 So maybe I should start. All right, so this is a third lecture. On the classic I saw a black hole, so I discussed this so-called tests of reality for black holes, which were to those rigidity. Then stability and collapse. in ležite. Zato they all have to do, so rigidity is a statement, so they are all statements in fact about the Einstein equations in vacuum, rigidity is a statement about the fact that stationary solutions, so one looks at stationary solutions of the Einstein equations, Veselo in tkaj je tkaj asymtotik, je tkaj zelo in tkaj je vsi vsega tkaj asymtotik, izvajske soličje in vsega, vsega izvajske soličje, vsak je zelo vsega, in zelo vsega doma, ki so nekaj soluzon, besedjega kaj kaji svijet, zelo ste, da je nekaj soluzon, kaj je očet od početnih predsupetih, vsega zelo je, ki je to nekaj soluzon. Početno se je začilo, ki se zelo izgodno izgleda, in različi iniski, tako vse zelo je pogleda z nekaj zelo zvali vsega. Zato se vseče inšljajte data, k0, k0, vseče konstručnoj vseče. In v sej staviličnih različnih, je bilo vseče, da se vseče vseče, zelo vseče solusi, inšljajte data, da je vseče. In vseče, da se vseče, zazbondvenje. Zobr, ki je vsega vrlasti, našli je, da je našli, zato, da je zelo vsega, začine je zelo vsega, bom je zelo vsega, začine je, da je vsega, začine je, začine je, da je začine. Zelo vsega, začine je začine. Zelo vsega, blakol. Allright, so anyway, so these are the things which I did before. And then I started to talk about, I talked a little bit about rigidity and now we're talking about stability. So so this is the conjecture stability of external care. So again you see you see here the care solution. This is the external part of the care solution. Right, starting at the event horizon. So R equal R plus is the event horizon, which is the boundary of the blakol region. That's a blakol. And this conjecture is interesting only in the outside of the blakol. So in other words, you start up with a care. You take a space like hypersurface. You look at the restriction of the care on the space like hypersurface. That will give you an initial data set. You make a small perturbation of it. In other words, you change the initial data set by a little bit and you look at the evolution and the conjecture is that the evolution will converge to another care solution. So you are starting with the original A and M care and you get a new one at the end. There will be two care solutions at the end? No, at the end, just one. So in other words, you have started with something here, right? But at the end you are going to get a different one. So it's not going to be the same care, right? So the horizon will change a little bit. It will not be the same horizon and so on and so forth, right? Okay, so this is a statement I made last time, which I think Slava was not happy with it, so I made it a little bit more clear. So what I am saying here is that, sorry, this was not the statement actually. This was fine. I meant that, okay, so these were results which we will discuss later on in more detail, so I am not going to get into it. And I said that lack of exponential growing modes is not enough to conclude anything about the nonlinear stability and of course, as an example, I mentioned the emergence of black holes or the emergence of turbulence. I will say more about this later on. We talk about the care solution, of course, again. You see the explicit solution, the fact that it is stationary and axisymmetric, it's very explicit in the case when the parameter a is equal to zero, get schwarzschild. Here is, again, the way the care solution looks like. And I mentioned that there are important things to remember. First of all, the values so far, there are interesting values so far, which give you, when exactly when r plus is a solution of this delta equal to zero, you get the horizon. Then r larger than r plus is exterior, r less than r plus is the interior of the black hole, r equal to infinity is this boundary at infinity. So this is done by conformal compactification. And r minus doesn't appear here because it's something that has to do with the interior of the black hole. So you see it's here. So of course, there are lots of interesting things to be said about the interior, but they're not going to get into that. So again, here you see the exterior in more detail. You see, again, the horizon. And the sky, which is infinity, the null infinity. You see the vector field t, which corresponds to stationarity. So this is in those coordinates, bojalinguist coordinates. It's exactly d over dt. And you see what happens here is that as you approach the horizon, t actually becomes space like. And this leads to all sorts of phenomena, both physical and mathematical. Another thing that I mentioned last time and I'll mention again later on is the presence of trapped null geodesics. In other words, geodesics that a typical geodesic, a typical null geodesic in this picture will be at 45 degrees and is moving either towards infinity or it's moving in the black hole. In both cases, you are not going to see them anymore, right? So if it moves in the black hole, it will never come back. If it moves at infinity, it never comes back either. So those are good in some sense. Unfortunately, there are some other ones, which are called trapped null geodesics and which sit here for all time. So they sit in a region of bounded r for all time. So they don't go to infinity and they don't go into the black hole. And they lead to all sorts of issues that have to do, that can be seen already from the point of view of the region r3m in the Schwarzschilder. Right, so in Schwarzschilder it's exactly r equals 3m, right? So in Schwarzschilder it will be a hypersurface r equals 3m, which is here, but in care it's a little bit more complicated. So you can have many trapped null geodesics in entire region of r. Okay, anyway, so let's, so, okay. And then I mentioned and maybe I'll repeat very fast, I mentioned a general discussion about stability. We have a nonlinear equations. We have some stationary solution, which is phi0 and we perturb it. Orbital stability we discussed that is the situation where the perturbation stays small for all time. As in total stability means that the perturbation actually is going to zero, right? Linearized equations we discussed when you look at the first time in the expansion. So you look at essentially what is called Fréchet derivative of n. Apply to psi, this is a linear equation. Linearized around phi0 and then again you can have all sorts of discussions about mod stability, boundedness and quantitative decay. Mod stability is, for example, the statement that there are no exponentially growing modes. That had to do with decomposing the linearized equation. The solutions of the linearized equation decomposing them into modes in some kind of eigenvalue expansion. And then we can show for every mode you can show that you have stability. In other words, you can show that the modes don't grow, right? They don't become, for example, exponentially growing. Then just having no growing modes does not even imply boundedness. In other words, you can have no growing modes for psi and yet psi doesn't stay bounded for all time. That will, of course, create huge problems from the point of view of nonlinear stability. To prove nonlinear stability you need what is called quantitative decay and I'll explain that later on in more explicitly. Okay, so then we looked at the possibility that there exists a family of stationary states around phi0. So phi0 is just one among many continuum, in fact, of stationary solutions. And then we saw that at a linearized level d over d lambda of i lambda, evaluated at lambda equal to zero, is actually an eigenfunction, corresponding to zero eigenvalue. The same thing happens if you look at the thermomorphist, which keep the equations invariant. In other words, psi lambda, if i0 is a solution, phi0 of psi lambda is also a solution. Again, you differentiate and you get a huge kernel as a consequence. So this is what we discussed last time. Now, okay, so this I'm not going to repeat. So to prove nonlinear stability you have to do many things. But in particular, you have to really understand gauges. In other words, you have to, this deformer, the fact that the deformer is, that the fact that an equation is invariant and the deformer is leads to the need to actually find the correct deformer. So that's one issue. Final state, you have to find the correct final state. And this can only be done dynamically. And anyway, we'll discuss about this again later on. All right, so this is care stability. Now, in the case of the actual Einstein equation, so I discussed that the issue in general, in the case of the Einstein equation, you can see that the linear Einstein equations, of course, are reach equal to zero. If you linearize around a stationary solution, which depends, in other words, a care, which depends on these two parameter, then you, so these are the linearized equations and you see that the derivative of g with respect to m. In other words, if you vary the parameter m, you get the whole family of eigenstates corresponding to zero eigenvalue. And the same thing if you differentiate with respect to a. So you get a two parameter family of solutions. And if you also look at the fact that the Einstein equations are deform office invariant, in other words, invariant relative to any deform office, then you also see that we have a huge kernel, which corresponds to that. The full dimension of the kernel is four times infinity plus two. Okay, so now let me start talking a little bit, let me be a little bit more concrete and start talking about about the geometric framework that one needs in order to understand this problem. Okay, so to start with, and I mentioned this earlier, it's in general activity, in Lorentzian geometry, more generally, but certainly in general activity. The directions, which are important, are null directions. So they are important. Why? Because they correspond to null geodesics and because because most of the energy is transmitted along null geodesics. All right. So so null geodesics are supposed to be much more important than say time like directions. Right, especially from the point of view of stability. It's extremely important to follow somehow the way waves, the decay of waves, for example, is very much dependent on the behavior along null directions. All right, so null directions are very important. So because of this, when you talk about frames, you start with two null, two such null directions, e3 and e4, they are both null and you also take ge of e3, e4 to be equal to minus 2. Okay, so in other words, you normalize the frame. Okay, now, once I have, once I pick up a frame, I should say it's not a frame, but a null pair. Once I pick up a null pair, I can take the orthogonal complement to the null pair. So at every point, I'm talking about something at every point. So at every point I'm going to have, I'm going to have a distribution, if you want. Right, so at every point p, I take the space perpendicular to these two. If I am in four dimensions, this is a two-dimensional plane. So it's a two-dimensional plane at every point, which is, of