The gravitational waves detected by LIGO were produced in the final faze of the inward spiraling of two black holes before they collided to produce a more massive black hole. The experiment is entirely consistent with the so called Final State Conjecture of General Relativity according to which generic solutions of the Einstein vacuum equations can be described, asymptotically, by a finite number of Kerr solutions moving away from each other. Though the conjecture is so very easy to formulate and happens to be validated by both astrophysical observations as well as numerical experiments, it is far beyond our current mathematical understanding. In fact even the far simpler and fundamental question of the stability of one Kerr black hole remains wide open. In my lectures I will address the issue of stability as well as other aspects the mathematical theory of black holes such as rigidity of black holes and the problem of collapse. The rigidity conjecture asserts that all stationary solutions the Einstein vacuum equations must be Kerr black holes while the problem of collapse addresses the issue of how black holes form in the first place from regular initial conditions. Recent advances on all these problems were made possible by a remarkable combination of geometric and analytic techniques which I will try to outline in my lectures.
 Ladies and Gentlemen, we ought to be talking about Temar L дог המomic예 letings. I want to thank the IHES for its very kind invitation to give these lectures. And I will be talking about general activity, concentrating on problems which are connected with black holes. So I apologize maybe at the beginning I will start with an introduction. So for people who are already very familiar with the subject of general activity, it might be a bit boring. But I hope to get something of interest to everybody. So this is an introduction. So let me talk a little bit about general activity. So this is a physical theory which has a very strong mathematical background in the sense everybody knows. Of course that Riemannian geometry, the notion of curvature, played a fundamental role. And what's remarkable about general activity is that just about any concept which has physical meaning can also be merely translated into something which is purely geometric. So for example, you have Lorentian matrix, Lorentian geometry is at the heart of general activity. So that means you have manifolds. I should take some chalks and also put something on the blackboard. Is there a chalk somewhere? Below. Oh, here. Yeah. Okay. So obviously the starting point of Lorentian geometry and therefore general activity, the fundamental object is that of a manifold with a metric. And of course the only difference is that, so the manifold is the same definition as in Riemannian geometry. The only difference is that the metric of interest is not Riemannian metric, but rather Lorentian. So it has signature minus plus plus. You can also have higher dimensions if you want. I'll put here 1 plus n. And therefore physical concepts like inertia translates into just the fact that the tangent space at every point on the manifold is a Minkowski space. Events, so physical events are nothing else than points in the manifold. Observers are what are called time-like curves. So that means, well first of all, I should say at every point in the tangent space, you have a light cone where the metric degenerates. And you can immediately distinguish between time-like directions, space-like directions, and null directions. So the division is simply based on looking at vectors at the point where the metric will be negative, zero, or positive. So this is space-like, this is null, and this is time-like. So therefore you have a notion of time-like curves, which simply means it's a curve on a manifold such that at every point the tangent space to the curve is time-like. So time-like in the sense that it verifies this. So this corresponds to observers. So observers in general think they are nothing else but time-like curves from mathematical point of view. So again, this remarkable translation between things which are purely geometric and things which are physical, light rays, so this is where light is supposed to propagate, are nothing else but null geodesics. So the geodesic, again the concept of the geodesic exactly the same as in the Devanian geometry, and null simply means a geodesic, which is such that the tangent at every point is, it keeps you g xx equal to zero, or x is a tangent to that point. Proper time, which is nothing else, so the proper time along a curve is nothing else than the affine parameter, just like in the Imania geometry. Tidal forces in whichever, a clear physical meaning are nothing else but curvature, right? So again, the curvature meaning exactly the same, it has the same meaning as in the Imania geometry, given the metric g, you can associate the so-called Riemann curvature tensor, so Rijkl, let's say, and the definition, the formal definition in Lorentzian geometry is exactly the same, there is no difference, right? So you have tidal forces, you have then isolated systems, again a physical concept, which means that you imagine, you imagine that something interesting happens at the local point in space and time, say a galaxy or just a star, and then the isolated means that somehow the remaining part of the manifold is really just flat space. So somehow you focus on the part where something interesting happens and everywhere else things become flat, in other words, there is no additional physical input into the system. So these are called isolated physical systems, so again, it's nothing but asymptotic flatness. Asymptotic flatness, the definition, the formal mathematical definition is just, again, you are on a manifold and typically the manifolds of interest, particularly if you talk about isolated system, is the ones which are, say, defomorfic, at least outside a sufficiently complex space, they look like R1 plus N from a topological point of view, and you want the metric also to look Minkowski and if you are sufficiently far away from this isolated system. All right, so that's, as you see, everything translates. I mean, this is a remarkable thing about general activity, is that everything interesting from a physical point of view has a very simple geometric meaning. Equivalence principle, which is at the heart of general activity from a mathematical point of view, is translated simply in saying that all physical laws can be expressed only in terms of the metric and its induced connection. So if I have the metric G, so again, in Lorentzian geometry, it's exactly like in Emanian geometry, you can associate the connection, which is the so-called Levy-Civita connection. So exactly the same properties as in the Emanian geometry. All right, so more concepts of interest in Lorentzian geometry, which come all the time. So first of all, time orientation simply means that I have a manifold, which is Lorentzian, and orientation is given by a vector field, say, maybe I'll call it T, a vector field, which is globally defined on the manifold, and it's time-like at every point. So that gives you a time orientation, which allows you to separate between things which moved forward in time and backward in time. Curves which are time-like or causal, I already said what time-like means. So time-like simply means that you have a curve where the tangent is time-like at every point. Cozal would mean that you allow the tangent also to be null. So future and past sets, well, let me go here. Future and past sets, again, a very simple concept, but it's quite important in Lorentzian geometry and, of course, in general activity, is that if you have a set S on your manifold, you are interested in all points. So the future set of S is a set of all points which can be connected to S by a time-like curve. And causal set will mean that you also allow the curve to be causal. So for example, if you have an alkan in Minkowski space, then the future of this point, the time-like future will be all the points inside the light cone. The causal will also allow the points on the boundary. So future and past sets, hypersurface, so this is another thing that, of course, you discuss in Lorentzian geometry, you can discuss in Lorentzian geometry. The difference is only that now you have three types of hypersurface of interest. The ones which are space-like. So space-like hypersurface is such that if you look at the unit normal at every point, it's time-like. So you need normal time-like at every point, which means that if I look at the Lorentzian metric and I restrict it to the space-like hypersurface, I get the Riemannian metric. So that's Riemannian geometry as a consequence is part of Lorentzian geometry. And part of general relativity, you really have to know Riemannian geometry very well in order to succeed doing anything of interest in Lorentzian geometry, I mean in general relativity. All right, so we have hypersurface is such a space-like, hypersurface is such a null. So this is a null hypersurface, right? So it's, as you see, it's ruled by geodesics such a null or curves such a null, let's say. And finally, you can have something time-like. Time-like would simply mean that, for example, if I take a section like this in Minkowski space, this would be a time-like section, right? So in that case, the metric induced is still Lorentzian. So you have the original Lorentzian metric. It induces a Lorentzian metric on a time-like surface. Another concept of importance which shows up a lot is that of a Cauchy hypersurface. So this is a space-like hypersurface with a property that if you take any point in the manifold, so you say that the manifold has a Cauchy hypersurface or that this is a Cauchy hypersurface in the manifold. If you take any point on the manifold, you can, and you look at any time-like curve, it will have to intersect the Cauchy hypersurface in, always it has to intersect the Cauchy hypersurface, and intersect it just exactly at one point, right? So obviously, T equal constant in Minkowski space is a Cauchy hypersurface because obviously every curve which is time-like will have to intersect this. But if I look at this hypersurface which is space-like, it's still space-like, it's not a Cauchy hypersurface because obviously I have all these time-like curves here which is not intersected hypersurface, right? So anyway, it's a very important concept and we'll talk more about this. All right, so the next topic, we're having a eraser somewhere, so the same thing, I have to find it here I guess, but it's not. On this side, okay, good. All right, so, all right, so frames. So in the Emanian geometry, of course, the kind of frames that you see always in the Emanian geometry, so if I have a manifold with an Emanian metric, typically a frame is a set of vector fields, say u1en on g such that the metric gEiej is equal to delta ij, right? So for all i and j, and this is a Kroniker symbol. In Lorentzian geometry, of course, you can define the same thing. You can have frames which are called orthonormal, right? So for example, if you are in Minkowski space, you can take a time direction, say E0, and then u1en-1, or en, let's say if I am in a plus or not dimensions, will be space-like curves, space-like vector fields which are perpendicular to E0 and E0 is time-like. But you can have another more interesting type of frames which play a much more fundamental role in general activity, which are so-called null frames. So null frames. So null frames instead are frames which consist of, say, vector fields l, l bar, and then E1 up to en-1 if I am in n plus one dimensions. So these two are null, right? So gll is gll bar l bar is equal to zero. You also normalize them. Of course, you cannot do anything like in the minority geometry where you take one of them to have lengths one because they are null, right? But I can normalize l and l bar so that this is equal, say, minus two. So for example, if I am again in Nikovsky space, I can take vector fields, so this is a null cone, at the point, right? I can take this to be l, this to be l bar, and I normalize them such that the length, the scalar product between them is minus two. And then the other ones, the other ones will have to be space-like because they are orthogonal to these two. They will have to sit here somewhere. They will be space-like. And I can pick them like in minority geometry to be orthonormal. So this will be orthonormal among themselves and perpendicular to l and l bar. So null frame, which I'll talk a lot about null frames in these lectures, is really defined like this, right? So for example, here is an example of a null frame in Nikovsky space. If I take dt minus dr, where r is a radial vector field, so I call this l bar, and dt plus dr, I call it l, and the other ones will be perpendicular to these two and orthonormal among themselves, right? So that's a null frame. And as you see, these null frames play much more fundamental role than the orthonormal frames. All right, so Cartain formalize, well, again, this is exactly like in minority geometry. If I have a frame, I can define, so if I have the frame, let me do it here. So if I have a frame, I can define the connection, right? So I have, say, in the minority geometry, e1, en, and I define the gammas, which are the derivative of the frame taken with respect to the frame. So the components of the derivative with respect to the frame, where the derivative is, of course, a covariant derivative. So it's induced by the Levy-Civita connection. And then the Cartain formalize tells you that somehow derivatives of gamma plus gamma, so you get equations which are the Cartain equations, which will flex it, sorry, equal to curvature. So on the right-hand side, you have the Riemann curvature tensor. So I'm writing it just very formally. Whenever we need, I'll write it more explicitly. But these are the fundamental equations in minority geometry. And of course, they play a fundamental role on some of the Lorentzian geometry, except again that I will be using, instead of orthonormal frame, as in Cartain formalize, I'll be using null frames. Foliations. So another thing which is extremely important in problems in general activity is to have a Foliation. So one kind of Foliations would be given by Minkowski space. For example, it's just equal constant. And for a general Lorentzian manifold, the role of t is placed by what is called the time function. In other words, a function such that the exterior derivative, so dt, the differential of t, is time-like at every point. So that's a time-like function. And then of course, I can talk about t equal constant. So this gives me a Foliation by space-like hypersurfaces. So this will be on an arbitrary Lorentzian manifold. Another kind of Foliation, which is in fact even more important, is given by so-called optical functions. Optical functions. So instead of time functions, I'll talk about optical functions. So these are functions given by u. So optical function is a function u, which verifies the I-colonial equation. So gij diudiju is equal to 0. In fact, actually, to distinguish between Riemannian and Lorentzian geometry, I'll use instead, most of the time, I'll use Greek indices. So this will be d alpha of u, d beta of u is equal to 0. So this means nothing else that if I take the exterior derivative of u, so if I take the differential of u, then g du du is equal to 0. In other words, if I look at u equal constant, the differential of u is in the direction of the gradient of u. But this means that somehow the gradient of u is also orthogonal to itself, which means, of course, that du itself is tangent to the hypersurface. And this, of course, can only happen if u equal constant does not look like this at all, but looks like a null hypersurface. So u equal constant is, in fact, null. So in other words, if I solve the I-colonial equation, which is, of course, an equation extremely important in all sorts of branches of mathematics and physics, if I solve the I-colonial equation, I get the solution u, the level surfaces of u will give me a foliation by null cones. So for example, again in Nikovsky space, I can have a foliation given by, for example, like this, where all these are null cones. So I can foliate part of the spacetime by null cones. And these foliations, again, are going to play a