into terrible, so in this concrete case by using terrible coordinates, what's hard is to take a bad object and make it to good, so but in this case you can do it and the coordinate singularity and in this case with a equals zero we just have to change T so an appropriate change of T, so if you let T star be T minus some function of r, where this f r, all it has to do is f r should be behaving like plus or minus one over mu r at r equals r plus minus, so if you take something like this and you compute the metric in this T star coordinates, then you see that on r T star cross, so you started or from from here r minus r plus across the sphere the metric extends smoothly and as a solution of Einstein's equation to still r T star or in this case actually I can just take the full half line across the sphere, and so you would think about this and you would think about your space as something like this or at least so we'll actually consider only consider a thickened version of the sort of original region so so that is to say what I really want to think about is maybe region omega on which I work so it's a region in the manifold m as this and then maybe I have a slight thickening, so I just go beyond r minus and r plus I think that the significance I'm just saying the significance of r minus and r plus is that this r plus is the cosmological horizon and r minus is the black hole event horizon and in terms of geometry of geodesics they play so classical geometry they play a completely analogous role okay so yes thank you f prime is that exactly so f has a logarithm so this mu r has a simple as the picture is sort of correct that this mu r has a simple zero here and so the derivative is this and so f we have a logarithmic alright so so very good and so now very concretely the question is you pose initial conditions at t star equals constant so say t star equals zero say doesn't really matter t star equals zero and then ask the stability questions so let me just draw the picture honestly as we're in here so what we have I have a real line in t star we shouldn't over interpret pictures the saying is that the picture is worth a thousand words so I think the correct way of saying this is the picture is worth a thousand words if you want to deceive someone it's really very easy to deceive people by drawing pictures because somehow we connect emotionally so much to pictures so you shouldn't believe pictures in general or not take them seriously anyway so this is the real line in R and I have an R minus I have an R plus and so these are my horizons and then I slightly thicken them so I went a bit beyond the horizons and this is my region so in between here is my region omega okay so that's what that's what we are interested in and so this is my t star equals zero and I want to post data here so this is only for t star greater than or equal to zero so I put the plus here so that's the the manifold picture of course if you want to draw something like the Penrose diagram of this this would not look the same because here things like angles mean nothing it's really just a manifold picture no extra structure whatsoever for the Penrose diagram then you want 45 degree lines so this is really across the sphere to be not geodesics so in a picture like that these two horizons should be converging at this 45 degree angle but that's not important for us right now so we are imposing initial conditions here and we are trying to solve the wave equation and I sense equation in this domain okay so the the theorem this is with hints is that that for for initial data I'm just going to initial conditions close to those of let me say gm0 so this is just Schwarzschild's the sitter with mass m so let me point out maybe right away that this includes gma for small a okay so in particular things that are close to a slowly rotating curve are included in this statement okay so the theorem is that for initial conditions close to those of gm0 specifically this would be in some some L2 sands including derivatives a number of derivatives derivatives I'm not going to write out precisely so for initial conditions close to those of gm0 this Einstein's equation has a global solution has a solution I mean this is our omega is our road nothing exists outside the omega so it has a global solution on omega and it tends to in fact even exponentially tends to so at an exponential rate this is in T star to g let's say m prime a prime where m prime is close to m and a prime is close to a and I mean you get an estimate also this how close in terms of how close the data it's a quantitative statement here so right so the initial conditions before have to satisfy constraint equations and let's hope I didn't forget anything in this statement so the point is that for the square desicter black holes if you perturb the initial data a bit you will get a global solution good because nonlinear differential equations usually don't have global solutions so that's one thing it stays close to to a shorter desicter or if you want to occur desicter but it's even better it will tend to occur desicter and then one important thing is that even so of course if you start with a gma with a small doesn't have to be small if you start the gma that itself solves Einstein's equation so you can't expect that and it's not the same as gm0 you can tell the difference between them so even if you start with a gma with a very small but nonzero you will not tend to gm0 so it's not like you tend to the shorter desicter rather you have to think of a whole curdesicter family at the same time and the statement is that as time goes on as t star goes on the solution will tends to at an exponential rate to to a curdesicter metric where if you were say close to like put up slightly an initial curdesicter metric with some parameter m0 a0 what you get is that m prime would be close to m0 and a prime would be close to a0 but changed so both the mass and the angular momentum change in general I mean yeah so that's the the statement so there are no symmetry of any kind hypothesis in this so let me say that the