course, space-like. And this is what I call a horizontal structure. So this is horizontal structure. Now, this horizontal structure can be intergalba or it may not. So if it's intergalba, typically, if it's intergalba, it might generate two surfaces. Like, for example, if you, if you deal, if you look at the intersection between two null cones, a null cone going this way and a null cone going this way, then at every point at the intersection you have a two surface and you have automatically a natural null pair, which is given by this. So one, which is tangent to this null hypersurface and the other one which is tangent to this null hypersurface. So a very natural way to define foliation is to take the intersection between, between a null hypersurface and another one or the intersection between a null hypersurface and say, a space-like hypersurface. It will also give you an intersection and again, you can talk about a vector going this way and another vector going this way, which is, they are both orthogonal to these two surfaces. So anyway, what I wanted to say is that this horizontal structure can be intergalba or non-intergalba and we'll see examples of both. Okay, once you have, once you, you have this horizontal structure, I can also take in the, in the space perpendicular to these, I can take vectors E1 and E2 which are perpendicular to both of them, so they are in the horizontal structure and I can, so they are perpendicular to these two and also let's call this EA is one or two and I'm going to assume that g of EA, EB is delta EB. Okay, so this is again a normalization that I pick and as a consequence, I get what it's called now, a null frame. So null frame consists of this pair of E3, E4 and then this as a vectors EA here I wrote it was capital A but there I wrote it because it doesn't matter. Okay. All right, so now once you have the frame, we do what's done always in geometry. You look at the connection coefficients. Connection coefficients, right? Okay, so how do you define the connection coefficients? Well, typically you take the derivative, the covariant derivative with respect to say E alpha so E alpha can be E1, E2, E3 and E4, right? So alpha in azos stands for this indices and I take the alpha of E beta and then another one E gamma I take g of this, right? So this is a vector field and I pair it with another vector field and this is what it's called the coefficient gamma beta then alpha gamma or gamma alpha, actually, right? Christopher symbols. So these are Christopher symbols. All right, so the you get the Christopher symbols in this in this particular case when you talk about null frames you can identify various Christopher symbols. So this is something quite different from Imani in geometry where typically in Imani in geometry the all directions are the same. So it doesn't matter too much. You don't need to you don't need to identify specific specific connection coefficients typically but here you do and so let me write down some which are extremely important. So for example if I look at if I look at this say E4 if I look at E4 which is a null vector and I take I take say EA k, so I take EA is in this direction. So this is remember A is one or two. If I take EA of E4 and then EB so again EB is like the same B is one or two. So this is this is a connection coefficient as you see here which is called chi A B it's called the null second fundamental form because typically whenever you have a two surface in other words if this distribution here is integrable and it corresponds to a two surface then this is null second fundamental form in the I mean it's a second fundamental form in the universal sense and it's null because it corresponds to an E4. So I can do the same thing symmetrically I can do the same thing with E3 and then I get what it's called kappa bar AB. And again so this is the null second fundamental form so you have a null second fundamental form in E4 direction and a null second fundamental form in E3 direction. So these are kappa and kappa bar which you see there you have kappa and I didn't write kappa bar but that's by symmetry and then you have many others so for example I wrote there say xi this xi is G of D E4 E4 of EA so this is a one vector so yeah by the way I should I should say here which is very important so you see this A and B can you can have the indices one one one two two one and two two now in the particular case when E4 is when the distribution is integrable in other words when these two things are integrable here then actually this second fundamental form like any null second fundamental form like any suck like sorry like any fundamental forms like any second fundamental form I should say they have to be symmetric it's easy to see that the symmetry comes because of the fact that this is integrable in general it's not true so in in most cases these components will be equal by symmetry but not necessarily the other one and not necessarily always because as we will see examples in a second in interesting situations you may not have integrability okay in any case that's a situation this is a this is now a vector on the two sphere or on the horizontal structure if you don't have integrability and the same thing for psi bar where you put here E3 E3 D3 E3 so for example if E4 is geodesic if D4 E4 is geodesic equal to zero then this coefficients zero which is again something that you can choose to do in in various situations you can do you can make them to be zero or not alright so as you see you can do many other combinations and you get all the other connection coefficients we call it connection coefficients or Richie coefficients gamma okay now what about the curvature so so these are Christopher you said Richie sorry? Richie yeah, but Richie is usually Christopher is used for coordinates right and Richie is used for it's the same thing more or less but here it's a frame and in the other case are coordinates, right? okay so next you go to curvature and the curvature has four components I mean four tenths it's a fourth tensor right and because of the fact that Richie is equal to zero it means if you are in one plus three dimensions it means you have here exactly ten components relative to the frame they should be exactly ten components and this is the valtense exactly alright so then you write down again various possibilities I can take for example r ea e4 eb e4 right so this is one component which is called alpha so again it depends on two indices and it's very easy to see that it's symmetric because of the Richie condition this is symmetric and therefore it's not only symmetric actually it's also traceless so if I look at if I look at delta ab alpha ab I get also zero and again this is because of the Richie because of the properties of the Riemann curvature tensor in Richie flat similarly I can take alpha bar ab if I replace c4 by 3 and then I can also do this I can take r ea e4 e3 e4 and I take one half and I call this beta right so you see it's another so these are two components here two components here two components here right because this has two components because it's symmetric and traceless in in two so it's a matrix a two by two matrix which is symmetric traceless this is a vector so it has two components then I can do the same thing where I replace e3 e4 e3 so by symmetry actually this has a minus but this doesn't matter minus beta bar a so I put an underline here to illustrate the symmetry and then I have to have two more components which are r e3 e4 e3 e4 so that's what I call rho actually four rho so this is one over four rho and then finally there is another one where I do exactly the same thing with a hodge dual so I take the hodge dual and I do the same thing and I call this rho star so these are the ten components as it should be and these decompositions are crucial because every component behaves differently right and so it's extremely important to get familiar with this kind of decompositions alright so so then you can write down main equations so I have curvature I have connection coefficients main equation well the equation I'm just going to write symbolically maybe I'll save more later but normally you are going to have some the typical equations look like this d gamma plus gamma times gamma is equal to curvature and then you have Bianchi so dr is equal to zero these are the Bianchi identities so in fact actually I should be careful I should write it like this if I look at components and I take d delta and I take cyclic permutations of this this is I get zero this is Bianchi identities so this is what I write here not the contracted one these are the full ones and this one of course is the usual relation the cartab say relation between gamma richer coefficients and the curvature so again because we are decomposing these components I have to be careful to also decompose relative to components and I'll get a lot of equations this way and to work in this business you really have to understand very well these kind of these kind of equations all right now I think I can turn off the light this is here this one or it's fine okay all right so now I go to the care family again just to remind you but the new thing here that I want you to remember somewhat at least is that in this coordinate so this is explicit formulation of the care metric you find out these vectors e3 and d4 which are null so it's easy to check that they are null and they are called principal null directions for the reason that you'll see in a second okay so these are the kind of e3,4 that I want to pick okay so in care this is a pair that I've interested in right now observe that if I were in Minkowski space if I were exactly in Minkowski space in other words a and m and m would be 0 then e3 would be precisely dt minus dr and e4 would be dt plus dr right so these are very simple null directions that obviously are important to understand the radiations of say linear equations in Minkowski space this play a fundamental role from that point of view otherwise they are much more complicated but they are still null okay so now again I'm just repeating what I said before I have e3 and e4 I have the span of e1, e2 which are perpendicular to e3, e4 I define the connection coefficients and you see some of them okay so they all play an important role and then you have a curvature components which I mentioned which are these ones all right now okay so now here is an important thing a crucial fact is that if I look at the principal null frame down in Bore-Linckis coordinates for the care solution if I look exactly in that frame I find out that all the components of the curvature are zero was the exception of the so-called middle components rho and i rho star which can be complexified like this so if I put them together I get this very simple expression minus 2m divided by r plus i a cosine theta to the power 3 so here there is some miraculous thing happening if you complexify and then and then this has a component psi psi bar chi-hat and chi-bar-hat are zero so these are these are rich coefficients I didn't tell you what chi-hat is maybe I should say it now so chi-hat AB is a chi where I subtract the trace so chi AB and I subtract delta AB times a trace