major role in our discussions. Finally a little bit about the role of Riemannian geometry. I already said that if you have a space like hypersurface, you immediately have equations, which are typical to Riemannian geometry. So you have Riemannian geometry because a metric induced on a space like hypersurface is actually Riemannian. So the geometry on a space like hypersurface is Riemannian. And therefore, lots of concepts that come in Riemannian geometry will be of great use also in general activity. But there is another kind of variation, which also is extremely important, maybe even more important, and will play a role in our discussions, which is, say, foliations, which are obtained either by taking an intersection with t equal constant. So imagine that you have a time function, t equal constant on a manifold, and you intersect that with null hypersurfaces, like this. So the intersection then consists of two surfaces, which are Riemannian, because the metric induced to be Riemannian. So we call them, so this is u equal constant. Then these surfaces will be called S of t u. So if I am in 1% dimension, this will be 1n minus 1 dimensional. There will be the metric induced Riemannian, and this kind of the geometry, so this is also has the advantage of being compact, right? So these are compact Riemannian manifolds, and again, they play a major role in our discussions. So finally, to finish with this introduction, I want to say a little bit more on null hypersurfaces. So again, the null hypersurfaces can be given by these optical functions, in other words, function u such that g du du is equal to 0. But they also rise naturally as boundaries of future and past sets, right? So if I have a set here, and I look at the future, the boundary of the future is null, at least in the portion of the boundary, which is regular. So if there are, so these are typically ruled by null geodesics, right? So the boundaries of future are ruled by null geodesics, right? So in other words, they can be constructed by putting null geodesics together. So for example, if you start with a two surface, you take the null geodesic perpendicular to the two surface at every point, and that gives you a null hypersurface. Of course, you could have the problem that these null geodesics, which are perpendicular to s, at some point will intersect because you can have conjugate points or cut locus. And then the boundaries become more complicated, but I'm not going to discuss it. So as long as in the absence of this conjugate point and cut locus points, these null boundaries are smooth and they are given by null hypersurface. So let me end this with an interesting observation of Poincaré, who said, parmi ses actions en plusite, il est un qui semble mériter quelque attention, parce qu'en l'abandonnant, on peut construire une quatrième géométrie aussi cohérente que celle de Euclide, de Lobachesque et de Riemann, je ne citerai qu'un de cette hauretme et je ne choisirai pas le plus singulier. Une droite réelle peut être perpendiculaire à elle-même. And of course, this is what happens if you take a null hypersurface, if you take the null geodesic, which is a straight line if you are in flat space, in Minkowski space. And this is of course perpendicular to itself relative to the induced Lorentzian geometry. So the Lorentzian geometry induced on a null hypersurface is actually singular and is very different from what happens in the Manean geometry. And this leads to a lot of interesting mathematics, which has been somewhat ignored by geometers. I would say that Poincaré is always his visionary and he's understood that this could be very important and indeed this null hypersurface is extremely important in general. Okay, so with this let me go now to the beyond Lorentzian geometry, what is the geometric framework of general activity? Well you have field equations, so this is now, obviously you are looking at Lorentzian, I'm always losing my, we are looking at Lorentzian manifolds, which in addition to being just Lorentzian, they verify an equation. So general activity is distinguished from Lorentzian geometry by this fact that you are interested only in Lorentzian metrics, which verify an equation. So the equation is just the Ricci, so as you know if I have the Riemann curvature tensor and I take a trace with respect to the metric I get the Ricci tensor and so this is a two tensor and the equation looks like this, minus one over two G, times the scalar curvature, so the scalar curvature, so this is Ricci, scalar curvature is obtained by taking one more contraction with a metric G and on the right hand side you have T which corresponds to the input of some matter theory which is present in our spacetime. So these are the famous Einstein equations and I'm not going to talk directly about them, partly because so this part of the Einstein equations is sufficiently interesting in its own right and in fact most of the complication, there are lots of complication which we have to do of course with the matter fields which are present, but in a sense in mathematics we always want to separate difficulties, so these are a set of difficulties like for example you can have fluids and you have all the difficulties that have to do fluids or you can have Maxwell equations, all sorts of other things that are on the right hand side and to simplify I'm simply going to ignore that part, I put this equal to zero which is of course makes perfect sense, so this will be if you are to make an analogy with Maxwell's theory is studying electobar, pure electromagnetic waves without sources, right, makes perfect sense and it's already very interesting, but in general activity this is also very hard. So in this case it's very easy to see that the equations actually immediately imply that the Ricci has to be zero. So the equations of interest in my lectures will be this one, so you have Ricci equal to zero, sorry, sorry, what do you mean conversely? Yeah, correct, so if t is equal to zero these two are equal to zero indeed. Okay, so these are the equations that I would be interested in and again mathematical is extremely rich and you see essentially all the important, most of the important parts of sort of black or seori have to do these equations, right? All right, so let me talk a little bit about the character of these equations, what's special about these equations, so again we have Ricci flat, Ricci of G is equal to zero and first think of interest is what you mean by a solution, so we have to solve this equation, this is a complicated equation, I mean if you think about how the Riemann curvature tensor is defined it involves two derivatives of the metric plus nonlinear terms, so two derivative plus nonlinear terms, but these two derivatives also come up in a complicated fashion and if you don't take precautions you don't see any interesting partial differential equations showing up here. To make it into an interesting partial differential equation you have to take into account another important fact about this equation which is that there are different morphism invariant, in other words if I have a solution G, if I have a metric G I can all distinguish between phi star of G where phi is any deformorphism of the manifold to itself, so one plus n, one plus n, right? So a solution is actually a class of equivalence of solution modulo, this huge group of deformorphism of the manifold, right? So these are generally different, that's why the Einstein equations are called generally covariant. So if you take that into account then solving the equations you have an additional, the initial bonus that comes from it which is that I can try to solve it relative to a specific coordinate system or a specific deformorphism, so I look for solutions together with a class of, together with a specific, with a specific kind of deformorphism which will allow me to solve and the simplest way is, the simplest observation of this type is that if I assume a coordinate system, so you see this covariance allows me somehow to say well there are, I can solve them by equations in a specific coordinate system, right? So if I pick up the coordinate system to verify this very simple equation where this down version here is nothing else but the Laplace-Beltrami operator but associated to the metric which is now of course not Riemannian, so you don't get the usual Laplacian but you get what is called the wave operator, right? So this is defined exactly the same way as in Riemannian geometry except that because of the metric, so these are covariant derivatives applied to X-alpha, and this operator is what we call the wave operator, right? It's exactly like the Laplacian but it's much more complicated because of the signature. Is that what people call also harmonic coordinates? Yeah, this, in Riemannian geometry I call harmonic coordinates. It's interesting that actually harmonic coordinates were discovered earlier in physics than in geometry, right? So these were known, in fact, even Einstein himself had some knowledge of them but they were certainly understood by people in the French school like Yvon Trocadro which used it in her famous result on existence for solutions of the answer equation, right? I'll talk more about this in a second. Okay, in any case, once you fix up your coordinate system then these equations take this form and as you see here you have once again some kind of wave operator applied to all components of the metric. So here I'm fixing the components of the metric. I apply this and I get something on the right hand side which is a complicated expression which will involve the metric and first derivative of the metric and say it's quadratic in the first derivative of the metric, right? So in any case, it's recognizable. The character of the equation is now recognizable. This is what we call nonlinear system of wave equations. It's clearly hyperbolic, right? So this is what one calls the hyperbolic character of the equations, right? So I'll talk more about things like this later but for the moment I just want to give a general introduction to these things. Okay, so equations are hyperbolic, right? That's what we discover. We discover that if we use the gauge dependence of the Einstein equations, if you do it, you can change it to our advantage, you can actually show that the equations are hyperbolic. Now being hyperbolic, it means you have to solve sort of similar problem as you do, say for example, if you are in Minkowski space, right? So the metric is then just a Minkowski metric. Then say I want to solve just the inversion of phi is equal to zero. So the inversion being again, this is minus dt squared plus Laplacian of phi is equal to zero. And typically one solve this by choosing initial conditions, let's say t equals zero. It can be any space like hyper surface. You can take initial conditions on any space I have, but in post-implicity you just take t equals zero and then you get phi of zero x is say f of x and dt of phi of zero x is g of x. So this you prescribe and then you show that there exists a unique solution that solves this equation, right? So this is for the simplest possible equation. Of course you can do that for much more general classes of equations. Yes? So does the formalism of general relativity is the same that the formalism, the eminator used to demonstrate theorems about physics? There is of course a connection obviously, but I'm not going to get into this now, right? Yeah, but of course there are conservation laws and all sorts of things that you have to do. There is a whole theory, but I will not talk about this now. Here I want to talk about something very simple, which is that you can solve the way of equation, unique solutions for the way of equations you find if you prescribe these two initial conditions, right? So you want to do something similar in general relativity. So in general relativity the role of this f, so the role of this t equals zero hyper surface and these two initial conditions is taken by what is called initial data sets. So an initial data set will consist into a space, some three dimensions, so let's say we are in four dimensions for simplicity, right? So we are in four dimensions, so this will have to be three dimensional, starting one dimension less. This will have to be a Riemannian metric because I know that if I can solve the Einstein equations, I will get a Lorentzian manifold and this original hyper surface sigma zero embedded into it will have to be space like. And therefore this as an input will have to be a Riemannian metric and this will correspond to the time derivative of the Lorentzian metric. This is called the second fundamental form of the original space like hyper surface, right? So typically I prescribe G zero and K zero and sigma zero and I solve for the constraint equations. So then there is a unlike in this situation, you have to do a little bit more work about the constraints. And as those constraints cannot be, the constraints cannot be prescribed arbitrarily. They have to be something which is called the constraint equation, which is very much like in Maxwell's theory. In Maxwell's theory, when you prescribe the electric and magnetic field, you prescribe it together with the constraint, which in that case is very simple because it's linear. In this case it's much more complicated. It's a nonlinear constraint and this actually the study of solutions and the character of these constraint equations has led to a lot of developments in Riemannian geometry. This you could view the constraint equation, initial data, constraint equations can be viewed as a subject in Riemannian geometry, right? And I won't talk about this at all, but I just want to tell you, for example, that the so-called positive mass theorem, which is a famous theorem in Riemannian geometry, played a major role not just in connection with where it started in general activity, but in many other applications of problems in Riemannian geometry. And they have to do somehow with this, with the geometry of these initial data sets together with those constraint equations. Once you have set up the initial conditions, you can talk about solving the equations in the same way as you solve it here. Of course, this is a linear problem. That one is nonlinear. It's much harder, but nevertheless it turns out that there is a very general theory of nonlinear hyperbolic equations, nonlinear wave equations, which allows you to solve the Einstein equations in the wave coordinate. So you have to use a wave coordinate conditions. To start with initial data, you construct a spacetime at least for some time. It will have to be local to start with, because typical nonlinear problems can develop singularities. But you can solve it for some time. You can also talk about the maximal future development. In other words, this would be something similar to what happens for simple ordinary differential equations. So x dot is equal to x squared. If I want to solve it, some initial data, x of 0 is given. Then obviously I can only solve it for a local amount of time, because you can have singularities. But you can talk about the maximal time of existence of solution of this equation. And the analogous of this notion, and this, by the way, is not just for this equation, for any nonlinear ordinary differential equations, you can talk about maximal time of existence. And the same concept can be discussed in the context of these initial data sets. So in other words, I can start with an initial data set, and I can talk about developments, and I can talk about the maximal development. And then somehow the entire subject of general activity, you could say, is really nothing else but the discussion of the character of this maximal development. So the theorem, which was proved by Ivonsho K. Brown in 1952 and complemented by Gerroch in 1969, is that to any initial data set, you can associate a maximal future global hyperbolic development. Global hyperbolic development, what does it mean? It simply means that the space time that you have constructed, by the way, all this concept came in the French school, by the way, I should say. Then the formulations of the initial value problem as a hyperbolic problem was really not well understood until Loré, Schokebouin, and many other people in the French school. So this is really a good place to talk about it, because it's