analytically harder problem again is the lambda equals 0 case and a lot of people have been working on this so analytically harder and I hope to say something about this why it's analytically harder so it's the lambda equals 0 problem so that would be cur and here the state of affairs is the current best best results are statements about linear stability and there are so slight variations of maybe significant variations but anyway some sort of linear stability and so so mention a few names so the thermos actually is going to talk about something else next week but maybe I will actually get to say something connecting to what he'll talk about anyway and Ronjanski then there's Kleinermann and Schaftal so this is a restricted Kleinermann and Schaftal so there are symmetry restrictions here so this is a work in progress and the thing that's done that's an archive is the a equals 0 case and some partial things to cause the equation about which is not quite the same thing about nonzero a and then there's Anderson blue ma and then forgetting someone am I forgetting Obedda blue and ma so again there's a conditional statement for the full sub-extremal range of a's but the unconditional is a small a statement and then also so these are all done in different ways so recently with Hefner and Hintz so this is all the last few years Hintz and myself have a version that I mostly talk about for a curse so it's again the small a the small a enters in these problems for us in a relatively minor way and it's a relatively minor way that if you like computations then maybe you can remove this restriction so it might be quite easy to extend these theorems if you are good at doing certain kind of computation that they will mention so that's the so Slendik was there is harder basically because it's the Euclidean infinity Euclidean infinity is let's say is big in quotes that's really somehow the ultimate cause of problems so you might say what's nicer than Euclidean space or Minkowski space let's say Euclidean space what's nicer than Euclidean space I mean RL could be well the answer is a compact manifold so any compact manifold beats from an analytic perspective Euclidean space so that's basically the reason why KDS is nice because here you are effectively working in a sort of quote spatially compact region that helps a lot analytically in ways that I will hope to get to ok so my plan is to say a few words about the initial value problem from our perspective because it will enter into the analytic things I want to say so let me first say that the initial value problem has a long history but but the sort of general statement I mean the general treatment is due to Shokei Bruin and sort of the global the local theory and global symbolic settings Gero and so the idea is here the local solubility so when you are working locally because of this domain of dependence issues for wave equations what you really you can really work in coordinates in coordinates and what you want is that the coordinates should satisfy wave equations so this is the gauge is that so this has come up before I believe coordinates should satisfy wave equations so I mean this sounds sort of crazy right that you are trying to solve an equation you don't know how to solve it and so what you do is impose extra conditions but this and then you claim it's easier and it actually really is and the reason why it is from a certain perspective is because how do you actually solve equations so I hope to say something about this in a bit is by separation variables blah blah is really by doing estimates for the adjoint problem that's how you really solve an equation alright so anyway I say something about that in a bit and so the geometric version I mean without referring to coordinates is that you have a background metric which may or may not be the metric you are perturbing though you could say why it takes something different but background metric a metric g0 and what you want is that you introduce this gauge one form which I'm going to call epsilon g so it's going to be a one form so dual to a vector field a one form and what it is is if I get it right it's something like this gg0 inverse so this is divergence or maybe negative divergence gg0 and g is trace reversal right so this is the thing that takes the richie and produces Einstein so in higher dimensions different dimensions not quite trace reversal but reversal so gg of anything minus half trace gr times g so g0 is a background metric that you fix it doesn't have to be something that you're perturbing just something you fix if you fix the Minkowski metric as the background then this is exactly going to give you the wave coordinates gauge and so what you do is this gg just so you start with this metric this gg produces you another two tensor the divergence will produce a one form and then this inverse and back to this will just produce again a one form so this is the gauge one form and your gauge will be so gauge is that this epsilon should vanish so that's the gauge you want to have so this is the harmonic gauge or wave gauge in general so not coordinate but and this is equivalent to the statement that the identity map that the identity map going from M to M but you really think of M equitment g to M equitment g0 so satisfy a wave equation so I want to get into this but anyway that's a fully geometric way of saying this so this now sounds very nice and geometric I can just write down this expression maybe it doesn't sound so geometric but it's purely written in terms of geometric operators okay so you think of this and then what you're going to do so there are two things you need to do one is you chose a gauge but that's not enough because why I mean if you just have a gauge again you have this problem you have an equation you don't know how to solve add an extra condition why should you know how to solve it the point is that now you can modify the equation so you're going to have a gauge a gauge fixed Einstein equation which is that I'm going to add the term to Einstein maybe I'm going to call it phi gg0 equals 