so the trace of chi is delta AB chi AB ok, and this this plays also a important role ok so so if I am exactly in care then you get a lot of cancellations and it's because of these that the principal now frame is so important these are the components of what of the curvature so these are all the curvature components that we discussed these are the ones that we discussed here and in care they are exactly all zero is the exception of the the row and row star which are these two here and in addition you get a lot of richer coefficients to be zero now it's interesting however and this is important to point out that in care e3 e4 the perpendicular to it in other words this distribution perpendicular is not integrable is not an integrable distribution right so in other words you don't get two surfaces no e1 e2 are not right so this is the perpendicular is exactly the e1 e2 so they are not integrable so which is quite a remarkable fact about the care solution and again another reason why it's so complicated so this is not integrable so it doesn't fit into sort of normal geometric pattern somehow right clearly the principal direction are extremely important and yet they are not integrable so which is kind of bad so in particular this chi 1 2 and chi 2 1 are different so they are not the same right so you don't have the symmetry that you usually get into for null for null for second fundamental form I should say so second fundamental form I remember for any surface if you have any surface not necessarily a two surface any surface maybe three dimensions you can define the normal you take the normal and you take the second fundamental they do second fundamental form the second fundamental form is always symmetric if this is a the true hyper surface if it's not if it's just a distribution it's not going to be symmetric alright so that's unfortunate but that's the way it is in actually I should say it's fortunate because it makes us do interesting things so in Schwarzschild we have in addition so if I am in Schwarzschild now and I look at the same null pair but in Schwarzschild so I take a to be zero in other words you get that actually this is integrable so in that case it's you get integrability so you get spheres yes right in addition you get the draw star is equal to zero in other words this last component that comes from the hodge dual this is actually zero and you also get additional components in addition to this you also get this other components which are zero and in fact the only non vanishing components of the of gamas are trezka, trezka bar, omega and omega bar right so trezka is what I just defined trezka bar, omega and omega bar so these are the only things that that I should say omega is defined like this is d4 d4 e3 e3 I think one quarter right so that's and omega bar is defined by symmetry okay, so that's that's what it is now if you are in in Minkowski in addition to all that so in addition to Schwarzschild you also have that omega, omega bar and rho are zero so all curvature components are zero obviously in Minkowski space the curvature is zero it's a flat space and you also have these components omega bar equal to zero so therefore the only thing which is not zero are these and these of course are very simple geometric meaning these two expansions they are called so these are called expansions actually and they played a role in in what I talked about last time I mean on what was it on Friday because they were connected with it was a definition of a trap surface a trap surface was defined in terms of these two quantities which are called expansions ok so ok, now you want to talk about perturbations so I want to take a care solution and perturb it a little bit and the expectation is that that somehow you are going to get something which at least in the least will stay close to the original care solution you started with, right? so for that point of view it makes sense to start talking the simplest level to talk about all of epsilon perturbations of care so what is all of epsilon perturbations of care? well everything that vanished in care I now assume that is all of epsilon in other words I assume that there exists a frame E3, E4, E1, E2 right which is close to the frame of care in some way right and I assume that relative to this frame all components which were zero in care exactly zero are now of epsilon right it's reasonable right you expect that things are not going to deviate too much and epsilon is a parameter which I I control ok now what is the problem with this definition the problem is that there are lots of frames which achieve this right so if I have a frame for which I have this in other words if I have a frame of that type in which these components are all of epsilon but then I have a lot of others which are also of epsilon because I can take any frame transformation which takes a null frame into another null frame so I start from E3, E4, E1, E2 and I get to E1, E2, E3 the prams E3 prime, E4 prams so and I I can write down all the possible frame transformations I write here all of epsilon squared terms because actually the formulas are more complicated there are many other terms but in any case I just want to concentrate on the all of epsilon terms so in other words if F, F bar and log lambda are all of epsilon then if I transform the original frame into this new frame then this conditions will be preserved so it means I have infinitely many such spaces which are all of epsilon so what frame do I choose and this is of fundamental importance because of what I said the frame is going to play fundamental role if I don't understand what is the correct frame in which I do my calculations and I prove my stability result I'm not going to be able to do anything so this is the first important fact that that there are lots of frame transformations which preserve these conditions and lambda F and lambda F bar scalar well lambda is a scalar and F as the synthesis F A so this is A is 1 and 2 and F bar A is A equal 1 and 2 so there are 5 parameters in some sense alright so now the one thing which is important to remark this is the first important remark in connection with this stuff which is quite remarkable actually is that the curvature components alpha alpha bar so these are the components which I define here if you remember so these components so they are components which are obtained by taking 2 e force these components are all of epsilon squared invariant this is a huge observation in some sense I mean it's trivial to prove but it's huge because it tells you that that at least some of the components of this of the curvature are actually invariant up to all of epsilon squared so they are not just all of epsilon but all of epsilon squared so if I make if I take any frame and I take a transformation like this the difference between alpha and alpha prime is going to be all of epsilon squared then the same thing with alpha bar so that's a huge fact because it tells you that at least I can put my hands on some facts which are almost invariant being all of epsilon squared invariant it means that at least at least at linear level they are totally invariant I mean in other words the choice that I make on my frame does not going to affect alpha and alpha bar at a linear level so this is clearly a huge observation even though it's trivial I mean you can really write down I mean it's not a big deal it's just a simple calculation to show that but it's clearly very important ok so this is what it is now oh sorry there was another observation here another observation is that if in addition I'm dealing with perturbation of Minkowski space so I'm not just so this is a general perturbation of care if I look at perturbation of Minkowski space then in fact all curvature components so everything here including rho and rho star they are all invariant quantities in that case so clearly the stability of Minkowski space is simply because of this because the composition of curvature you can think of the composition of curvature into components does not depend up to terms quadratic in the particular choice of frames I make right so these are invariants from the point of view of nonlinear theory I can view them as invariants so in other words the full curvature tensor is invariant if I do perturbation of Minkowski space if I am perturbation of care I only have alpha alpha bar to be invariant by the way so this is true about a curvature if I look at the other components like and they are far from being invariant so all these other connection coefficients are far from being invariant they definitely change in a major way whenever I make a transformation and this will play an important role when I make my final choices of gauges to first order in epsilon it is invariant to first order of terms in epsilon second order epsilon squared up to these terms oh, these ones so in other words they are not invariant because they change even at a linear level so now what I want to do now is to actually spend a little bit more time about stability of Minkowski space and then I'll come back and talk about the care solution so what time is it so stability of Minkowski space alright, so for this I have to in order to discuss it I want to start with very simple things thank you so I want to start with very simple discussion so first of all if you remember when we talked about the Einstein equations for each of g equal to zero remember that I said at some point if you pick up coordinates which verify the wave equation equal to zero the wave equation relative to g in other words if you look at so these are harmonic coordinates or wave coordinates we call them if you use this kind of coordinates then the equation take this form alpha beta d alpha d beta of g component of g and is equal to going to be an F mu nu of g and first derivative of g and it's quadratic so this is quadratic in first derivatives of g so this is the kind of equation you get so you see it's a nonlinear system of wave equations so obviously you have to understand this if you are to do anything that has to do stability you have to understand the long time behavior of solutions of this equation so let me simplify a little bit and look at somewhat something simpler but which symbolically is very similar so suppose I look at Lorentzian metric g alpha beta of phi d alpha d beta of phi and here I write just the function of phi and first derivative of phi so you see I replace this g I replace it by a phi so that I get a scalar equation just to simplify things a little bit so this is the kind of equation I get I can simplify it even further so this is let's say this is type 1 this is type 2 this is type 3 I can assume that actually this is just a Minkowski metric in which case I'm getting the Minkowski metric so the long version of phi in Minkowski space is equal to n of phi and first derivative of phi so that's a reasonable a reasonable model problem if I want to understand this I first want to understand this to understand this I have to understand this I first have to understand this because it's much simpler so this is a kind of equation that you should be able to control if you want to prove stability so for example if I am to prove the stability