a... So anyway, so this was... The hyperbolic character of the equations was definitely an important part of the work of Loré and certainly Ivonsho K. Brua, and then Bruha Gerroch. You associate with initial data, you can associate this maximal future hyperbolic development. That simply means global hyperbolic development. By the way, global hyperbolicity was defined by Loré, in fact. But for us, it simply means global hyperbolic, it simply means that the space like you are starting with, when you construct your space time, this three-dimensional, limanian surface together with a first and second fundamental form of this, is what you want to say. So I want you to say that relative to the space time that you have constructed, this is a Cauchy hyper surface. So global hyperbolicity is something very simple. When you solve the Einstein equations using PDE techniques, you automatically construct a globally hyperbolic development. So that's part of the construction itself. So again, you have this maximal global hyperbolic development, you can associate with any initial data and therefore what's left is to discuss the character of this maximal future hyperbolic development. It sounds simple, but certainly not. So let's talk now about the next step after this. So the next thing that needs to be said before going into real problems is to talk about what kind of solutions you could have in general activity. Of course, solving the equations in general is very complicated and it's really the role of mathematicians to solve the equations without necessarily producing explicit solutions. But the physicists who started first to work on this, they wanted to look at explicit solutions. So that's natural and we don't blame them. In fact, it's extremely important. So how do you look for specific solutions? You have to look for symmetries. You have to assume that somehow you have all sorts of additional symmetries. So what kind of symmetries you can have? Well, the symmetries in geometry are given typically by vector fields, which are key link or conformal key link. In other words, they generate, so these are vector fields, which generate one parameter group of deformal fields, which correspond to isometries or conformal isometries. So that's how you look. So when you look for symmetries in general activity, you follow the same idea. You look for solutions which have additional key link or conformal key link vector fields. The simplest case is actually when you have a lot more, not just one, it's spherical symmetry. So you look for solutions with spherical symmetry. So in that case, you have an action. It's more than just one or two key link vector fields. It's an action of the SON group. I'm not going to say much more about it. There is a very simple definition of spherical symmetry, in which case once you impose spherical symmetry, the Einstein equations become much simpler, and you can actually solve the next velocity in certain cases. So I'll say more about it in a second. But of course, you can also look for solutions which have just one symmetry or two symmetries. So say x1, x1, and then x1, x2, and so on and so forth, and you get more and more simplifications as you go. So if you impose enough symmetries, in fact, you reduce the Einstein equations, which are partial differential equations, you reduce it to just ordinary differential equations, for example. But you can also, if you have one, for example, typically, you have to think about having one such key link or conformal key link, typically key link, having one such key link vector field reduces the dimension. So suppose you are in one plus one dimension, and you have one symmetry, you reduce it to n dimension. So you reduce the problem from n dimension to 1 plus n dimension. Maybe I should say 1 plus n minus 1, for example. So you simplify the equations quite a bit, but they could still be very complicated. That symmetry makes life much easier. And in the case of the Einstein equations, of course, the simplest solution is the Minkowski space itself. The Minkowski space certainly verifies this equation. So Minkowski space is a spatial solution of the Einstein equations in vacuum. But then there are more such solutions. The most interesting class of such solution was discovered by Schwarzschild. So here is an example. So this is Minkowski metric in polar coordinates. So you see the difference between this and Schwarzschild is that you add these terms here, where m is some constant, and you want to take it positive. So m can be also 0. The case equal to 0, it's exactly the case of Minkowski space. But you see that the moment you introduce a norm 0 m, the solution becomes somewhat more complicated. The character of the solution was actually, even though the solution was discovered by Schwarzschild in 1915, immediately after Einstein, it took more than 50 years to really understand the character of the solution. But the next, so this was discovered in 1915. The next one was discovered much later in 1963, which is a famous care family of solutions. So let me now go to here. So this is how the care solution looks like in coordinates, which are again like polar coordinates, T, RT times 5. These are called the Boyer-Linckes coordinates. And you see the metric is more complicated now. But it's still amazingly explicit. This is one of the, probably the most remarkable solution. I mean, we should discuss, but I think it's one of the most remarkable solution in all of physics, because it's clearly a solution of a nonlinear problem, highly complicated nonlinear problem. It's explicit, right? Because all these coefficients are calculated by this very simply trigonometry formula. So it's explicit, and it has huge impact, right? I mean, basically, whatever we think about black holes today are essentially based on these solutions, right? So showing that this is a solution is not so easy at all. I mean, if you try, you probably spend at least a few days to try to calculate and show that to indeed the solution. Of course, if you use a calculator, it's much smarter. You will probably get it in a few hours. Sorry? You use the fracture forms. It makes it easier, certainly, if you introduce correct. All right. Anyway, so this is the family. Let me make some remarks about it. First of all, you can see that this is asymptotically flat. You see that asymptotically flat from the character of these coefficients here. You see that when r goes to infinity, right, that r goes to infinity, you get closer and closer to Mikovsky space. So that's just standard to see immediately. Yes? So these equations are homogenous. Excuse me? Homogenous. Homogeneity. Sorry. All those in units. I'm not looking at units now, right? I mean, in terms of physical units, you mean? Yes. No, no. I'm certainly, I'm writing it as purely a mathematical solution, right? At this stage. You can always put units. But I'm not going to do it. All right. So anyway, so the first observation is that, again, this is asymptotically flat. So it's an asymptotically flat solution. It has symmetries because you can see immediately that since these coefficients do not depend on time, right, you can see it here. They don't depend on t. It means that d over dt in this coordinate, so the vector field d over dt in this coordinate, is killing, right? OK? So this gives you that these solutions are in fact stationary. Stationary, by the way, I didn't say. I should have said it earlier. These solutions means, I guess I didn't. Yeah, OK. Anyway, I'll discuss about stationarity later on. The important thing is that this vector field, which is killing, is also time-like. At least as you go far away in the r direction, as r goes to infinity, it becomes, certainly it's time-like. We'll discuss about the actual global behavior later on, but for the moment, it's stationary because this d over dt is time-like. The second observation is that it's also axisymmetric in the sense, again, that if I look at all these coefficients, they don't depend on, the coefficients don't depend on phi, which means that d over dt phi is a killing vector field, right? OK? So this corresponds to a rotation. So this corresponds to a rotation. So these are, we say, are black holes which are rotating. You see that there is an extra parameter instead of m that you have now two parameters. You have a and m, and in fact, this set of solutions are supposed to be physical only in the regime a is strictly less than m. Actually, sorry, less or equal to m. Equal to m is called the extremal case, right? All right. So again, you can see that if a is equal to 0, you are reduced to Schwarzschild, and of course, if m equal to 0, you are reduced to Mikołski space, right? So this is the famous two parameter family of solutions of the Einstein-Vakum equations, which verify all the interesting assumptions that you want. It's asymptotically flat, and it's stationary, and it's also axisymmetric. All right. So now, come much time. Maybe I should take a few more minutes, and then we'll take maybe a break. So here I want to talk about the D major conjecture in general activity. It's probably, I mean, there are some others, but I would say that this is sort of the dream conjecture that you'd like to solve. So what is this conjecture? So it's called the finite set conjecture. So it says the following thing. It says that generic means not all, there could be some exceptions to this conjecture, right? So I say generically, of course, you don't know what that means at this stage. Of course, whenever you solve a major conjecture, you also have to define everything. In particular, you have to define what you mean by generic. But leave aside generic, any asymptotically flat initial data set that you take, right? So it has to be asymptotically flat, has to be smooth, of course, but I make no other assumptions. So this is global future development. So in other words, there is this global maximal development that I talked about before, right? So you go as far as you can go. And here you are saying that this maximal future avivory developments, which are maximally extended solutions of the n-stand vacuum equations, so that's the same thing, look asymptotically in any finite region or space as a care solution, right? So let me make it a bit more graphic. So you take an initial data set. You take a very general sigma 0, g0, k0. You take a very general initial data set, which is, of course, asymptotically flat and smooth in whatever sense. And you look at the maximum future hyperbolic development. So this tells you that the maximum future hyperbolic development is not finite. It doesn't terminate in finite time. It goes all the way to infinity, right? So this is the first major statement, which is included in here. The notion of maximum hyperbolic development, when we talked earlier about maximum future hyperbolic development, it could have been that if I look at a time like curve, the time like curve terminates in finite time. In other words, observers will die after finite time, right? So that means from a physical point of view, it's not a very interesting solution, right? So this is not an interesting scenario. So here, it tells you that in reality, what happens, you can go for infinity time. But it's even more interesting is that what you see asymptotically will be just a finite number of care solutions, right? So asymptotically in time, whatever that means, of course, you have to define the two. But say intuitively, it's pretty clear. So asymptotically in time, you are only going to see a finite number of care solutions, which are black holes. And they are going away from each other, right? So because in principle, they could have interacted before, when they interact, they form only one black hole. So if you have two that come together to form only one black hole, so asymptotically, you will see only a finite number. In any compact region, in other words, you see only one care solution at most, right? And they could rotate. So that's why care, that's why the final states will be care. So what is? Rotate two black holes rotating around each other. Well, the conjecture says that this is not possible, right? But I'm not saying that the conjecture is true, of course. And generically, right? Yeah. Because then they will evaporate, they will emit traditional waves in the area. Correct. Yeah. Correct. So that genericity is also very important in this business. But of course, you also can have gravitational waves, which goes to infinity. Right. So what is the basis of this conjecture? Intuitively, it's very clear what it says. Somehow, any solution would radiate energy at infinity. So these gravitational waves would be generated at infinity. And therefore, you will be left to stationary solutions. But the only stationary solutions, this is another conjecture we'll discuss in a moment, any stationary solution has to be a care solution. That's another belief that people have. Right. So anyway, this is a conjecture. Now, I want to show you a little bit, before taking the break, I want to show you a little bit what is behind, what are... I mean, if you really try to understand this conjecture, you will see that it consists on many, many things, which by themselves are extremely interesting. So for example, the first thing that you will have to connect it to this is that if the data is sufficiently small, in other... Of course, if I am in Minkowski space, the Minkowski space obviously has no black holes. You don't see any care solution at the end of the day. Right. It's just flat Minkowski space. So you... In order to solve this, the first thing you like to do is to see that if I have small data, in other words, if I make a small perturbation of Minkowski space. So by the way, the initial data of Minkowski space will be... The metric here will be the Euclidean metric. So maybe I'll write it Euclidean. And this will be zero. Right. So I start with Euclidean, and the second fundamental form is zero. Right. And then I want to make a small perturbation of this. The fact that a small perturbation leads to... It doesn't lead to any black hole is what it's called the stability of Minkowski space. Right. So it's a statement which in itself is highly non-trivial. The second statement that is hidden inside that conjecture is that large data, in other words, if I take the data now sufficiently large, it may concentrate. So it actually can lead to a black hole. Right. So it can lead to a stationary solution. So you could produce stationary solutions if the initial data is sufficiently large. So this is called the problem of collapse. Right. So for example, from a physical point of view, you know that if you have enough matter, then stars can at some point create neutron star. And if you have enough energy, you can create a black hole. Right. So that's the problem of collapse. You can see that it can be formulated in pure mathematical terms, in a sense, and without matter. So this is just in vacuum. Right. But these sort of things can happen, and we'll discuss it. The next thing which is hidden in that conjecture is that all stationary states are care solutions. Right. So you see, the care solution is just a family of solutions which happen to be stationary. But how do you know that all other stationary states are care solutions? Right. So the conjecture tells you that you see at the end of the evolution, you see only care solutions, which means that somehow all stationary solutions will have to be care. Right. So this is by itself a highly non-trivial statement. There is no reason not to expect other solutions. So this has to be also studied as a mathematical problem. Care solutions are stable. So this is another major thing, is that if the conjecture were to be true, then care solutions themselves will have to be stable. Because if I make a small perturbation of a care solution, I can't get something crazy. Right. So the stability of the care solution is also part of this conjecture. And this is just the soliton resolution conjecture. Exactly. Exactly. It's a soliton resolution conjecture in the class. For relativity. But with the difference, I mean relative to the one that you guys are talking about, there is a small difference. But I'll talk about it. I mean stability. Stability is a major difference, because you do have stability of all these states. So the solitons are all stable. Right. So