0 but the symmetric gradient of this epsilon so this is the thing whose are joined is the negative divergence so this is the symmetric, the symmetrized gradient so it's called symmetric gradient right so what it does it takes this one form and produces a symmetric two tensor okay and so at least this equation now makes sense everything is a symmetric two tensor so the first when you write an equation it doesn't even make sense so it makes sense so that's a good thing alright and so now the reason why this is good is because this actually is a nonlinear wave equation the reason why you like this is because you know at least locally in time you can solve nonlinear wave equations so in other words it looks like it's going to look like box g of g plus lower terms so first derivatives equals to 0 get some nonlinear equation like this but it's at least a wave equation so that's why you like this there's at least a local theory so that's in general for solving this exists so that's good news okay so of course you have to do a slight bit of juggling that I'm not going to really get into because your initial data were these tensors on the sub manifold so what you really need to do is you start from gamma k and then what you need to do is cook up initial data for this gauged Einstein equation and you cook them up so what's the cooking up I mean one thing is obvious when you restrict back you should give the right things but the other bit is so that epsilon g is epsilon g and it's the first derivative derivative at sigma is equal to 0 okay and this is where you can do this and first derivative this is where the constraint equations come in but I think you have a whole mini course on this so I won't get that okay so what this means is that you can now solve this equation at least locally in time but okay this is a different equation it's very easy you know if you don't know how to solve an equation you could just change it and then solve the new equation and claim victory if you could do that right so why is this not cheating this is not cheating because of a second Bianchi so second Bianchi tells you that if you take the divergence of the Einstein tensors so apply this apply so remember g this turns reaching to the Einstein tensor so apply this to both sides of g and what you get is that so the first two terms go away and so you end up with there g there g star epsilon g is equal to 0 and now this thing here this is a one form the symmetric gradient produces a two tensor still symmetric two tensor divergence back to a one form and what this is if you work it out this is a one form wave operator maybe half there's a half involved okay and so now the main point is that if you have so it's a wave equation on one forms if you have vanishing initial data there's a Cauchy data epsilon and it's normal derivative so vanishing initial condition implies that epsilon is identically 0 and so this term is gone and then I mean this term is this is 0 0 and so you solve Einstein okay so the reason why one did this is was to make sure that that you have a PDE that you have techniques for solving in general and so that's what this achieved it turned this crazy equation Einstein's equation is a terrible equation from a purely analytical perspective to a nice well behaved wave equation okay so in the rest of the of the talk I I want to indicate how to so what are the issues with just doing this globally and how can you fix these issues okay so okay so I already said it's a non-linear wave equation non-linear equations so how to solve such a thing how to solve these how to solve non-linear equations equations well the answer is one of the answers is probably the general answer is there are different things you can do in certain specific situations but the thing that you can generally do is linearize I mean that's the whole point of linearization linearize it turns out that so wave equations have a not so pleasant attribute relative to equations like Laplace's equations elliptic equations and they're not so pleasant attributes is that so compare the following so this is the Lambertian and I want to solve an equation like this for example to F versus this is going to be just Laplacian so it corresponds to a Riemannian metric so it's a Riemannian rule I mean this is really let's say Lorentzian but in fact you can change the signature to anything non-Riemannian exactly the same thing is going to happen so in both cases it's a second order operator so in the ideal world what you'd like to say just a linear equation ideal world what you'd like to say is that if F has some regularity measured in you have to decide on what notion you're going to use typically in analysis it's nice to use L2 L2 based notions so say F has some derivatives in L2 say two derivatives in L2 well this is you're taking two derivatives of U to get to F you'd like to say that U has now four derivatives so two extra derivatives in L2 well you can't say that but here it's true so if F has so easiest thing if I say in L2 basis relative to L2 it has k derivatives in L2 then so say I'm doing this in a compact manifold if I have a boundary I should have some boundary conditions blah blah blah anyway so then U has k plus 2 derivatives in L2 and so here if F has okay so here of course my initial condition right so here's my sigma my initial surface and then I'm solving this equation on this space time you should specify initial conditions what you can always do and what's analytically much nicer than anything else is to break down your problem where you have zero initial conditions on sigma so zero initial conditions so I want to mention them I mean just that there are zero initial conditions and then you ask that F is supported here in the future of zero and then you want the solution U which is also supported here and get U so that's a forward solution so this is what gives you the what's given by the retarded propagator in the future of sigma okay and then the sad state of affairs is that if F has k derivatives you will have just k plus 1 okay so you gain back 1 but