of Minkowski space then I have to start with the initial data which are close to the Minkowski metric m mu nu is a Minkowski metric so in particular in relative to this model problem I'm interested then in discussing the initial value problem say the simplest initial value problem which is when phi is exactly equal to zero so phi equal to zero is of course a solution of this this is quadratic here quadratic in d phi so this is clearly a solution phi equal to zero is a solution so this solution corresponds in some sense to the Minkowski metric in terms of this analogy is the equation here so here I want to perturb the Minkowski metric where all the components are so these are minus one one one and zero everywhere else and here I want to I want to perturb phi equal to zero so it's a very reasonable approximation very reasonable model problem so I want to prescribe now t equal zero I want to prescribe phi to be say epsilon some small parameter f of x and d t of i I want to be epsilon times g of x well let's say f and g are some smooth functions with compact support for simplicity so anything very reasonable functions and I want to show that if epsilon is sufficiently small I would be interested to prove stability I would like to show that if epsilon is sufficiently small in other words if I perturb around the solution phi is equal to zero let's call it phi zero if I perturb around this I want to get a global solution which converges back to this one for example ok so this is a kind of question that you want to ask is it true that this problem is stable and there is small perturbation in other words for small epsilon so this is the kind of thing that I want to talk about it so ok so I want to deal with this kind of issue so let me actually simplify even more and look at down version of phi it's just d t of phi squared so this is simplification number 4 which is the one I have it there and again you start with initial conditions which are this one small so for small epsilon I want to know I want to understand what happens so how do you deal with this problem and if you don't have this term here then the only thing that allows me to understand the solutions of the wave equation are energy norms energy so the simplest kind of energy identity is that if I take e zero of phi to be one half integral of d t phi squared plus d one phi squared plus d n phi squared I assume that I am in Minkowski space of n plus one dimensions so if I take e zero of phi evaluated term t so I am integrated on t equal constant so I call it sigma t sigma t mean t equal constant so I am integrated with respect to dx sorry there are n coordinates so it is dx one dx n dx n so I am integrated with no time coordinates so the time is fixed so I am integrating in other words the picture is that this is t equal zero this is t equal say t zero and this is the t axis and this is rn axis alright so I am integrating at fixed t and the conservation law tells you that e zero of phi time t is the same as e zero of phi time zero so this is just conservation of energy which is something very easy to deduce ok but of course my problem is that I have something on the right hand side so just this conservation law by itself is not enough and in fact what one does is you look at higher derivatives you commute these equations with derivatives so you get this is a flat wave I can commute here I commute here and then I apply the energy estimate for the new field here and is what I call es of phi so this is the energy for s derivatives which is the same thing it is e zero of phi let's say d alpha of phi so it is a sum for absolute value of alpha less or equal to s in other words I take all derivative up to order s and this is my a new norm which is the generalized energy norm which I call es of phi so es of phi again if I were exactly in Minkowski space this would be true es of phi time zero so this would also be conserved if I am exactly in flat space but if I am not in flat space I have to do something about this term so you have to do it's not a big deal one can prove the statement that one can prove so I can prove the statement that the energy the full energy for the entire system the nonlinear system remains bounded by the same energy at time zero provided that I have a certain condition satisfied which is very easy to see when you do this energy estimates now with the right hand side it's easy to see that you can control all the energy if you can control this quantity in other words if you can control the integral from zero to t of dt phi in n infinity is bounded by say one so as long as you control that you are fine but you see this is highly non trivial because you need in particular the n infinity norm decays so in order for this to be integrable it has to decay at least like t to the minus one minus something so the n infinity norm has to decay so is it true that you can show that the n infinity norm is decay and here it's the main the major technical complication in all these business even when you do perturbation expansions ala t-boa I was hoping that t-boa is here but it's not if you do those you have the same kind of difficulties so in order to prove anything it's not enough to control the energy you also have to control this quantity so how do you control that quantity traditionally and this I'm sure is done by t-ball so when he does his calculation traditionally this is done by looking again at the original linear equation in writing down the fundamental solution so using the fundamental solution combining the fundamental solution with a nonlinear part it's a mess, it's very complicated in any case in linear if you have just this equation it's actually not too difficult to see from the fundamental solution if you write down the fundamental solution it's not so difficult to see that the solutions will decay like t to the minus n minus one over two in the finite of the parity of n in every dimension you get exactly t to the minus n minus one over two that's the optimal thing optimal decay of course there is a superposition of waves some waves behave fast decay faster but you will always get waves which decay only like t to the minus n minus one over two and nothing better and then of course this is a problem and in particular it's a problem because you see that if n is equal to three here which is an interesting dimension this will be divergent that integral will be divergent so you will not be able to do anything but even more complicated is how do you ensure that this cell is not decay at all because it's extremely complicated now if you use a fundamental solution it will be a huge mess but there is another way of doing it which is I think much better and so that's the one that I want to describe very fast this has to do with what's called the vector field method so the vector field method is based on the idea that somehow you should not commute only with normal derivative you have in Minkowski space you have in addition you have in addition to do so you have in addition to the usual derivatives which commute so we have that dT, dx1 so dxn so I'm going to call it d1, dn they all commute with a wave equation so you have the d alpha that are virtually zero and that's why you could form this higher energy estimate but there are other vector fields which commute in fact a lot of vector fields for example there is the vector fields xi, dj minus xj di or xi dt plus tdi and there is also tdt so xi di see here I'm summing over I so I is from 1 to n so this vector field I'm going to call s this is actually a Lorentz boost so this is what we could call Li from Lorentz and this is an angular so these are generated by rotations these are generated by boosts and these are generated by scale transformations it's very easy to see that all these vector fields commute so any of these vector fields let's call it x commute to the del inversion either is zero or actually in the case of this one you get minus 2 del inversion so in particular it takes solutions to solutions so because of the sorry these are symmetries of the del inversion but they are in fact symmetries of course of the space they are killing so all these vector fields are actually killing so these are all killing in mykowski space and these are conformal killing so they are all very useful so they are useful because they commute and therefore I can put these vector fields here and instead of looking at generalized energy norms of this type I can take any vector field here any number of vector fields so this allows me to define some kind of sobole space this would be some kind of sobole space which is a generalization of the usual solar space because it contains all sorts of vector fields ok so the consequence of all this is that this generalized energy norm which I have there this generalized energy norm which I have here is actually conserved for the wave equation it is conserved so if I take the solution of the wave equation then this e s of i is also conserved so it is another conserved quantity and the remarkable thing about this quantity is that it allows you to show that the solution decays in fact so if you remember I said that the solution should decay like this minus n minus 1 over 2 but you see that here you get even more than that since e s of i is conserved it means that if it starts by being bounded at time t equals 0 it is going to be bounded at later times therefore I can assume that e s of i is bounded I am using s larger than n over 2 the s test for the number of vector fields I take here so I take s derivatives integer larger than n over 2 it is exactly the n over 2 and I am able to control the infinity norm in terms of the sobole norm which is larger than n over 2 but instead of having just boundedness I also get decay now and the decay comes from this vector field so anyway so this is the whole point because it allows you to reduce decay to energy so instead of doing decay using the fundamental solution which is extremely complicated and almost never works I can do the decay I can incorporate information about decay in my basic energy norms energy is much more robust energy estimates are much more robust we use it all the time in PD so anyway so this is what happens and conclusion is that now you get this kind of decay rate you see that in this picture when t is exactly absolute value of x so if I look in Minkowski space sorry I have t equal 0 here and this is t equal r t equal absolute value of x absolute value of x equal to r so if I am if I am looking at solutions of the wave equation along null direction so if I look at the behavior along null directions I see that the best behavior is exactly the one given by so this will not be useful because t is like absolute value of x so this is b of 4 to 1 and I am getting just t like minus n minus 1 over 2 so I get exactly the decay which I have here but this gives me additional information because it tells me that if I am inside the light cone so in particular if this is if absolute value of x is much less than t then I get another t to the minus a half from here so I get the decay even better I get t to the minus n over 2 so as a consequence this is a much better way of understanding the decay