that could be a, anyway, we can discuss it. So, okay. So other things that are included in the final state conjecture, which are huge conjecture in their own right, which is the cosmic censorship conjecture. So if this final state conjecture were to be true, then you should also verify the cosmic censorship conjecture. Now the cosmic censorship conjecture says something which again can be made more precise, but I'll do it intuitively. You see what happens in the, when you look at the, say, Schwarzschild metric. The Schwarzschild metric has, it can be drawn by a diagram, which is called the Penrose diagram. I'm not going to talk, I mean there is something here at infinity, but I don't care about. There is a black hole region. So this is the separation. So this is black hole. And we'll talk more about it in a second. So this is the part outside the black hole. So this is the exterior region. This is a black hole. And once you are in the black hole, you reach a singularity. So this is something that the character of the solution that we wrote down will tell you immediately that once I start up as an observer, in other words, I started with a time like curve in the black hole, I will necessarily have to hit a singularity in finite time. Finite time means relative to the affine parameter of the corresponding time like curve. So in finite time, I hit a singularity. So in other words, it's something very objective that you can, it doesn't make sense. It's a proper time of the observer. On the other hand, if I'm here, and of course if I'm stupid enough to move in this direction, I will also hit a singularity. But I have a chance to escape because I can go this way. And I don't hit any singularities. So because of this fact that you can see that somehow the expectation is that outside black holes, you can't have singularities. So the only singularities are hidden by horizons, which are the boundaries of these black holes. And so this, of course, happens in the particular case of Schwarzschild solution. The generalization of it is called the cosmic censorship conjecture. So the generalization is that any initial data you take, again, generically, because they are counter examples, but generically, any initial data you take, it will have to have be such that you can define a black hole region where you can have black holes, and therefore you can have singularities in those regions. But outside the black holes, you can't have any singularities. So this will also have to be part of that conjecture, of the finite state conjecture. And by itself, it's as difficult as any other conjecture in mathematics. And in fact, maybe even more so. And probably more interesting than many other conjecture in mathematics. Alex, you believe that the Riemann hypothesis is the most interesting conjecture in the world? You have to admit that there are some other, very hard and extremely interesting. All right, anyway, the other thing that is also hidden here is a two-body problem, two and maybe more, because in fact, actually, this corresponds to the LIGO experiment. The recent LIGO experiments have to do with two black holes interacting, right? So they get closer and closer to each other, they rotate around each other, and at some point they coalesce, and they form a new black hole, right? So there is no mathematical theory of such a thing at this stage. There is a lot of numerical evidence and asymptotic expansions, but there is no mathematical theory, right? But this is clearly also part of the conjecture. I mean, you will not be able to solve the conjecture if you don't understand how black holes interact, right? And of course, this has a clear and time, I mean, clearly clear physical significance right now, right? Because of the LIGO experiments. All right, so there is a lot of evidence for the conjecture. There is astrophysical conjecture, numerical, mathematical, but we are extremely far from solving anything like this. The mathematical evidence, I am not going to talk about these ones. Of course, this we know from a lot of, I don't know, newspapers talk a lot about it, right? So we know that people believe that there are black holes all over the place, and therefore somehow this conjecture has to be true. I think the most montreal part of the conjecture is the absence of singularities. Suppose that there are some singularities of mild type, which is the physical evidence we have again. Yeah, I agree. I agree, but they are all connected in a sense. But they are all connected in a sense, right? I mean, this lack of singularities connected to stability, for example. Stability is also, I mean, obviously, if you don't have stability, you don't have anything. Right? You don't. Okay, well, we'll talk more about this. Anyway, so there is simplifications. Linear theory, one can do linear theory and one can be quite rigorous in linear theory. But that's a huge simplification because the equations are nonlinear, highly nonlinear. There is a symmetries. You can take, look at various symmetries and you reduce the problems to something much simpler. And what I will talk about is the emergence of stronger mathematical techniques to deal with this, at least to some of these issues. Right, so then, I don't know, we should take maybe a three-minute break or? Okay, so I talked about, so introduction and I mentioned this main problem in general relativity, which as it stands, is much too difficult and it contains a lot of deep problems in their own right. And therefore, as mathematicians, the only way we go when we have a huge problem is to divide it into smaller steps, right? Smaller problems. And so that's what I want to talk about now. I want to pick, in particular, I want to pick three issues, three questions where I think one can make progress. By the way, here was something else that I should have done in the first hour, which has to do with the picture, the standard picture of gravitational collapse, which is at large energy concentrations, right? May lead to the formation of a dynamical black hole, which settles down by gravitational radiation to a care or schwarze in black holes. So that's the kind of picture that you see here. But anyway, I'll make some of these things more precise. Anyway, the important, what I want to get out of the discussion in the first hour are three questions which we believe are difficult, they require new mathematical techniques and a lot of new ideas, but they are not impossible. I mean, they could, in principle, be solved and they would be relevant in their own right. So the first one is that the issue of collapse. How can you show, can you actually produce this situation where you start with data which are, as I said, not that small, so they're sufficiently large, but still regular in the sense that there are no black holes in them to start with? And how can I form a black hole later on? So this is the problem of collapse and there are now a lot of interesting results of this type, but we are very far away from a general understanding. But at least we do understand in some very interesting examples how collapse can happen. So that's the first question. The question of rigidity, which I mentioned earlier, are all stationary states of the Einstein-Vacuum equations, are they care black holes? So this is the problem of rigidity and it's the easiest in a sense. You'd think that it's the easiest problem. There are lots of problems like this in elliptic, in Riemannian geometry. And you'd think that at least this could be solved. Unfortunately, it still isn't. And then finally, there is a problem of stability, which is among them, I would say, the hardest. And we made a lot of progress, I think, but we are still relatively far. But in all these problems, you can see that there is an emergence of mathematical techniques that dilutes them and fortunately, hopefully, many of these problems will be solved. So this is, by the way, these three problems is what I call the reality of black holes. Because obviously, if any of these statements will be wrong, then physically black holes will not exist. So because they exist, we believe that all these results have to be correct. In other words, we should have the ability to collapse, because after all, black holes have to form somehow. We have to have rigidity. In other words, the black holes are really care black holes, or maybe care Newman if you have Maxwell equations involved. But right now, I'm doing everything vacuum. Or in the stability, of course, if the case solution is unstable, then obviously it's not an observable thing. So you think that in the minimum, these three things should be true. So first of all, what is a general black hole now? So a general black hole has to be a stationary solution, stationary asymptotically flat Einstein vacuum equations. So you are solving the equations we are talking about. And we are talking about the black hole, but the external part of the black hole. Not so the visible part. So black holes are not visible, but the part that communicates with the exterior. And the definition is that the external black hole is an asymptotically flat globally hyperbolic Lorentzian manifold with boundary. Defo-morphic to the complement of a cylinder in R1 plus 3. And so this is one very mild requirement. The most important one is that the metric has to have an asymptotically time like killing vector field T, namely vector field such that the leader of the vector field of the metric G is equal to 0. So that's the definition of a guinea vector field. And it has to be asymptotically time like because as we shall see, the care solution has such a stationary solution, but which actually tilts and becomes space like near the black hole. So it's only far away from the black hole that you can expect to have real time like. And then there is another thing which is called completeness of null infinity, which means basically that there are no singularities in the exterior of the black hole. I am not going to use. In the outside. Sorry, in the outside. There are no singularities in the outside. So that's basically, so the only important definition is really this one. Just understand, so you stop the black hole at the horizon. Exactly. So the external part stops at the horizon. So you don't look inside the horizon. You have a completely different behavior. By the way, this conjecture that I mentioned, the final state conjecture involves only the exterior of black holes. Inside black holes, you have lots of other conjecture that are equally difficult, maybe. But how do you understand that you might want to avoid it? But how do you avoid this mathematically in terms of definition? Because the fact that your space time is not complete because there are some, the basics would go inside the horizon. Yeah, but you have causality. You see, you still have causality. So when you solve partial differential equations, you are interested in causality. I mean, you, you know, so it's up to the boundary, everything is causal. Right? So, you know, it's like here, right? Right? If I'm in a region like this for the wave equation, right? This is not complete, of course, because I can go this way. But from the point of view of solving the, I understand, solving the wave equation, all you care is what happens here, right? And you construct this. I want to take a black hole and stop it even earlier before the horizon. You know, you, you, you remove. Actually usually stop beyond the horizon, actually. Inside the definition of black hole. Sorry? If you, if you, if you stop before the horizon, well, it will be just this business of, of this competence of null infinity if you want. I mean, that will not be solved anymore. It will not be satisfied, yeah. All right? Okay. So this, this is, all right. So now, a care family is, of course, an example, right? So again, I'm looking outside the region where, of the horizon. I'll say it in a moment where the horizon is. Anyway, this is just to remind you how things are. The important thing in the discussion is this delta, which is r plus a square minus 2mr. Now, Schwarzschild's solution, first of all, this is Schwarzschild. This is a complete picture of the Schwarzschild's solution, in the so-called conformal compactification, the Penrose diagram. So what the conformal compactification does for you is to allow to go to infinity with r. So, and r goes to infinity, but there is a way of conformally changing the metric so that all the directions, all the important things, the causal structure is preserved. But this way, you get something finite. So r is finite here, right? So, so this corresponds, in fact, to points at infinity in the physical, for the physical thing. This is also now hyper-safety. By the way, everything at 45 degrees here is null, right? So if you have an observer, observer, which is time-like or causal, will hit, will hit a scribe, but null hyper-safety, null geodesics at 45 degrees will hit scribe. The other ones, at time-like, will hit there at the point at i plus infinity, which corresponds to another type of infinity. But the important thing to see here is that there is a horizon. So there is r equal to 2m. You see r equal to 2m comes, of course, in the metric. r equal to 2m is not a singularity, as you might think, because you see you have 1 minus 2m over r to the minus 1 dr squared. It's not a singularity. It's actually what is called a horizon. Now this is what I mentioned earlier, that even though this metric was discovered very early on in 1915, it took about 50 years to understand the character of that singularity. We're extremely confused, in fact Einstein was really confused about it. Apparently he had some discussions with Zad Amar when he was in Paris. At Amar was bugging him exactly about that singularity, and Einstein was very upset because he couldn't answer. Anyway, so there was a lot of questions, apparently, in the mathematical community in Paris about that singularity. It took a long time. It was understood by the sixties it was understood. It is understood in that that singularity has only to do with the coordinates, is because you are using a specific coordinate system, which is good away from this horizon. It's used in this region. By the way, it's also good in this region. If you go for R less than 2M, you can also put this metric is still relevant in that region, but of course it's not relevant, and they're equal to 2M. But you can change, you can write down another system of coordinates in which that singularity is not there anymore, and in fact this is completely irregular. So actually the complete extension of the metric was found much later, and we now have these beautiful pictures. You have a part R larger than 2M, another part R larger than 2M here, and you have the region R less than 2M, and the singularity at R equal to 0. You can see again at the metric that the R equal to 0 is a problem, and indeed R equal to 0 is a true singularity, you have that the Riemann curvature tends to actually become infinite. In fact, we got scalars quantities which are formed from curvature, which becomes infinite at R equal to 0. Yes? Does the black hole rotate? No, no, no, this is a Schwarzschild solution. It doesn't rotate. So this is the external part of the black hole, and you see that everything is organized around R. So R is less than 2M corresponds to the black hole. R equal to 0 is a singularity. R less than 2M is black hole. R larger, event horizon is exactly R equal to 2M, so that's the boundary between the exterior region and the black hole region. If you are in black hole, you necessarily fall into a singularity. In other words, any time by curve in the black hole, if you extend it towards the future, it will go to that R equal to 0, which is singular, and therefore everybody dies in the black hole. If you are here and you are careful, you survive for all time, you go for infinite time, of course nobody lives for infinite time, but that's for other reasons. Anyway, so this is a picture, right? So R larger than 2M. By the way, R equal to Cm is also something very important. There is something very important happening at R equal to Cm, but maybe that's not the time to say it now. Maybe I'll say it later. What happens at R equal to Cm? So in other words, it's another important value of R. And then of course, it is R equal to infinity, which corresponds to these