not the full 2 and so what this means is that you have to be slightly careful this linearization so the implication and so this is all I will say about the nonlinear aspect when you do the for the nonlinear iteration so you have to do a Newton type iteration for linear iteration you need to linearize at not just at g0 so not just at g0 but at perturbations of it so for the wave equation for Laplace's equation I mean for nonlinear elliptic equation which starts at Laplace's operator you wouldn't have to do this that's why elliptic equations are nicer in this respect okay so so good so anyway the option is to solve these nonlinear equations for at least perturbations of a metric g0 that's solved Einstein's equation what you have to do is sort of linear equations but you have to do it for perturbations not just the linearization at g0 but the linearization at perturbations of g0 okay so it's not enough to know say what happens exactly on Kurdish but also what happens close to Kurdish alright so and then so here's the linear analysis part so what I'm going to say now is sort of goes against the spirit of what goes on in general in evolution equations so this is not the typical way people think about evolution equations so you really want to think about you want to take this analogy seriously so what you are doing here is really taking u is equal to box inverse f you're inverting box on an appropriate space of functions I mean it's an honest inverse this would be box would be a matrix and you're inverting this matrix just like here if you discretize Laplacian is in a matrix and you're inverting that matrix okay you're doing it in infinite dimensions so it's not a matrix I mean you can pretend but that's not a very wise idea in any case you're really thinking of inverting things it's a global operation that you are doing so what you are going to do is you want to invert inverse box of g so g is something close to g0 so maybe I mean you really want to of course a tensor version but that actually makes very little difference going from scalars to tensors basically nothing happens the key the only way in which tensors are annoying is these gauge issues but it's not strictly speaking because of tensors in some sense anyway so what you want to do is to invert data g for g close to g0 and I'm just going to put appropriate function spaces right I mean that's what this amounts to right so you want to write u is equal to box g inverse of f that's what you want to write okay and what this needs how do you actually do this I mean you can't write down a formula for something like this so how does an analyst deal with something like this what you actually do is to prove estimates so it turns out that the sort of estimates you like to prove are following so I'm going to write down both sides for an existence really only the second part is relevant you really want estimates u in some function space x which I'm not specifying to be controlled ideally by pu in some function space y and the dual estimate so they are two separate estimates you want to control v in the dual of y by a constant times constant times p star so the adjoint operator this will be the adjoint matrix if you discretize p star v in x star okay so right so this says for one thing that p is one to one right so this says that if it's a linear map so one to one just says that one is zero goes to zero well if pu is zero this says u is zero okay and this says that p star is one but so p here is the linearized linearized gauge fixed Einstein okay so you want to really prove estimates like this now the chances of being able to do this are around zero okay so we just don't have super good techniques like this what you actually so this if you do this actually this is the key part this but easy very easy I mean I could do it if I had extra 10 minutes I could do it easy the function analysis argument gives that one can solve p equations like this p equals f so it says that this p is onto that's basically so now I'm on right so like basically an algebra right this is the the one one thing one learns is that in basically an algebra it's honestly like this that the surjectivity ontoness of p that's what the statement is that you can solve it is related to the injectivity of p star so that's not directly true in infinite dimensions but having an estimate like this does it so um oh so it's just a duo space so the star is right okay let me just say I didn't say it so it's not nothing has a duo space of x so in other words linear functionals right and so the way you can think about it I mean this is like the easiest thing if x were a Hilbert space you can think of x star is just being x but with the identification there is representation here but what's a much better way to think about this is that if x so then it's not a Hilbert space and you really want to be a Hilbert space setting because you want norms but if x was something like smooth functions maybe compact it's worth smooth functions we have non-compact space x star with dual it's a space of distributions okay so the basic things like if you want to do QFT right that would be the basic thing so so in general if you want to linear algebra if you want to have surjectivity onto statements you really want to have they really come down to one-to-one statements but for the adjoint operator on duo spaces right that's a p star really goes between the duo spaces so p goes x to y then p star goes backwards between the duo spaces okay so in real life you cannot do this what you can do and so again I am emphasizing that what we are doing is we are solving these equations globally on omega and in the nonlinear iteration we just keep doing this in just step by step to solve away the nonlinearity which is not what's usually done usually you solve evolution equation you solve it for a short time where it's relatively easy because something is small and then you hope to control what you get so that you can keep going that's not how you think about this globally just like an elliptic problem alright so by the way if I change the