if I am to deal with nonlinear problems and once I use this type of estimate by the way there is even more which is extremely important in what I am going to say if I see phi decay is only like t to the minus n minus 1 over 2 but if I take the frame so remember that I have this frame e3 is L bar which is dt minus dr and e4 which is L which is dt plus dr so if I take if I take e4 of phi or if I take the other elements of the frame e4 or ea so I have the other e1 e2 here for the null frame so these are orthogonal to these two if I take the derivative in this direction in fact I get t to the minus n minus 1 over 2 minus 1 so I gain another decay for both of them and the only one which does not improve is e3 or 4 so e3 or 4 is t to the minus n minus 1 over 2 so in fact in fact e3 derivative improves but improves in this component so instead of being 1 plus t minus x minus 1 over 2 it will be minus 3 over 2 but it means that near the light cone it still does not improve so that's a remarkable amount of information that you can get from these very simple functionalistic methods which are based on symmetries once you have that now you can go back to this type of equations that we discussed derivation of phi is f of phi derivative of phi and second derivative of phi so I look at in other words I look at very general perturbations of just the wave equation derivation of phi equal to 0 and I'm looking at the vacuum state phi equal to 0 I'm looking at the stability of the vacuum state phi equal to 0 and again if you remember my discussion of Slava last time that he couldn't believe that exponentially growing modes is not good the fact that you don't have exponentially growing modes is not good for stability so here you see if you look at the derivation of phi equal to 0 in other words if you look at linearization around 0 this is a question you get which is of course stable not only that but it also decays if I am in dimension 3 it decays like t to the minus 1 if I am in dimension 1 it decays like t to the power 0 so it's bounded in any case but if I perturb it if I look at the nonlinear problem and I look at general preservations in fact these are unstable for example if I look at these equations in other words this would blow up in finite time and it forms in fact shock waves so the shock waves can be the result of a very of a perturbation of a very simple state like phi equal to 0 and of course I mention also turbulence in the case of the Euler equations again you start with u equal to 0 and you can end up in finite time you can end up with solutions which have which you have absolutely no control which are extremely unstable nevertheless the state u equal to 0 is a bound the linearization around it gives you a bounded solution so there is no issue of exponential growing modes or anything like that it's much less and you still get alright so anyway in dimension n equals to 3 typically most equations are unstable because of this equation in dimension less than 2 it's even worse dimension 1 is terrible if you are in dimension 4 it gets better you can actually prove something dimension is critical but of course since we are interested in dimension 3 dimension 3 most equations are bad you need equations which verify structural conditions so in order to have existence to have stability of this vacuum state I need some kind of condition on this structural condition on this this is called the null condition so if the null condition is satisfied which I call here if I verify the null condition then phi is structurally stable so what is the null condition so I'm not going to go into a formal definition I'll just say very simple which I mentioned here you see, relative to a null frame if you look at the composition with respect to null frames you see that derivatives in these directions improve and it's only this direction is very bad and therefore you expect a nonlinearity of this type to be bad but any nonlinearity where you have say e3 multiplied by e4 or ea this will be ok so the null condition is just a way of saying that the worst possible directions are not present when you do a decomposition of the nonlinearity relative to the null frame so it's a very simple procedure you take the null equation you look at the nonlinear terms you do an expansion in terms of the null frame and if you see the presence of these bad states you are done it means the null condition is not verified and if you don't see it then you say the null condition is verified now of course things are more complicated but that's roughly the simplest way to say that now let me actually this goes I'm sorry I should have realized that I can do it this way so dimension n is equal to critical so null condition is something extremely important which will play a role in what I'm going to say and I'll finish the first hour so what time is it so it's a good time to take a break so I'll finish with this fact geometric nonlinear wave equations verify some gauge dependent version of the null condition so this is a remarkable fact many equations of interesting mathematical physics that are derived from a geometric Lagrangian do verify the null condition so for example the Einstein equations verify the null condition but and there is an important but the null condition you see it's a condition of the nonlinearity it's not about the linear equation it's a structural condition and the complicated equations like the Einstein equations do verify the null condition but only if you take into account the gauge condition so the gauge condition is essential and only if I if I mod out the equations by by the deform office group and I look at in other words I look at the correct framework correct gauge dependent framework I will see the null condition so for example in what I mentioned earlier here if I look at the Einstein equations in this kind of harmonic ordinance they don't satisfy the null condition which is just simply not true there is something else which is called the weak null condition which is still verified in this context but the null condition is not verified and never the lazy Einstein equations definitely verify the null condition and that's the reason why why the Mikovsky space will be possible all right let me finish with this vector field method before taking the break so the vector field method of studying nonlinear equations which to some extent you could say is it's certainly not dependent on perturbations so it's a it's a robust method of deriving estimates, decay estimates by reducing it to energy type estimates so integral estimates which is based on symmetries or approximate symmetry or other geometric features to derive generalized energy bounds in other words generalized energy bounds I mean these kind of norms which have those vector fields but it could be even more complicated than that to derive energy bounds and robust and infinite quantitative decay because as we saw you are also going to have an infinity quantitative decay and it applies so this method applies not just to the wave equation as I showed here but it applies to tensor field equations like the Maxwell and Bianchi type equations in Mikovsky space so for example the Maxwell equations are such an example where you can still use exactly the same techniques in order to get decay you commute this equation by taking a leader derivative relative to the same vector fields you can commute and you get the same equations verified by the leader derivatives then you create norms based on these leader derivatives and from it you read the decay and then you can treat nonlinear problems as a consequence so I'll stop here so the vector field method so again the vector field method you could think of it as a non-perturbative tool to study classical field equations something that does not require anymore to talk about expansions and fundamental solution it's non-perturbative and it's very general it can be applied in many situations in particular it applies to Maxwell equations as I said but it also applies to the Einstein equations in the following sense so if you if you look at a solution of the Einstein equations so reach of g is equal to zero then if you look at the Bianchi identities so you take cyclic permutation equal to zero it's very easy to see from here if you take the divergence it's easy to see that this also implies the divergence so there is a derivative with respect to alpha equal to zero which I can write as delta of r is equal to zero but at the same time it's not so hard to see that something similar happens with r star and I can derive also so in other words if I take the hodj dual and I remember the hodj dual is defined alpha beta gamma delta is epsilon where epsilon is a volume form epsilon alpha beta mu nu and r mu nu gamma delta was a one half here so there is a very simple way of defining the dual so the hodj dual verifies the same thing delta of r star is equal to zero only with respect to alpha so you see that formally you can write down the Bianchi identities, you can write it like this dr equal to zero this corresponds to this but then you have also this other one delta r is equal to zero and this is very similar to what you had so so and now issue you can start doing the same thing that we did for Maxwell equations you can start doing the same thing by taking the derivative with respect to vector by commuting with various vector fields so I take x1, xn and I hope that this goes inside and therefore after that I treat these equations very much like the Maxwell equations so the Maxwell equations in Minkowski space of course there is a problem because I need it in order to do this in Minkowski space I need the symmetries of Minkowski space so I need keeling I need the axis to be keeling or conform a keeling vector fields and of course if I take a general solution of the Einstein equation there is no way I am going to have keeling so the only thing I can hope and this is what I am going to talk about in a second is that this x1, xn are approximate keeling vector fields so you have to so in Minkowski space you have the keeling plus conformal keeling so in perturbations of Minkowski space I have to take approximate keeling plus conformal keeling so what does it mean? well it's very simple to define in a way because in Minkowski space so in general keeling vector field means that the derivative with respect to the vector field of the metric is equal to zero right so this in general I am going to define it as pi of pi so the deformation turns out of x so pi x alpha beta is in fact d alpha x beta plus d beta x alpha so this is from the keeling equation right so this is equal to zero implies that x is keeling but of course I cannot expect to have keeling vector fields if I do a general perturbation of Minkowski space I won't have them but I might hope to have this sufficiently small so of course Minkowski say d of d x of r is not going to be zero is going to be here some complicated expression involving r and pi something involving a product between curvature and the pi the deformation tensor so let's call it pi x the deformation tensor of this vector field and this now is of the same order of complication that what I had here when we treated