boundaries here and here and here and there. The care solution is similar in the external part. So the external parts are very similar. You don't terminate at R equal to 2M. You terminate instead at R plus, which is the root of the delta polynomial R plus A square minus 2MR. You see that there are two roots. One is this R equal to R minus. That's a smaller root. And the higher root is R plus, which corresponds to the horizon. Now inside the black hole, you have all sorts of new phenomena. And there are all sorts of interesting new results in mathematics. They have to do the character of this R equal to R minus. It's called the Hoshi horizon. It's not a singularity. And the question is whether it's stable or not. And there are all sorts of very interesting issues connected to this. But again, we are only interested in the external part, so I'm not going to talk about it. So the external parts, again, the boundaries are equal to R plus. You see that this is a causal region. And this will be a space like hypersurface. And of course, you'd like to, for example, if you do stability, you'd like somehow to start the zinitial data here and see the evolution. And we'll discuss about it later on. So that's external part again. This is a picture of the time-like stationary Keelink vector field that I mentioned we have in care. There is this d over dt, which looks like this. It now looks exactly like a Minkowski space. This is the part of the external part of the black hole, which looks like Minkowski space. So because of asymptotic flatness, as you move, we are going to infinity, you become more and more Minkowski. And therefore, this t also looks perpendicular, just exactly like Minkowski space. But as you approach the horizon, you see the t tilts and becomes time-like. In the Schwarz's solution, this doesn't quite happen. T actually remains tangent to the horizon. So in other words, it's not time-like anymore. It's not strictly time-like. It becomes null. And that has consequences. So this is the region where t becomes space-like. It's called the Ergo region, and it has a lot of physical and mathematical significance, particularly it means that somehow the energy associated to t, but we know that energies are always associated to time-like Keelink vector fields. So this has to do with no theorem. The fact that this is space-like in this region means you can extract energy from the black hole. So this actually has huge significance, both from a physical point of view. Mathematically, it creates huge problems because typically the energy which is associated to a time-like Keelink vector field has coercivity properties. And this coercivity is unfortunately violated when t becomes space-like. So there are all sorts of issues about this. There are also these Strapnal geodesics, which I'm not going to talk about it now. It's connected with what was before r equals 3m, I mentioned earlier, but I will not talk about it now. OK, so these are the tests of reality. Again, the rigidity, stability and collapse, rigidity does a care family exhaust all possible vacuum black holes. So again, stationary, because black holes are the definition stationary. Stability is a care family stable and the arbitrary small perturbation. Then collapse can black holes form starting from reasonable initial configurations. So this has to do with formation of Strap surfaces. So I hope to talk about all of them. I will not talk in detail. So my hope is that I will later on talk in details about each one of them. But for the moment, I want to give you sort of a general view of each one of them. So they all can be viewed from the point of view of the initial data formulation. So we start with initial conditions. For example, the problem of collapse, you can formulate it as a problem in which you start up with initial data, which are very reasonable. There are no black holes initially. And then you want to produce black hole later on. Stability again is a problem that has to do with the initial data, because you can start with the initial data set of care and make a small perturbation, make a small change, and you want to know what happens in the long range. Particularly, you don't want to have singularities. So this, you see, the problem of stability has also a problem of singularities, because even this could have singularities. Anyway, so and then I will start now talking a little bit about each one of these problems. So let me talk about rigidity first. So the rigidity is a care-family exhaust or stationary asymptotic-ready-flat. So what do we know about this? So first of all, there is a case which is completely understood, which is a static case. The static means, so we have a stationary solution. In other words, there is a Keating vector field, right? So there is a stationary Keating vector field, which is T. So in other words, we have a solution of the Einstein equations, which has a Keating vector field, which is time-like, at least far away, right? In other words, in the asymptotically-flat region or in the region that becomes more and more flat. Now if you make another assumption about T, which is that T is hypersurface orthogonal, so it's orthogonal on hypersurface, it's at every point you can find a hypersurface, regular hypersurface on which T is orthogonal. This is sort of an integrability condition, and it's always very hard in Schwarzschild, but not in care. So it's not really in care, but it's true in Schwarzschild. And with that condition, with that extra condition, you can actually show that the solutions have to be Schwarzschild solutions. So the only static solutions are Schwarzschild. So this is well understood, right? The next thing is axially-symmetric case. There is a beautiful result of Carter Robinson. It goes back a long time ago, and there have been many other developments, simplifications and extensions, or that result. In fact, it's a result in Riemannian geometry, right? Point being that if you assume not only stationarity, but also axially-symmetric, you take solutions of the Einstein-Vakumov equations, which are static and axisymmetric, right? So of course, care is axisymmetric also. Then you are in care, right? So this reduces to a problem about harmonic maps, in fact. So it's a purely Riemannian geometry problem of harmonic maps, rigidity of harmonic maps, and that's why it's true, right? Okay, and then there is another result, which is due to Hawking. It's really not a result. It's just an observation. So this is an observation that if you, in addition, assume an analyticity, then you can reduce the general case. In other words, the stationary case without axisymmetry, you can reduce it with this case, and therefore, because of Carter Robinson, you are in care, okay? So that's his result. So this was viewed by people in physics as being the definitive result on rigidity, because once you have this, you are done, right? And of course, this assumption of analyticity is kind of no justification for it whatsoever. But how does it work? Why is it elitist? You cannot assume a elitist. How can you use an elitist to reduce like this? Okay, so the proof goes like this. Let me tell you roughly how the proof goes. So you take that T, so you have a stationary solution. You have a T, you can define the horizon, right? That can always be defined starting from infinity, right? So remember that has something to do with completeness of scribe, right? So this, anyway, so this is a very, very weak argument. I mean, a very soft argument that gives you the horizon. Okay, then you can also show, it's also a very soft argument to show that T has to be tangent to the horizon. So it's not null anymore, it's actually space-like, but it's tangent to the horizon. Once you have a key effect of being on the horizon, you can say that it generates a rotation on the horizon, okay? So you find a rotation exactly along the horizon. Now you use the Einstein equations and analyticity, in other words, Koshy Kowalewski, to extend this in the interior, and that's it, right? So in other words, you show the existence of a second Keeling vector field, which is a rotation based on the fact that you can, by a soft argument, you construct one exactly on the horizon, right? Okay, but the problem is, the problem is analyticity. Why are you allowed to assume analyticity? So well, okay, you can say, if I am away from the Ergo region, this is a very reasonable assumption because stationary solutions, so if I have the Einstein equations together with a time-like Keeling vector field, then the equations become an, the equations become elliptic. So the Einstein equations can be reduced to elliptic equations in the region where T is time-like. I mean, very much like the wave equation, right? If I have a solution of the wave equation, dt squared plus Laplace N of i, equal to zero, and if I know that twice stationary, that doesn't depend on time, then I'm left with just the Laplace equation. So it becomes elliptic. The same thing here, this becomes elliptic, and therefore we know that even nonlinear elliptic, very general nonlinear elliptic equations have analytic solutions, right? So this is a reason of argument. In fact, it can be made perfectly correct mathematically. But this argument only works in this region where T is time-like. When it becomes space-like, it's not true anymore. In fact, the equations become hyperbolic again. So in that region, the Einstein equations under the assumption that there is a Keeling vector field, since T is not time-like, but space-like, the corresponding equations would be hyperbolic. No, no analyticity whatsoever. In fact, it's even worse because there is a transition between elliptic and hyperbolic. This is the hardest problems in partial differential equations, the ones where you have these transitions. In fact, they are not understood even in much, much simpler cases. So his argument is completely faulty. In fact, it's not entirely without merit because it definitely shows that there can be no as explicit solutions of the Einstein equations. Because we know that the Keeling solution is analytic. So if you are looking for as explicit solutions, you are not going to find them. That's basically what Hawking tells you. But certainly it does not answer the problem. So let me tell you what we know now. So analyticity is not reasonable. It's an assumption. And therefore, you have to do something else. And it turns out that, so what you have to do is, of course, forget about analyticity. Yeah, by the way, there is another. Yeah. So forget about analyticity, then analyticity makes no sense. So you want to, it turns out that the problem becomes much, much more difficult. I mean, once you don't have an analyticity, it's infinitely more difficult. And you have to use a full Einstein equation somehow. You cannot anymore do these kind of soft arguments. So there are a few results. Actually, maybe I'll mention the most interesting result. There are some results that I have in collaborations with Yonescu and also with Alexakis and Yonescu. This one, maybe I'm not going to explain it now. But this one can be easily explained. So I'll do that. So the results, and this is, we have many versions of this result. I mean, with weaker assumptions, in a sense. But roughly what the result says is that if you have a stationary solution, which is sufficiently close to care, in a sense that you can make precise. In fact, you make precise by using the so-called mass-simon tensor, which I will maybe talk about next time. So you can make it precise. And so again, if you are close to care, and you can make it precise what it means by closeness, then you must be in care. So it's, perturbatively, you are in care. All right. So here is a conjecture. So this is connected to a question raised by Slava. So the conjecture we have. So we have a lot of methods. I mean, this required completely new methods in terms of the kind of things that are used in general activity. And maybe I'll talk about it next time. But for the moment, I just want to say that at the end of our work on these things, we came up with a conjecture, which we think is very reasonable. So the conjecture says that rigidity conjecture is true, provided there are no T-trap null geodesics. All right. So let me mention something about this trap null geodesic. So a trap. So if you remember the picture that we had, the Spendros diagram of Schwarzschild, let's say. So remember that I said that there is a region here, which corresponds to R. So this is R equal to 2M. But there is a region R equals 3M. And then R larger than 3M. All right. So if I am in the region R larger than 3M, any null geodesic will have to move towards infinity. Or if I am here, they will have to move inside the black hole. So in other words, if I forget about this region, any null geodesics either moves towards infinity or it moves inside the horizon, therefore in the black hole. Once it goes into a black hole, I don't see it anymore. It's finished. I don't care about it. Once it goes to infinity, I also don't care about it. But unfortunately, you can have null geodesics which stay arbitrarily close to R equal to 3M. In fact, there are null geodesics, complete null geodesics, which actually rotate around R equal, rotate exactly on R equal to 3M in the case of Schwarzschild. So this creates huge problems. So it creates huge problems because somehow all the intuition they come from geometric optics is wrong. And that leads to issues in stability but also rigidity. OK. So anyway, this presence of trap null geodesics is very basic. Now what we say here in this conjecture is that we don't care about the trap, in general, about trap null geodesics. We only care about those trap null geodesics which are perpendicular to T. Now you could say where does it come from? Well, it comes from our intuition, mathematical intuition. But more importantly, it comes from the fact that in care, there are plenty of trap null geodesics but there are no trap null geodesics perpendicular to T. So this is something that you have to check. I mean, there is no particular reason. It just simply happens to be true. So in other words, this explains this result because if there are no T trap null geodesics close to care, I mean, in care, there are also no T trap null geodesics close to care. And therefore, it explains the result here. But more importantly, the conjecture is that this should be true in general. If there are no T trap null geodesics, you can actually do it. So the question is, can there be T trap null geodesics? I see absolutely no reason to exclude them. I mean, in other words, if you just think in terms of the original formulation of the rigidity problem, there is absolutely no reason to exclude such things. And of course, if such a, if these things exist, it's probably true that the conjecture is wrong. So in other words, I conjecture that in fact there are solutions of the ancient question such as which are not care. But they will probably be very unstable. They will probably be very unstable, yes. But they are there. I mean, if you want to solve the conjecture, you have to take that into account. So they will have to be unstable. And maybe, you know, somebody should try to prove this conjecture. I think none of us at this point, none of the people who were working. Sorry. Yeah, I think they, yeah, correct. Yeah. So in other words, the issue of stability should be measured in terms of the initial value problem. You see the rigidity conjecture has nothing to do with the initial value problem in some sense, right? It's just a problem about all possible stationary states. And but if I also connected somehow to the initial conditions, and if I know that there is one stationary solutions which is, which is for you to have the state trapped in our geodesics, I look at the corresponding data on a space like a surface and I make a small perturbation that would be unstable. Yeah, but I don't see why. It seems to me that you're simply emotionally attached to the first conjecture that you made, which is this maximal global. You want to save that one, but here you are really... Well, yeah, okay. So give me a chance to respond. Yeah, give me a chance to respond next time because you have to go through a little bit more. But I believe it's unstable. Of course, you know, if it's stable, it's even better because it will be more