function spaces here I just said that I just want things that are supporting the future of sigma if I did it in a different setting at least in different kind of function spaces the inverse you would get will be the Feynman and Enti-Feynman propagators the honest inverse is in appropriate spaces alright so here we are now you can do this what you can do instead can do is estimates like this and maybe I write down the duo version just write down this and the duo is similar that's what's actually needed now if I don't say anything about z it's not a very exciting statement you can just take z equals x and then c equals 1 will do the job the key point is that x so somehow this is supposed to be a very weak norm ok and what this means mathematically is that the inclusion so x should be sitting inside inside z and this inclusion map should be what's called compact ok so what a concrete thing you can think about is that eg we have eg we have I don't know one derivative in l2 on a compact manifold so let's say the sphere or torus and this for example is just l2 of the compact manifold so then this inclusion is compact the point being that so what does this say it says that when when you have unit size elements of x in the x norm so apart from a finite dimensional space they are tiny in the z norm ok so that's what this is so if you think about say the torus then you have Fourier modes that give you you can describe or circle just simply you can describe functions in terms of their Fourier series and this would have a weight corresponding to the derivative that multiplies by the Fourier mode number and if that has size 1 in some l2 sense then if you get rid of that multiplier k well it becomes tiny so apart from finitely many modes so this says that apart from finitely many modes things are great and as an analyst you think finite dimensional problems are somehow very easy just linear algebra so the analysis itself is infinite dimensional so it's really hard and infinite dimension things can go very wrong that naively sound ok ok so that's what you can really do so this is the global analysis part and how so plus the adjoint problem and then how can you do this so this is where the other part of the title comes in so which was micro analysis now this sounds very fancy what this really means is phase space so it's phase space analysis plus plus at least in this GR setting quasi normal modes ok that's those are the two things that enter and why are those the two things that enter so let me say something about the phase space part so the micro local analysis part ok so when I had this compact example so you have a compact manifold and you think of this one derivative to zero derivative and this is compact so that's the statement about what happens at high frequencies this is a high frequency statement there are only finitely many low frequencies and you say who cares ok maybe intermediate you say who cares from a perspective of infinite dimension analysis that's not a problem of course for other reasons it can be a problem anyway so the phase space this really corresponds to the high frequency analysis and what it described by so this is of course a classical problem but what goes into this is exactly what goes into quantum mechanics so this is doing analysis up to what the uncertainty principle allows so position and momentum don't commute and you are looking at the operators you are looking at the PDE position and momentum operators don't commute so you can't just simultaneously work it too but you can do it as well as possible and that's what this does and so what this will say for wave equations this will tell you that energy so this is some sort of L2 based estimates unfortunately I don't have enough time to go into more details energy L2 based estimates including for derivatives propagate along let me say lifted null geodesics so you really want to think of the Hamiltonian dynamics in phase space but I mean you take phase space seriously here so your underlying manifold is the four dimensional space your phase space is the phase space of that so it's an eight dimensional space and then you have you have a function on this the function is just the symbol so like you write in the highest derivatives and interpret them as so second order derivatives interpret them as a function on phase space so you have a function say P the function on phase space so that's the cotangent space and then you get the vector field the Hamilton vector field the symplectic structure in T star M produces you a vector field the Hamilton vector field and then what happens is that of course in most of phase space you are off the energy shell P is known zero so things are really elliptic there so it's just as if you had the Laplace's equation then everywhere things are elliptic but in P equals zero which is the only delicate place energy propagates along integral curves of HP so energy in this sense so if you project these you get the geodesics I think actually that's my favorite description of geodesics anyway so this is how you get the null geodesics so that somehow is the high frequency part of this now so I have six minutes so so now the issue is that as nice as compact manifolds are sometimes things are not compact so why if manifold so if we had a compact manifold if M was a compact manifold then this would take care of high frequencies so this is geometric optics by the way as always this is geometric optics so this would take care of high frequencies and if in a compact underlying space like the torus then this is all that there is up to finally many modes the estimates don't care about that but if you are a non-compact space and this is what happens in the GR settings then the other thing that you need so say Rn, Rn is a beautiful non-compact manifold it's not enough to gain derivatives you also have to get in these estimates like this Z cannot just have fewer derivatives it also needs to have less decay so you need to gain decay at infinity so this Z needs to be more vaguely more permissive more growth at infinity than X how