the ambition of i is the derivative of i squared because now this is a complicated term that I have to control and in order to control it I need decay now so there is no way I can prove anything any stability result if I don't know how to control this one but to control this one I need to control the decay at least of one of the factors in fact I need to control both factors I need to control the decay of both factors both the curvature and the pi that's the only way I would be able to actually control the curvature so the crucial thing when you do stability of Minkowski space is to control the curvature you have to control the curvature without which you don't control anything so let me now go through a discussion of that so let's talk about stability of Minkowski space so I'm trying to solve the gauge flat I start with the initial data set which consists as usual a three manifold a metric which is now it's a it's a Riemannian metric and a second fundamental form and they verify the constraint equation which I'm not going to write because it's not relevant but the constraint equations of course are by themselves very interesting so now I also impose a gauge condition at least initially which is sorry not only initially I impose a gauge condition which is k equal to 0 so you see I can impose 4 coordinate conditions you have to think about when I do wave coordinates I had 4 coordinates here because alpha can be 0, 1, 2, 3 so there are 4 possible coordinates so here I'm using the full coordinate here I only prescribe one coordinate which is in fact t it's a time coordinate so in other words I'm trying to construct I'm starting with sigma 0 and I'm constructing a foliation by sigma t which is a time function so by time function I mean a function defined on the manifold such that the level surfaces are so first of all this is the different level surfaces are space like exactly but I can think of this t as being 1 coordinate condition I have 4 conditions to make and I can make 1 and I assume that this is maximal so assuming that it's maximal it means that if I look at the second fundamental form trace of the second fundamental form relative to the induced metric here should be 0 so that's in maximality condition so this is what is done in Stavidiopin-Kovski space it turns out that it's not so important but at the time we saw that such a time function is fundamental ok, so then you have the constraint equations plus trace k equal to 0 and now you look at the initial data set forming Kovski which is exactly R3 this is the Euclidean metric second fundamental form so that's how you start and you assume asymptotic flatness you assume asymptotic flatness so in the definition of asymptotic flatness now you have to be careful so in other words asymptotic flatness is always taken relative to a system of coordinates outside a large compact set so I am at sigma 0 let's say I take a sufficiently large compact set and I look outside and I look at the system of coordinates and now the initial data alright and I look at the components of the metric relative to that initial data which are this gij and that's where you see the mass 1 plus 2 m over r delta ij so gij minus minus delta ij but you have to subtract also a term which is like r to the minus 1 so this is long range for those people who know the Coulomb the Coulomb potential that's exactly this r to the minus 1 component which is sort of a very slow decaying component and in front of it there is mass and the positive mass theorem tells you that for general initial data this m is positive this is a famous theorem of Sean and Yao so this positive mass theorem is only tied to the constraint equation only it has nothing to do with the evolution it's just an issue about the constraints okay so these are the assumptions and in addition I want a smallness condition in other words I want to make a preservation of the flat initial data so I want to be initial data which are close to this one and you can make that precise there is no point in going through this so I start up with this initial data set so it's a small initial data set so it's close to the set of Minkowski space and then I look at the maximum global hyperbolic development of this and I want to asymptotic flatness is on the initial data asymptotic flatness is the initial data but it's carried by the evolution so the evolution will carry this asymptotic flatness so the question is what is the character of the maximum development in other words we know that there is a maximum development in the distance result that tells me that I can go up to something but it could be that my spacetime terminates in finite time relative to proper time of a given parameter of a given observer and that of course is unacceptable and stability should mean that at least should mean that the spacetime construct should be complete so this is a theorem which is a theorem of 1993 and it says that any asymptotic initial data set close to the flat one has a complete maximum development which converges back to Minkowski so here you are not converging to another parameter you could have converged to another black hole for example you can go from Minkowski to a Schwarzschild this doesn't happen fortunately in time it could develop a black hole later on and it could be converging to Minkowski so the statement is that actually you stay in Minkowski's space and this is very very important because remember that this example is dt phi squared this blows up in finite time which means that there is an instability here which may lead to a black hole in the case of the Einstein equation in this case it's a shock wave but in the case of the Einstein equation it might lead you to another black hole or it could lead you to another singularity who knows because of these kind of examples it is not at all clear that such a statement is correct the physicists as always they have their own way of simplifying the problem and they would say that yeah well of course of course it should have been stable but you still you still have to give a reason for that and the kind of reasons they usually gave were not satisfactory so for example one of the reasons that was talked about is that since the mass is positive you cannot have you cannot the Minkowski space has to be stable but in reality you can give lots of examples where you have positivity of mass and it's not you don't have stability the simplest one is again the Berger equation that is equal to zero then I see that u equals zero corresponds to the energy u squared so if I look at the energy this energy at t equal constant is actually conserved and you start and it's positive so you have positivity of the energy for any initial data and yet of course u equals zero so this forms a shock wave in a very fast in a very very short time so perturbations of all the epsilon form a shock wave by the time epsilon to the minus one so this is very very far from being stable and of course also piece Slava he's not here you can see that the linear case is just ut equals zero which obviously it doesn't have exponentially right modes u equals zero the linearization is exactly ut equals zero which of course all solutions are bounded so so you are very far away so the fact that you have some kind of linear stability doesn't tell you anything about the non-linear equation ok so the Mikovsky space is this one so now how do you construct it you construct it together with a gauge condition let me make a little bit let me come back to something I said earlier so remember that I said that at the level of the curvature so we have the frame so you have the frame that is the c3, e4 and the alpha ea or e capital A and then we have the gammas which are the crystal principle or the Ricci coefficients and then we have the curvature here we have components like chi, chi bar and so on and so forth and here we have components alpha, alpha bar, beta beta bar rho and rho star and and we think in terms of perturbations of this so these are these all of epsilon perturbations that I discussed earlier and remember that we said that these are invariant when I do the stability of Mikovsky space invariant, which is a very important fact so because of this curvature itself the components of the curvature do not depend much on how I pick up my frame but on the other hand the gammas are going to depend in a fundamental way and I cannot control the curvature if I don't control the gamma because in the Bianchi identities, if I write down the Bianchi identities because the covariant derivatives depend on the gamma which is essential also in controlling the curvature but somehow the good thing is that I don't care too much when I treat the curvature I don't care too much about which gauge I choose right ok, this is after the fact that the time we are not that was not exactly the way we saw it alright so so you have to pick up a gauge right and the gauge consists on two things this time function that I already mentioned which was maximal but more importantly this optical function which is properly initialized so this is much more important in fact because in order to in order to understand decay so you start up with initial conditions you construct a time function so a foliation by a time function but remember that the decay properties of curvature of waves in general depends on this null directions because in null directions most of the energy is transmitted in null directions this is you get the worst decay and outside you get much better decay so it's extremely important to construct to construct some way by which you keep into account what are the null directions and this is where you construct this time function this optical function you construct an optical function which is defined like this so you construct the space time to get those foliation by a second function so you have t and now you have a second function u which should be viewed as like u equal constant should be like light cons in Mikowski space the way to make that sure is to solve the so-called iconal equation so this is a metric d alpha of u d beta of u is equal to 0 so this if you remember from the very beginning of my lectures I said that's the way to see null hyper surfaces u equal constant if I solve this equation u equal constant is a null hyper surface so that's how I'm going to construct the u but of course by itself is a non-linear problem because this is quadratic in derivative of u and of course it also depends on the metric so when you actually solve the equations you have to think about solving r alpha beta equal to 0 together with this one these two are the fundamental building blocks in the construction of the space time you construct both this and this and from this once and from the t you get the intersection so the intersection that you see here which I'm going to call stu so these are two surfaces and from here I can construct the null frame so once I have t and u I have stu and then I have the null pair e3 and e4 e1, e2 which I construct very easily I mean in this one is generated by u so it's a null geodesic null vector field associated to u the other one which is like this I construct it based on this one and the fact that I already