interesting. It will definitely lead to something very interesting. But I... And of course, it would mean that the cosmic censorship conjecture is also wrong. No, sorry, not the cosmic censorship, but the final state will be wrong, right? It will be wrong. Yeah, okay. So anyway, let's leave it as it is. It's an interesting issue, both from the point of view of proving the conjecture. I think that is probably... It's difficult, but probably not impossible given the techniques we have today. And then the other thing is to show that there are space times which are stationary and which have this T-trap nanogenetics. And then finally, to show that they are unstable, which will... I think will make Slava happy, right? You'd rather have it unstable. No? Yeah, well, it's hard for me to eliminate the evidence. All right. Okay. So let's go to the issue of collapse because we don't have that much more time. So collapse can block form starting from reasonably initial data configurations. And this has to do with trap surfaces. Namely, you cannot... It's not so easy to show that there are known black holes because the concept of a black hole is kind of a tautological concept that you can only understand once you have a global picture of the space time. So in other words, you have to understand that whatever you do, you'll have to have a global theory in order to understand formation of black holes. So that's complicated. But there is a concept which was introduced by Penrose, which in connection with this incompleteness theorem, which allows you to reduce the problem of formation of black holes. And if you assume the weak cosmic censorship conjecture, which of course is a huge conjecture, but if you assume it, then the problem reduces to the connection to the fact that there exists trap surfaces, right? The existence of trap surfaces. And the notion of a trap surface, which I'm going to explain in a second, is a local notion. So it doesn't... You don't care about the global picture, right? So that's why it's very powerful. The concept introduced by Penrose. Here is what it is. It's very simple. Imagine a space time and imagine a two surface in the space time, right? So if I am in Minkowski space and I have a two surface, then I can generate two types of null cones, right? So I can look at null geodesic perpendicular to the surface moving in this direction, in other words, moving out, and the ones moving in, right? So you get light cones this way and that way. So if I look at the corresponding area, in other words, if I change, if I move a little bit in this direction, the surfaces, and I look at how the area changes, I see that the area, of course, is decreasing in this direction and increasing this direction, right? So that's a typical situation in... Now, of course, you can have a more complicated situation where the two surface looks like this, which in that case, if you look at the corresponding light cones, you'll see that the ones here, in the region where the surface looks like this, then the areas will also decrease in this region, but they cannot... In other words, in the outgoing direction. One is the outgoing direction, the other one incoming. So this area can decrease locally in the outgoing direction, but globally, you can never make something which is outgoing decreasing at all points. And this can be measured by something which are called expansions. So you can actually write down some geometric quantities, which are called trace-ky and trace-ky bar. I'll talk more about these quantities later on, but for the moment, I'll just mention there are some very simple geometric concepts which measure exactly the change of areas in these directions or in the outgoing direction. So this is the incoming direction and this is the outgoing direction. So here, again, it's in the outgoing direction for trace-ky. So this measure and this measure trace-ky bar. And if this is positive, at points where trace-ky is positive, the area is increasing. At points where trace-ky is negative, it's decreasing. So here, trace-ky would decrease in this region, but will increase everywhere else. And now, with all this, a trapped surface is very simply is defined as being a surface, but if you look at the expansion, these geometric expansions which can be defined in general, both geometric expansions are negative. So in other words, the area is decreasing in both directions, both in the incoming and the outgoing direction. So this is obviously something that you cannot have in Minkowski's space. But you can have, in fact, if you look at Schwarzschild's, if you look at the picture of Schwarzschild's, if you look at the point, so points inside the black hole region are two surfaces, in fact. All those two surfaces are, in fact, trapped. So you see that in a black hole, you have trapped. On the other hand, if I'm outside, every surface outside is not trapped. So somehow, the trapping has something to do with the black hole. So it kind of, and in fact, you can show that if you have a trap, yeah. So this is a singularity theorem. So let me mention all the singularity theorem. So Penderos theorem is following. If you have, say, Ricci in any null direction, Ricci is a two tensor, and I apply it to any null direction L. If Ricci is positive, in particular, if Ricci equal to zero, that's obviously true. OK? So in flat case, this condition is trivial. If in addition, M contains a non-compact or she-hypersurface, this is typically our case, because we talk about asymptotic flatness. So we have something which is non-compact, as you go to infinity. So this also is a very simple condition to generate. And if it has a trapped surface now, so this is the hard condition, if it has a trapped surface, then it must have a, not a singularity, it must be incomplete. In other words, there will be, in that case, null geodesics, which will have to terminate in finite time, in finite affine parameter time. Right? So it doesn't tell you anything about the nature of the singularity. It doesn't tell you that the curvature blows up. It doesn't, it's a very, very weak statement, but it's an incompleteness, we call it an incompleteness statement, exactly for the reason that it only says something about the existence of some null geodesics which are incomplete. Right? But it's nevertheless a remarkable theorem, because using very little, you can prove something which is extremely interesting. So in particular, if you have a trapped surface and the cosmic censorship is true, then you must have a black hole, right? Because the cosmic censorship means you cannot have singularities outside black holes, but if you have a trapped surface, it automatically has to mean that there is also a black hole, because it has to be hidden under the black hole. So somehow, a trapped surface is detect black holes. Right? So now, so the questions which are raised by this term, yeah. Have you stare in? Yes. So if you want to have a nice initial data, so. Yes. And you want to have a trapped surface. What is this? It has to be B, yeah? Yeah, yeah. So of course, you cannot be close to Minkowski space. Right? You cannot have close to Minkowski space. Right? But this will be exactly what I'm going to talk about. Right? All right. So these are the questions. Can trapped surfaces form in evolution, in other words, starting with things which don't have trapped surfaces, or starts with an initial data free of trapped surfaces? I want to develop one in the future, right? This is sort of the analogous of saying that there are no blacks also originally, but later in evolution, you form a black hole, right? All right. So can they form from non-isotropic initial condition? So this is very interesting because essentially all the intuition that people have, including the one of Pendros, is that you have to produce energy from all directions in order to form a trapped surface, in order to have collapse. Right? So for example, if you have a collapse of a star, of a neutron star, to a black hole, the collapse in all directions, right? So somehow the intuition was, or is to many people, is that you have to pump energy from all directions in order to form a black hole because the trapped surface, obviously, it's a condition, a large condition in all possible directions. All right. So that's finally what is the nature of the singularity of Pendros. This is an open question. It's a fundamental question, very little is known. But I'm not going to talk about this. So OK. So here is a result. So this is one of the results which was proved by Christodulu in 2008. Right? Most of the results were proved in recent years, right? So all these developments, many of these developments happened in the last 20 years, let's say. So Christodulu showed that there exists an open set of regular vacuum, right? So in other words, this is for the vacuum equations. By the way, again, a lot of the intuition that physicists have is that you have to have matter in order to produce a black hole, right? Here this is actually happened in vacuum. You have in vacuum, in other words, you don't have any matter. So the formation of a trapped surface has to do with gravitational waves, in fact. All right. So this is a picture. The initial data here are not taken on a space like hypersurface, which is a little bit more complicated, to simplify, you take the initial data on two null hypersurfaces. So this is a characteristic initial value problem in partial differential equations, which is very well understood also. So it's not more difficult than the initial value problem. In other words, I can, instead of taking data on a space like hypersurface, I take data here and here. On two null, so these are null hypersurfaces. So you can see it in this picture there. There's this null hypersurface and this null hypersurface. So I want to take data here and here, which corresponds to here and here. So what he did was to take Minkowski data here. So these are trivial data. And on this side, in other words, on this side, you take what he calls a short pulse. So in other words, you take initial data, which give somehow a pulse in this direction, a strong pulse in this direction. And then he checked, of course, the following things, that you can construct data which is sufficiently large. So the pulse has to be sufficiently large to form a trap surface. So it has to be sufficiently large, but free of trap surfaces here. So initially, there are no trap surfaces. Of course, there are no trap surfaces here because this is Minkowski and data. And then he has to show with this initial data, you have to solve the Einstein equation, the full Einstein equations. By the way, in the case of Penrose, you don't have to solve the Einstein equations at all. I mean, the result of Penrose is based purely on the Sokol-Rachaduri equation, which is just one equation of many in the Einstein-Valkomine equation. So it's actually a very, very soft result. The result of Penrose is very, very soft, but it's remarkable, nevertheless. So here you have to solve the full Einstein equations and show that you have a solution sufficiently long time so that the trap surface can form. So this is a semi-local result. I mean, semi-global, sorry. It's a semi-global result. It's not a local result. It's not like in Montchocke-Brua. It's much, much more complicated. In fact, the proof is very much, is quite similar to the proof of stability of Minkowski space about which I'll talk next time. Yes? Do you take the disadvantage of the center of mass energy to show the existence of the final trap surface? There are similar setups in discussing physics where you collide particles and then a big black hole forms which can. Yeah. No. So this is actually simpler than that, in a sense, right? Because that one has to do, it's too black hole. No, it's not. I mean, in a way, this is actually simpler. What's remarkable here is that you can do it vacuum. You don't need anything else. We should just, it doesn't seem to me, yeah, I never understood this, because it's the first time I'm trying to understand what the equation should be. Okay. All right. Well, so please interrupt me. I mean, there is no reason to go fast. What I want to say is that, again, you could put the initial data here and here on two non-hypersurfaces, right? Minkowski and here, so it's flat. You don't have to do anything interesting here. But you choose a set of initial conditions here, which is sufficiently large so that you can prove a semi-global result. Semi-global means, for example, I have a parameter delta, let's say, and the size is, let's say, delta to the minus a half or something like that. So the size of the data here is delta to the minus a half, right? Which means typically in the local existence, for people who know how to prove local existence, you will get a solution only up to delta to the one half in time, right? So you can extend it only for a short time, right? So here you have to extend it for a whole, I mean, up to one, in other words, right? So data is of delta to the minus a half, but you solve it up to time one in this direction, right? So this is a... The physics, I mean, it's clear that you're taking that shape, central mass, because why do you throw things from two sides? If you just had a pulse coming from one direction, then by itself it would not form a flat surface. So clearly you have to have two pulses which come in from two directions, one curve space in one direction, by itself it would not be sufficient to fully on the other side start curving the... So actually they are intuitionists. And then they kind of... I should... They work together. Okay, so... In some regions where they come close to each other. So the two pulses propagate and they come close. But you are talking about the pulse from here and one from here, right? From that pulse to the right. Right, but here it's... There's no pulse coming from here. There are no pulses. There are no pulses from here, right? So this is... This is trivial data. So the only pulse is here. Right? It's just one pulse. Just one pulse, just on the left. One pulse on the left, that's all. But I'll say even more. Okay, so this is the first result. So just one pulse here, but the pulse is large in all directions, all angular directions, right? So you are still pushing things, you are producing a lot of gravitational energy from all directions. Right? Okay. So that's important. So in other words, it's uniform, a long null... Yeah, so anyway, so this is a general result, right? But then in addition, in that class... By the way, this is a hard part of the proof. You have to show that you can control the solution for long, long time is where you have to use a lot of analysis, right? But once you have that, once you have produced a space time which you control, then there is a second part which says that among all those solutions, all those initial, class of initial data which I control by my existence result, I pick up a special condition, an additional condition, which is in fact a condition of uniformity along directions. Then trap surface must form. And at the same time, you show that the data is such that you don't have a trap surface to start with. So you have to take this quantitative, right, relative to this parameter delta. So this parameter delta is a parameter that measures the strength of the pulse. But why does it have to be uniform? I mean, isn't this enough to... I mean, in physics, there are such situations, this kind of... You just throw in stuff from two directions, it comes close to each other, and you go trap surface. So let me show this as a result. Okay? That you might find more interesting. So this is a result that I have with Ronianski and on look. So it's actually... What's interesting is that it's based on the first part of Christodulus result. Well, Christodulus result was very long and it was simplified and we have much better understanding. Instead of 600 pages as it was originally, it's not only about 50 pages, let's say. This is one of the cases where long proofs have been simplified quite a lot. So anyway, but that result is still used here, except now the difference is that I take a pulse which is concentrated only in one direction. And everywhere else, I can make it to be Minkowskiian, say. Right? So in other words, it's the same situation, except that instead of taking a pulse which is strong in