can you get to that in a situation like this so the key thing that lets you do this is exactly this is where the quasi-normal modes come in so I don't think I can properly go into this because I want to say something about the actual relation this is where the quasi-normal modes come in and these are really the quasi-normal modes for what so remember that it was not enough to solve this equation for the background metric that we are perturbing we have to do it nearby but the quasi-normal modes of so we are worrying about now decay so what's happening at infinity so the only thing that should matter is what the metric is at infinity so this is the metric we are expecting to at this point we don't know the theorem yet we are expecting to to converge to at infinity so that would be the curdeseter so that's why even though for this way because you have to solve more generally this has two parts geometric optics which is okay in general so there are some funny things you really have to think of the spacetime the whole spacetime and there are these issues where the horizon hits future infinity but then it's the the phase-space interpretation of the actually really a phase-space thing already of the direct shift effect etc that's coming to do those but in addition the extra part is so okay so what you would like to say is that if um what you can say is that if all QNMs are decaying plus you have the phase-space the microlocal estimates high-frequency estimates so microlocal plus all QNM decaying implies that um the wave equation u equals f can be solved for decaying f in decaying function spaces and that means that the non-linear iteration works okay so again I want me it only but maybe probably give me two extra okay because I just want to say something about how okay so this of course is not true so this is okay this is not okay and cur but let me ignore that this is not okay okay so in curate it would be actually this is appropriately interpreted it's almost okay but not quite still um so so what happens so why do things go wrong and how can you fix it the reason why I want to say is because actually related to things that was one of the biggest advances in physics which is LIGO and the numerical GR behind it so what goes wrong in general so what's wrong so this is okay so the geometric optics is fine so what's wrong QNM so they're growing they're growing mode solutions incur the sitter of gauged Einstein now is this a surprise well I mean you know you can be lucky and so in curate itself you're lucky so okay there are two things to think about when you're solving PDEs one or inverting operators one is the analytic aspect so this is dealing with the infinite dimensional issues that's a huge problem the other one is algebraic aspects so lambda positive is much better analytically basically because at least spatially you have a compact problem quote on quote then lambda equals zero but algebraically it's much worse so algebraically most of things go wrong everything is way messier and in particular you have growing quasi normal modes now the the what's even worse is that this is not really a surprise in a sense because what you've done is you change the equation you really try to solve Einstein to this gauged Einstein equation and you did two things in the process first you implemented the gauge and second implemented it by putting the symmetric gradient in front now even if it was there was a good gauge the implementation need not be good so the key thing and I stopped with this key thing to fix the problem issue is to change how the gauges implemented so you have to do a bit more than why I say but but let me just say stop with this or gauges implemented how do you change it so unfortunately it raised the equation so it was like blah blah blah minus da g star epsilon g is equal to 0 that was the equation so what you do in the way you should think about this was good why was this good this is good for exactly one reason it gave you a wave equation to leading order so there's some first order terms but they are okay the important thing with the second order terms but you can change this appropriately by zero for their terms and why is this good this is good because remember how we got from from g e to e e what was the key thing it was apply second Bianchi and what does this give us it gives us that change this to by second order so something I'm going to call delta tilde star so applied second Bianchi you will get now that delta tilde star so g g da g sorry here's the delta tilde star da g that this thing of epsilon is equal to 0 okay now what happens if you change by a zero for your term so this is a wave equation but now it becomes a damped wave equation and so this is what's called constraint damping in numerical g r and why is this good it's good because you know what goes wrong there's one thing that goes wrong there's many things that can go wrong but one thing that can go wrong without anything really going wrong is your gauge is misbehaving so globally it's not a good gauge and in numerical g r of course what you do is when you're starting epsilon is equal to zero and you solve your equation numerically but numerically nothing is ever zero so numerically even if this was zero epsilon was zero to start with solutions it's a disaster in finite time and that's what happens if you just use the standard symmetric gradient here but if we change it appropriately it becomes a damped wave equation and at least your gauge modes are decaying and that's exactly what will sort of save this problem because in terms of the quasi-normal modes of the gauged equation they in principle have nothing to do with the quasi-normal modes of the ungaged equation they will have something to do only if you have this translation works that this does not give anything extra crazy and that's what constraint damping gives you so that's somehow the key thing both in LIGO and in black hole stability in the g r setting so constraint damping gets through the irrelevant behavior in an analytically very nice manner so thank you very much