know this derivative transversal to t equal constant so this is dt very symmetry so it's some kind of symmetry I can construct the other one so from this two I can construct the second one so I get both e3 and e4 now and then of course the e1, e2 are just tangent to these two surfaces so this is a frame so this gives me the frame e3 plus e4 is dt e3 plus e4 is essentially dt I have to normalize it also a little bit but it's in the direction of dt correct so now so this is my frame so this is my frame so you see once I have these two I can define a frame, a null frame once I have the null frame of course I can define the gammas and then I can write down equations for the gamma equal to curvature so somehow the way to think about now is that I have two type of equations I have equations at the level of the curvature itself and the equations at the level of the gammas right somehow if I know the curvature I can determine the gamma by just integrating these equations so that's sort of a big thing that needs to be done but the first and the most important thing is to actually determine the curvature and that's where the fact that the curvature that is all fepsion squared invariant is going to be very important so how do you deal with the curvature ok, so let me write it here ok, so the curvature is the crucial thing and there I have to I have to think about so we have the Bianchi identities dr equal to zero and these other equations which are divergence equal to zero so these are I can think of it as some kind of Maxwell equation right, it's more complicated because it has more indices but it's similar to a Maxwell it's formally similar to a Maxwell equation so it's always satisfied that yeah, yeah, yeah, so if Richie is zero then this is the Bianchi identities and this one is the divergence which is follows from this and this right, so you get something which looks very much like the Maxwell equation so the idea is actually to do this as the Maxwell equation so in other words, I have to start taking the derivative re t to vector but first of all how do I define my vector fields right, so it turns out that the vector fields can be defined using t, u and the frame so for example I mean just to give you an example if I want to define this t, dt plus x i, di as an example, I mean the analog of this this will be u times e3 plus u bar times e4 where this one is contracting by u plus 2t, so if I have in other words if I have u and t I can define this one e3 I have, e4 I have and u I have and this is the exact analog of s for example so this is a kind of construction and I can define other vector fields like this right, so I construct in other words I construct the vector fields to be intimately tied to the to the these two functions and the connections associated with these two functions the important thing is that you see there are only two functions not four that I have there I only need to construct two functions and it's a much more geometric construction because I know exactly what I construct and I know the reasons I construct so t is maybe not so important ok, so once you once you've done that you now have the vector fields and you can start commuting right, so you're going to get d of lex of r and d of lex of r so this delta but of course here you are going to get pi's, complicated expression involving the pi and curvature right, so this this is going to be again the hard part alright so now so the way to deal with this is in a first approximation you ignore you assume that somehow you are like in flat space you get the estimates for the curvature which are going to be energy type estimates from it you get the decay rate using this kind of global sobolev inequality that I mentioned earlier so you get the decay rates for the curvature so you get the various components of the curvature decay at different rates so for example alpha will decay like t to the minus seven half and beta t to the minus seven half then rho and so on alpha bar which is the lowest component dk is only like t to the minus one so see this is exactly the alpha bar if you remember contains e3 twice e3 so in other words this is a bad direction e3 is a bad direction for this reason alpha bar is the worst component behaves only like t to the minus one t to the minus one is exactly the behavior of a solution of the wave equation so you cannot do better than this so alpha bar is what you observe when you do LIGO experiments all the information I carry by z is because all the answers decay too fast to see so this is what you are going to see in LIGO experiments because it's a gravitational wave obviously it's a curvature that carries gravitational waves so that's sort of the general philosophies let me go through a little bit so we have the gauge condition which I explained robust decay based on the vector field to get decay for r and there has to be a null condition so the null condition has something to do with the structure of this term because you are forced because you want to treat these terms here you are forced to decompose everything you have to look at the components of this components of this and hope that the worst possible decay rates that come from here and they come from here are such that they are not similar to what I mentioned earlier this is 3 of i squared you don't have to have you have to avoid having components of this type and if your if the construction is geometric then you would avoid it and indeed my construction here is geometric so that's sort of the general philosophy let me go a little bit more into this ok so this is a theorem you construct this maximal variation you construct a null variation by this null hypersurfaces you construct an adaptive null frame which are these ones e4 is 3 and the ea's which are tangent to this intersection between t equal constant and u equal constant so this variation by two surfaces these are compact two surfaces so it's integrable in this case it's integrable because it's sabitiomikowski space the expectation is that it's integrable if I do stability of care I expect to get something which will be non integrable so so you get stu which are the intersections you define r to be the area radius of this stu sorry is that something to do with what I do or not no I don't do anything so it's probably not me ok anyway ok so you get r you define r as a geometric in a geometric way is being the area radius connected with the area radius of these two surfaces so r should be a long null directions r has to be like t because you expect this null you expect this u equal constant to be similar to the t minus r equal constant in minkowski space but of course there is a deviation here but the deviation you hope is not too big so in a case t and r have to be comparable and here is what you get in terms of r alpha becomes like r to the minus seven seven half r to the minus three r star r to the minus seven half r to the minus two r to the minus one this is the component that again that someone that you see it's being seen by lego so so this are this is called incomplete peening because somehow penrose had sort of a not hoc way of deriving the decay estimates for the curvature making certain assumptions so based on certain assumptions you are able to find much stronger decays so he would find for example r to the minus five here r to the minus four, r to the minus three and so on but in the stability of minkowski space we prove much less much more consistent with what is actually going on in fact the strong peening is non-generic ok it's a null hypersurface no stu is away from a tiny so you have this cu here so this is a null hypersurface which corresponds to equal constant and then you have t equal constant so and then you have stu equal constant which are these ones, the intersection so these are the wave fronts stu are the wave fronts so ok so that's the kind of decay you get this is again another picture initial data sigma zero t equal constant will be seen here these are the null cons which are here I call h but I call it c before but these are the null cons and these are the intersections and you see there are some I mean obviously these are not spherical anymore because the gravitational waves distorts the wave fronts but this this is how the spacetime construct looks like so the role of curvature again which I mentioned before all null components are relative to adaptive null frames in all of epsilon of the initial values these are these do not depend on the frame effectively gauge independent and that's why you can analyze it by this thing that I told you here so this this method which I mentioned here will not unfortunately work when you do stability of black holes and the reason is because not all components are for epsilon square invariant you get certain things which stay there forever and therefore you are not going to be able to get decay in this way you have to do something much more drastic but in any case in the case of the stability of mykowski space you have a uniform treatment of all components of curvature by using a tensor which is associated to this equation which I just mentioned I don't know why I mean this is a method of notation I mean this is not a very good notation in any case you can associate to this equation something which is called the bell Robinson tensor which is a remarkable 4 tensor which plays a role of an energy momentum tensor so you see if you talk about Maxwell equation df equal to 0 and df star equal to 0 then there is associated to this equation there is a energy momentum tensor which is called t alpha beta is f alpha mu f beta mu plus the dual so you can define the dual so you get something completely symmetric so this is maybe one half here this is the energy momentum tensor and the energy momentum tensor has this remarkable property that we all know this comes from notice theorem that this divergence is equal to 0 so it's symmetric and divergence is equal to 0 and once you have this then the way you get conservation laws according to notice theorem is that you look at a vector field x you take this contraction t alpha beta x beta you get a one form because you have just this index you take the divergence of this one form and you get here if d alpha falls on this you get 0 if you falls on this you get t alpha beta let me write it this way t alpha beta t alpha beta x beta plus d beta x alpha exactly so I put a one half and that's the symmetry of t and this is exactly the derivative of the metric g so in the healing you get 0 here and that gives you the conservation law by integration you integrate you get conservation law otherwise you have to take that into account this is exactly the pi x so in the for the answer in the question there is something similar but remarkable because it's not a 2 tensor it's a 4 tensor again maybe one half there is a I'm not going to put all the indices there will be a combination of indices r times r plus r star times r star in the correct way of putting the indices so you you have two contractions exactly you have two contractions and you are left so this is 4, this is 4, two contractions you are left with 4 so this will be a 4 it's fully symmetric and it's so this is for reach equal to