all directions, I take a pulse which is now strong in only one direction. Of course, I have to take a sufficiently strong pulse. And I form a trap surface later on. So the theorem is that it gives you conditions on the initial data here so that you form a trap surface later on. Now what's interesting is that this trap surface, you see, in the previous result, the trap surface, the proof of the result is based on what is called the Dublin alpha radiation. In other words, you foliate the spacetime by a family of light-cons going this way and light-cons going this way. OK? So that gives you a foliation which is represented here in this picture by these surfaces and these ones. Right? So and somehow this Dublin alpha radiation plays a very important role in the proof of the first result, right, that you have this one here. So the actual trap surface is actually in this foliation. So you find the trap surface at the end of your spacetime here. So the trap surface would be right here. So this is a trap surface. Right? In our result, that's not the case anymore, you have to find a deformation. So you see, you actually have to deform the two sides. So the surfaces of the double-nile foliation are this one, these regular surfaces here, and you have to find actually by an argument, which is a deformation argument, you have to find this one. Right? So this is the case when you can show that actually all you have to do, you don't need two pulses, as you said, you need just one pulse in one direction and you can already get the trap surface. So if you have two, of course, you'd think that it's a little bit easier, but... Does it have to be focusing with pulse? This one here? Yeah. Well, obviously you have to focus it in this direction. I mean, you have to produce a lot of energy in this area. But of course, you don't have a trap surface originally. I mean, it's obvious because everything is Minkowski and outside that region. Right? Yes? So the trap surface as maybe form like as a cow? No, no, no. So this is trapped. Trap surface is not yet a black hole. It detects a black hole. So you know, if you have a trap surface, you can say that the black form will form later once you have the trap surface, a black hole form. But yes? How long do you take in time to store the formation of the trap surface? Yeah. So again, if you have a pulse of size, say delta to the one half, delta, whatever. So if it's delta, as you know, existence will go... Sorry, delta to the minus one, excuse me. This is delta to the... I mean, sorry, the size of this is delta. And the strength of the pulse is delta to the minus a half. Delta to the minus a half, right? And then you can... this falls at time one. This time one, yeah. So anyway, so this is formation of trap surface. Now, there are many other results of interesting, but I guess I'll have to stop. I wanted also to talk about stability, which is maybe the main focus of what I'll talk about here. But I guess I'll leave it for next time. Right? And if there are any questions, I'll... Yes. Okay. So... Okay. Okay. Yeah. Okay. So I guess I'll have to stop. So I guess this is the end of my video. Thanks for watching. I hope you have enjoyed watching. I would like to thank you. I hope you enjoyed watching as well. I'm going to try it. Thank you. Thank you. If there are any questions, I'll... Yes. Thank you. You're welcome. Thank you. Thank you very much. Thank you. Thank you. Thank you very much.
 Ladies and Gentlemen, we ought to be talking about Temar L дог המomic예 le tings.  I want to thank the IHES for its very kind invitation to give these lectures.  And I will be talking about general activity, concentrating on problems which are connected  with black holes.  So I apologize maybe at the beginning I will start with an introduction.  So for people who are already very familiar with the subject of general activity, it might  be a bit boring.  But I hope to get something of interest to everybody.  So this is an introduction.  So let me talk a little bit about general activity.  So this is a physical theory which has a very strong mathematical background in the sense  everybody knows.  Of course that Riemannian geometry, the notion of curvature, played a fundamental role.  And what's remarkable about general activity is that just about any concept which has physical  meaning can also be merely translated into something which is purely geometric.  So for example, you have Lorentian matrix, Lorentian geometry is at the heart of general  activity.  So that means you have manifolds.  I should take some chalks and also put something on the blackboard.  Is there a chalk somewhere?  Below.  Oh, here.  Yeah.  Okay.  So obviously the starting point of Lorentian geometry and therefore general activity,  the fundamental object is that of a manifold with a metric.  And of course the only difference is that, so the manifold is the same definition as  in Riemannian geometry.  The only difference is that the metric of interest is not Riemannian metric, but rather  Lorentian.  So it has signature minus plus plus.  You can also have higher dimensions if you want.  I'll put here 1 plus n.  And therefore physical concepts like inertia translates into just the fact that the tangent  space at every point on the manifold is a Minkowski space.  Events, so physical events are nothing else than points in the manifold.  Observers are what are called time-like curves.  So that means, well first of all, I should say at every point in the tangent space, you  have a light cone where the metric degenerates.  And you can immediately distinguish between time-like directions, space-like directions,  and null directions.  So the division is simply based on looking at vectors at the point where the metric will  be negative, zero, or positive.  So this is space-like, this is null, and this is time-like.  So therefore you have a notion of time-like curves, which simply means it's a curve on  a manifold such that at every point the tangent space to the curve is time-like.  So time-like in the sense that it verifies this.  So this corresponds to observers.  So observers in general think they are nothing else but time-like curves from mathematical  point of view.  So again, this remarkable translation between things which are purely geometric and things  which are physical, light rays, so this is where light is supposed to propagate, are  nothing else but null geodesics.  So the geodesic, again the concept of the geodesic exactly the same as in the Devanian  geometry, and null simply means a geodesic, which is such that the tangent at every point  is, it keeps you g xx equal to zero, or x is a tangent to that point.  Proper time, which is nothing else, so the proper time along a curve is nothing else  than the affine parameter, just like in the Imania geometry.  Tidal forces in whichever, a clear physical meaning are nothing else but curvature, right?  So again, the curvature meaning exactly the same, it has the same meaning as in the Imania  geometry, given the metric g, you can associate the so-called Riemann curvature tensor, so  Rijkl, let's say, and the definition, the formal definition in Lorentzian geometry is  exactly the same, there is no difference, right?  So you have tidal forces, you have then isolated systems, again a physical concept, which means  that you imagine, you imagine that something interesting happens at the local point in  space and time, say a galaxy or just a star, and then the isolated means that somehow the  remaining part of the manifold is really just flat space.  So somehow you focus on the part where something interesting happens and everywhere else things  become flat, in other words, there is no additional physical input into the system.  So these are called isolated physical systems, so again, it's nothing but asymptotic flatness.  Asymptotic flatness, the definition, the formal mathematical definition is just, again, you  are on a manifold and typically the manifolds of interest, particularly if you talk about  isolated system, is the ones which are, say, defomorfic, at least outside a sufficiently  complex space, they look like R1 plus N from a topological point of view, and you want  the metric also to look Minkowski and if you are sufficiently far away from this isolated  system.  All right, so that's, as you see, everything translates.  I mean, this is a remarkable thing about general activity, is that everything interesting  from a physical point of view has a very simple geometric meaning.  Equivalence principle, which is at the heart of general activity from a mathematical point  of view, is translated simply in saying that all physical laws can be expressed only in  terms of the metric and its induced connection.  So if I have the metric G, so again, in Lorentzian geometry, it's exactly like in Emanian geometry,  you can associate the connection, which is the so-called Levy-Civita connection.  So exactly the same properties as in the Emanian geometry.  All right, so more concepts of interest in Lorentzian geometry, which come all the time.  So first of all, time orientation simply means that I have a manifold, which is Lorentzian,  and orientation is given by a vector field, say, maybe I'll call it T, a vector field,  which is globally defined on the manifold, and it's time-like at every point.  So that gives you a time orientation, which allows you to separate between things which  moved forward in time and backward in time.  Curves which are time-like or causal, I already said what time-like means.  So time-like simply means that you have a curve where the tangent is time-like at every  point.  Cozal would mean that you allow the tangent also to be null.  So future and past sets, well, let me go here.  Future and past sets, again, a very simple concept, but it's quite important in Lorentzian  geometry and, of course, in general activity, is that if you have a set S on your manifold,  you are interested in all points.  So the future set of S is a set of all points which can be connected to S by a time-like  curve.  And causal set will mean that you also allow the curve to be causal.  So for example, if you have an alkan in Minkowski space, then the future of this point, the  time-like future will be all the points inside the light cone.  The causal will also allow the points on the boundary.  So future and past sets, hypersurface, so this is another thing that, of course, you  discuss in Lorentzian geometry, you can discuss in Lorentzian geometry.  The difference is only that now you have three types of hypersurface of interest.  The ones which are space-like.  So space-like hypersurface is such that if you look at the unit normal at every point,  it's time-like.  So you need normal time-like at every point, which means that if I look at the Lorentzian  metric and I restrict it to the space-like hypersurface, I get the Riemannian metric.  So that's Riemannian geometry as a consequence is part of Lorentzian geometry.  And part of general relativity, you really have to know Riemannian geometry very well  in order to succeed doing anything of interest in Lorentzian geometry, I mean in general  relativity.  All right, so we have hypersurface is such a space-like, hypersurface is such a null.  So this is a null hypersurface, right?  So it's, as you see, it's ruled by geodesics such a null or curves such a null, let's say.  And finally, you can have something time-like.  Time-like would simply mean that, for example, if I take a section like this in Minkowski  space, this would be a time-like section, right?  So in that case, the metric induced is still Lorentzian.  So you have the original Lorentzian metric.  It induces a Lorentzian metric on a time-like surface.  Another concept of importance which shows up a lot is that of a Cauchy hypersurface.  So this is a space-like hypersurface with a property that if you take any point in the  manifold, so you say that the manifold has a Cauchy hypersurface or that this is a Cauchy  hypersurface in the manifold.  If you take any point on the manifold, you can, and you look at any time-like curve,  it will have to intersect the Cauchy hypersurface in, always it has to intersect the Cauchy  hypersurface, and intersect it just exactly at one point, right?  So obviously, T equal constant in Minkowski space is a Cauchy hypersurface because obviously  every curve which is time-like will have to intersect this.  But if I look at this hypersurface which is space-like, it's still space-like, it's  not a Cauchy hypersurface because obviously I have all these time-like curves here which  is not intersected hypersurface, right?  So anyway, it's a very important concept and we'll talk more about this.  All right, so the next topic, we're having a eraser somewhere, so the same thing, I have  to find it here I guess, but it's not.  On this side, okay, good.  All right, so, all right, so frames.  So in the Emanian geometry, of course, the kind of frames that you see always in the  Emanian geometry, so if I have a manifold with an Emanian metric, typically a frame  is a set of vector fields, say u1en on g such that the metric gEiej is equal to delta ij,  right?  So for all i and j, and this is a Kroniker symbol.  In Lorentzian geometry, of course, you can define the same thing.  You can have frames which are called orthonormal, right?  So for example, if you are in Minkowski space, you can take a time direction, say E0, and  then u1en-1, or en, let's say if I am in a plus or not dimensions, will be space-like  curves, space-like vector fields which are perpendicular to E0 and E0 is time-like.  But you can have another more interesting type of frames which play a much more fundamental  role in general activity, which are so-called null frames.  So null frames.  So null frames instead are frames which consist of, say, vector fields l, l bar, and then  E1 up to en-1 if I am in n plus one dimensions.  So these two are null, right?  So gll is gll bar l bar is equal to zero.  You also normalize them.  Of course, you cannot do anything like in the minority geometry where you take one of  them to have lengths one because they are null, right?  But I can normalize l and l bar so that this is equal, say, minus two.  So for example, if I am again in Nikovsky space, I can take vector fields, so this is  a null cone, at the point, right?  I can take this to be l, this to be l bar, and I normalize them such that the length,  the scalar product between them is minus two.  And then the other ones, the other ones will have to be space-like because they are orthogonal  to these two.  They will have to sit here somewhere.  They will be space-like.  And I can pick them like in minority geometry to be orthonormal.  So this will be orthonormal among themselves and perpendicular to l and l bar.  So null frame, which I'll talk a lot about null frames in these lectures, is really defined  like this, right?  So for example, here is an example of a null frame in Nikovsky space.  If I take dt minus dr, where r is a radial vector field, so I call this l bar, and dt  plus dr, I call it l, and the other ones will be perpendicular to these two and orthonormal  among themselves, right?  So that's a null frame.  And as you see, these null frames play much more fundamental role than the orthonormal  frames.  All right, so Cartain formalize, well, again, this is exactly like in minority geometry.  If I have a frame, I can define, so if I have the frame, let me do it here.  So if I have a frame, I can define the connection, right?  So I have, say, in the minority geometry, e1, en, and I define the gammas, which are  the derivative of the frame taken with respect to the frame.  So the components of the derivative with respect to the frame, where the derivative is, of  course, a covariant derivative.  So it's induced by the Levy-Civita connection.  And then the Cartain formalize tells you that somehow derivatives of gamma plus gamma, so  you get equations which are the Cartain equations, which will flex it, sorry, equal to curvature.  So on the right-hand side, you have the Riemann curvature tensor.  So I'm writing it just very formally.  Whenever we need, I'll write it more explicitly.  But these are the fundamental equations in minority geometry.  And of course, they play a fundamental role on some of the Lorentzian geometry, except  again that I will be using, instead of orthonormal frame, as in Cartain formalize, I'll be using  null frames.  