zero it's fully symmetric and it's traceless so the trace relative to any two indices is zero and it verifies a divergence so d alpha t alpha beta gamma delta is exactly identically equal to zero so in other words it plays the role of an energy momentum tensor except that now instead of taking just t alpha beta gamma delta you have you can play with three things here you can take x1 alpha x2 beta x3 gamma and then you do this and again you see the same thing because of the symmetries you are going to see the the deformation tensor of this vector field showing up on the right hand side so in particular all of them are keeling or conformal keeling you get a conservation law right of course you don't expect for solutions of the Einstein equations you don't expect to have keeling so you always have something on the right hand side but this is what I wanted to say here that you can define energy generalized energy estimates just like we did for the wave equation from which you get once you have those energy type estimates you derive decay for each component of the curvature you can derive the decay it has so this is sort of the effective invariant way to treat the wave character of the Einstein vacuum equation so again this is remarkable because again you don't need the fundamental solution I mean typically t-boy is not here but if we ask him we see that's what he does he uses the fundamental solution of course here you don't have to use the fundamental solution at all in the last way of deriving the decay and of course deriving energy estimates also ok so finally proof is based on a huge bootstrap with three major steps so first of all so it has to be a bootstrap because that's why you can't do anything so you start by making certain assumptions of the curvature norm so these are this invariant curvature norms involving these vector fields you assume that they are bounded for all time and from it you get precise decay estimates of the connection coefficients of the t-u foliation based on equations which I wrote here so which were these ones d gamma plus gamma times gamma is r so if I if I have if I make a bootstrap assumption if I assume something about r then I can derive estimates for the gammas once I have the estimates for the gammas I can use in to derive estimates for the deformation tensor of this keeling and the approximatic keeling vector fields because as you remember these keeling vector fields are defined based on on the geometry of of the foliation and therefore based on the based on the frame and based on the richer coefficients of the frame and so on and then once I have this then I can go to this step I can do this step which is the most complicated one I can do this step and show that indeed I can get bounds for the curvature but of course to do that I have to control the error terms which are generated here which require lots of things in particular decay for all components but also requires in addition the null condition if the null condition is not verified I am dead because I will not be able to control these error terms just like in the wave equation I recall in the wave equation if I have dt phi squared and I try to implement my strategy I will not be able to because this blows up in fact so the same thing here in order to work you have to have this null condition satisfied and fortunately it is and therefore this is how you get bounds on the curvature and therefore you can go back here and close the whole loop because this unfortunately takes quite a while original proof we had about 600 pages I think now maybe less 580 nowadays of course it can be done much faster so there are other proofs which are much faster ok, there is another result which I think would have been of interest to t-bow I am not sure since he is not here I am not sure that it was worth spending too much time this is something that that a question that he was I remember interested from the very beginning when I met him there the thing was that this peeling these components of the curvature which are r to the minus seven half r to the minus two and r to the minus one was being predicted by physicists before they had this strong peeling which was predicted by penrose or bondy socks also and so he wanted to know whether you can do better whether you can get the full peeling anyway so here I will mention very fast two results first result which is with Nikola in 2001 which was the following thing which was instabilitiv Mikovski space so he started with sigma zero gk the same condition the same initial condition but I don't assume any smallness I don't make any smallness assumption but instead I look at the domain of of influence of the future set of a sufficiently large compact set so I take a large compact set on sigma zero and I am only interested in what happens outside so here if I have large data of course I am in the regime of the final state when lots of things can happen here I can have extremely complicated things so this result tells you that if you forget about this part which obviously is going to be difficult and you are interested only sufficiently far from a compact set then this behaves like in the stability of Mikovski space in other words the data now is going to be sufficiently small because of asymptotic flatness because asymptotic flatness means that things become small and small as we approach infinity so things are going to be sufficiently small and therefore I can construct my spacetime all the way to the null hyper surface generated here so in other words I cannot go beyond but I can construct all the way to the null hyper surface and this is based on a double null variation so the innovation here is that you don't use t equal constant anymore of course you couldn't because the maximality of t will make it impossible because make it impossible to use it just in this region something maximal has to be global so it will have to go inside the region which you don't control so you construct only something outside but instead of constructing with t equal constant you construct with a double null variation so in other words you take in addition to this null cons you take another family of light cons so in other words you replace t equal constant you replace it by u bar equal constant which is another family of null cons now moving which are now incoming so moving in this direction so in other words in this region like this so I can the intersection see the intersection is still going to give me two surfaces which are s u u bar so instead of having s ut as I had before now I have su u bar as level surfaces and I can do the analysis more or less in the same way so the only innovation here is that it's what it's called the double null variation is given by two functions u and u bar verifying these equations in other words it's very similar and the result is very similar and finally the same power yeah so those would be the same power but I make also the same assumption in that result but now I can say more so this is another result in Nikola in 2003 where we show that so it's a matter of the of the assumptions on the initial data if you remember the assumptions were here these ones what does it mean ok plus one ok so let me explain so if you don't take any derivatives it's just g i j minus one plus two m over r delta i j is all far to the minus two so in other words if I subtract this is a schwarzschild part if I subtract the schwarzschild part from the metric what I am left to is terms which decay like r to the minus two k plus one means I am also taking a certain number of derivatives every time I take a derivative of this relative to the coordinates because I have a coordinate system in the neighborhood of infinity it decays better by power of r it improves by power of r the same thing here it improves by power of r so now in here you see I can so in this result I make stronger assumptions so if I take away the schwarzschild part then I can go all the way to r minus three half and plus a gamma in other words gamma is a parameter which allows me to make to make strong and stronger assumptions at infinity right so for example if gamma is exactly three half and r minus three here so I get much stronger decay than before so I take the schwarzschild I get much stronger decay so if gamma is larger than three half in other words if the initial data is decays sufficiently fast then I can get the pendulum's peeling in other words r to the minus five for alpha and r to the minus four for beta but you see this requires this requires so it's somewhat non-generic this was postulated by Penrose so this came from the analysis of Penrose, yes, yeah so Penrose sort of assumed that a space time can this kind of space time which are a flat solution of the Ricci flat equation can be conformally compactified by adding a boundary at infinity but this is an out-hawk assumption I mean of course it was never justified, in fact nobody was able to justify this conformal compactification picture but this estimate is still true, right, exactly yeah, I mean you can do it provided that you take enough decay which it's not, yeah, I mean the exact amount of decay is important in applications but that's something should have been discussed to Stiboli if you ask him what is g and gs? that's a schwarzschild part so you have g and k right and I take gs the schwarzschild in other words is 1 minus 1 plus 2 m over r and then what's left should decay fast you cannot, you see because of the mass the positive mass theorem tells you that if m is equal to 0 then in fact the g has to be exactly the Euclidean g so if you are to have any non-trivial perturbations they have to contain this one over r terms otherwise it's not right so you always have to have this long range but if you are close to schwarzschild but if you are close to schwarzschild in fact, yeah, exactly so in a certain sense is also tied to Stabilitio schwarzschild but schwarzschild of course much more difficult anyway so I think this is a good place to start so next time which is just the last two hours I'll really talk about this new result on black hole stability with Jeremy and that will be to show at least so the black hole stability is much more difficult in the case of schwarzschild and Ker the only thing which is known today is in linear theory so there are many interesting results in linear theory but non-linearity is much more difficult and the result I'll mention is the result on Stabilitio schwarzschild under restrictive perturbations those restrictive perturbations are such that they constrain the final state to still be schwarzschild because normally if you perturb schwarzschild you will not stay in the schwarzschild class you will go into a care with small a, with small rotations you always generate some rotations unless you make some restrictions so the restrictions we make is just so that the final state is still schwarzschild but nevertheless you have to take in you still have to work hard to adjust for the final mass to track the final mass because the final mass is going to be different and to track for the gauge is dynamically the correct gauge in which you have decay that's the hardest part in fact so I did with this, I'll stop