Foliations.  So another thing which is extremely important in problems in general activity is to have  a Foliation.  So one kind of Foliations would be given by Minkowski space.  For example, it's just equal constant.  And for a general Lorentzian manifold, the role of t is placed by what is called the  time function.  In other words, a function such that the exterior derivative, so dt, the differential of t,  is time-like at every point.  So that's a time-like function.  And then of course, I can talk about t equal constant.  So this gives me a Foliation by space-like hypersurfaces.  So this will be on an arbitrary Lorentzian manifold.  Another kind of Foliation, which is in fact even more important, is given by so-called  optical functions.  Optical functions.  So instead of time functions, I'll talk about optical functions.  So these are functions given by u.  So optical function is a function u, which verifies the I-colonial equation.  So gij diudiju is equal to 0.  In fact, actually, to distinguish between Riemannian and Lorentzian geometry, I'll  use instead, most of the time, I'll use Greek indices.  So this will be d alpha of u, d beta of u is equal to 0.  So this means nothing else that if I take the exterior derivative of u, so if I take  the differential of u, then g du du is equal to 0.  In other words, if I look at u equal constant, the differential of u is in the direction  of the gradient of u.  But this means that somehow the gradient of u is also orthogonal to itself, which means,  of course, that du itself is tangent to the hypersurface.  And this, of course, can only happen if u equal constant does not look like this at  all, but looks like a null hypersurface.  So u equal constant is, in fact, null.  So in other words, if I solve the I-colonial equation, which is, of course, an equation  extremely important in all sorts of branches of mathematics and physics, if I solve the  I-colonial equation, I get the solution u, the level surfaces of u will give me a foliation  by null cones.  So for example, again in Nikovsky space, I can have a foliation given by, for example,  like this, where all these are null cones.  So I can foliate part of the spacetime by null cones.  And these foliations, again, are going to play a major role in our discussions.  Finally a little bit about the role of Riemannian geometry.  I already said that if you have a space like hypersurface, you immediately have equations,  which are typical to Riemannian geometry.  So you have Riemannian geometry because a metric induced on a space like hypersurface  is actually Riemannian.  So the geometry on a space like hypersurface is Riemannian.  And therefore, lots of concepts that come in Riemannian geometry will be of great use  also in general activity.  But there is another kind of variation, which also is extremely important, maybe even more  important, and will play a role in our discussions, which is, say, foliations, which are obtained  either by taking an intersection with t equal constant.  So imagine that you have a time function, t equal constant on a manifold, and you intersect  that with null hypersurfaces, like this.  So the intersection then consists of two surfaces, which are Riemannian, because the metric induced  to be Riemannian.  So we call them, so this is u equal constant.  Then these surfaces will be called S of t u.  So if I am in 1% dimension, this will be 1n minus 1 dimensional.  There will be the metric induced Riemannian, and this kind of the geometry, so this is  also has the advantage of being compact, right?  So these are compact Riemannian manifolds, and again, they play a major role in our discussions.  So finally, to finish with this introduction, I want to say a little bit more on null hypersurfaces.  So again, the null hypersurfaces can be given by these optical functions, in other words,  function u such that g du du is equal to 0.  But they also rise naturally as boundaries of future and past sets, right?  So if I have a set here, and I look at the future, the boundary of the future is null,  at least in the portion of the boundary, which is regular.  So if there are, so these are typically ruled by null geodesics, right?  So the boundaries of future are ruled by null geodesics, right?  So in other words, they can be constructed by putting null geodesics together.  So for example, if you start with a two surface, you take the null geodesic perpendicular to  the two surface at every point, and that gives you a null hypersurface.  Of course, you could have the problem that these null geodesics, which are perpendicular  to s, at some point will intersect because you can have conjugate points or cut locus.  And then the boundaries become more complicated, but I'm not going to discuss it.  So as long as in the absence of this conjugate point and cut locus points, these null boundaries  are smooth and they are given by null hypersurface.  So let me end this with an interesting observation of Poincaré, who said, parmi ses actions  en plusite, il est un qui semble mériter quelque attention, parce qu'en l'abandonnant,  on peut construire une quatrième géométrie aussi cohérente que celle de Euclide, de  Lobachesque et de Riemann, je ne citerai qu'un de cette hauretme et je ne choisirai pas le  plus singulier.  Une droite réelle peut être perpendiculaire à elle-même.  And of course, this is what happens if you take a null hypersurface, if you take the  null geodesic, which is a straight line if you are in flat space, in Minkowski space.  And this is of course perpendicular to itself relative to the induced Lorentzian geometry.  So the Lorentzian geometry induced on a null hypersurface is actually singular and is very  different from what happens in the Manean geometry.  And this leads to a lot of interesting mathematics, which has been somewhat ignored by geometers.  I would say that Poincaré is always his visionary and he's understood that this could be very  important and indeed this null hypersurface is extremely important in general.  Okay, so with this let me go now to the beyond Lorentzian geometry, what is the geometric  framework of general activity?  Well you have field equations, so this is now, obviously you are looking at Lorentzian,  I'm always losing my, we are looking at Lorentzian manifolds, which in addition to being just  Lorentzian, they verify an equation.  So general activity is distinguished from Lorentzian geometry by this fact that you  are interested only in Lorentzian metrics, which verify an equation.  So the equation is just the Ricci, so as you know if I have the Riemann curvature tensor  and I take a trace with respect to the metric I get the Ricci tensor and so this is a two  tensor and the equation looks like this, minus one over two G, times the scalar curvature,  so the scalar curvature, so this is Ricci, scalar curvature is obtained by taking one  more contraction with a metric G and on the right hand side you have T which corresponds  to the input of some matter theory which is present in our spacetime.  So these are the famous Einstein equations and I'm not going to talk directly about them,  partly because so this part of the Einstein equations is sufficiently interesting in its  own right and in fact most of the complication, there are lots of complication which we have  to do of course with the matter fields which are present, but in a sense in mathematics  we always want to separate difficulties, so these are a set of difficulties like for example  you can have fluids and you have all the difficulties that have to do fluids or you can have Maxwell  equations, all sorts of other things that are on the right hand side and to simplify I'm  simply going to ignore that part, I put this equal to zero which is of course makes perfect  sense, so this will be if you are to make an analogy with Maxwell's theory is studying  electobar, pure electromagnetic waves without sources, right, makes perfect sense and it's  already very interesting, but in general activity this is also very hard.  So in this case it's very easy to see that the equations actually immediately imply that  the Ricci has to be zero.  So the equations of interest in my lectures will be this one, so you have Ricci equal to  zero, sorry, sorry, what do you mean conversely?  Yeah, correct, so if t is equal to zero these two are equal to zero indeed.  Okay, so these are the equations that I would be interested in and again mathematical is  extremely rich and you see essentially all the important, most of the important parts  of sort of black or seori have to do these equations, right?  All right, so let me talk a little bit about the character of these equations, what's special  about these equations, so again we have Ricci flat, Ricci of G is equal to zero and first  think of interest is what you mean by a solution, so we have to solve this equation, this is  a complicated equation, I mean if you think about how the Riemann curvature tensor is  defined it involves two derivatives of the metric plus nonlinear terms, so two derivative  plus nonlinear terms, but these two derivatives also come up in a complicated fashion and  if you don't take precautions you don't see any interesting partial differential equations  showing up here.  To make it into an interesting partial differential equation you have to take into account another  important fact about this equation which is that there are different morphism invariant,  in other words if I have a solution G, if I have a metric G I can all distinguish between  phi star of G where phi is any deformorphism of the manifold to itself, so one plus n,  one plus n, right?  So a solution is actually a class of equivalence of solution modulo, this huge group of deformorphism  of the manifold, right?  So these are generally different, that's why the Einstein equations are called generally  covariant.  So if you take that into account then solving the equations you have an additional, the  initial bonus that comes from it which is that I can try to solve it relative to a specific  coordinate system or a specific deformorphism, so I look for solutions together with a class  of, together with a specific, with a specific kind of deformorphism which will allow me  to solve and the simplest way is, the simplest observation of this type is that if I assume  a coordinate system, so you see this covariance allows me somehow to say well there are, I  can solve them by equations in a specific coordinate system, right?  So if I pick up the coordinate system to verify this very simple equation where this  down version here is nothing else but the Laplace-Beltrami operator but associated to  the metric which is now of course not Riemannian, so you don't get the usual Laplacian but you  get what is called the wave operator, right?  So this is defined exactly the same way as in Riemannian geometry except that because  of the metric, so these are covariant derivatives applied to X-alpha, and this operator is what  we call the wave operator, right?  It's exactly like the Laplacian but it's much more complicated because of the signature.  Is that what people call also harmonic coordinates?  Yeah, this, in Riemannian geometry I call harmonic coordinates.  It's interesting that actually harmonic coordinates were discovered earlier in physics than in  geometry, right?  So these were known, in fact, even Einstein himself had some knowledge of them but they  were certainly understood by people in the French school like Yvon Trocadro which used  it in her famous result on existence for solutions of the answer equation, right?  I'll talk more about this in a second.  Okay, in any case, once you fix up your coordinate system then these equations take this form  and as you see here you have once again some kind of wave operator applied to all components  of the metric.  So here I'm fixing the components of the metric.  I apply this and I get something on the right hand side which is a complicated expression  which will involve the metric and first derivative of the metric and say it's quadratic in the  first derivative of the metric, right?  So in any case, it's recognizable.  The character of the equation is now recognizable.  This is what we call nonlinear system of wave equations.  It's clearly hyperbolic, right?  So this is what one calls the hyperbolic character of the equations, right?  So I'll talk more about things like this later but for the moment I just want to give a general  introduction to these things.  Okay, so equations are hyperbolic, right?  That's what we discover.  We discover that if we use the gauge dependence of the Einstein equations, if you do it, you  can change it to our advantage, you can actually show that the equations are hyperbolic.  Now being hyperbolic, it means you have to solve sort of similar problem as you do, say  for example, if you are in Minkowski space, right?  So the metric is then just a Minkowski metric.  Then say I want to solve just the inversion of phi is equal to zero.  So the inversion being again, this is minus dt squared plus Laplacian of phi is equal  to zero.  And typically one solve this by choosing initial conditions, let's say t equals zero.  It can be any space like hyper surface.  You can take initial conditions on any space I have, but in post-implicity you just take  t equals zero and then you get phi of zero x is say f of x and dt of phi of zero x is  g of x.  So this you prescribe and then you show that there exists a unique solution that solves  this equation, right?  So this is for the simplest possible equation.  Of course you can do that for much more general classes of equations.  Yes?  So does the formalism of general relativity is the same that the formalism, the eminator  used to demonstrate theorems about physics?  There is of course a connection obviously, but I'm not going to get into this now, right?  Yeah, but of course there are conservation laws and all sorts of things that you have  to do.  There is a whole theory, but I will not talk about this now.  Here I want to talk about something very simple, which is that you can solve the way of equation,  unique solutions for the way of equations you find if you prescribe these two initial  conditions, right?  So you want to do something similar in general relativity.  So in general relativity the role of this f, so the role of this t equals zero hyper  surface and these two initial conditions is taken by what is called initial data sets.  So an initial data set will consist into a space, some three dimensions, so let's say  we are in four dimensions for simplicity, right?  So we are in four dimensions, so this will have to be three dimensional, starting one  dimension less.  This will have to be a Riemannian metric because I know that if I can solve the Einstein equations,  I will get a Lorentzian manifold and this original hyper surface sigma zero embedded  into it will have to be space like.  And therefore this as an input will have to be a Riemannian metric and this will correspond  to the time derivative of the Lorentzian metric.  This is called the second fundamental form of the original space like hyper surface,  right?  So typically I prescribe G zero and K zero and sigma zero and I solve for the constraint  equations.  So then there is a unlike in this situation, you have to do a little bit more work about  the constraints.  And as those constraints cannot be, the constraints cannot be prescribed arbitrarily.  They have to be something which is called the constraint equation, which is very much  like in Maxwell's theory.  In Maxwell's theory, when you prescribe the electric and magnetic field, you prescribe  it together with the constraint, which in that case is very simple because it's linear.  In this case it's much more complicated.  It's a nonlinear constraint and this actually the study of solutions and the character of  these constraint equations has led to a lot of developments in Riemannian geometry.  This you could view the constraint equation, initial data, constraint equations can be  viewed as a subject in Riemannian geometry, right?  And I won't talk about this at all, but I just want to tell you, for example, that the  so-called positive mass theorem, which is a famous theorem in Riemannian geometry,  played a major role not just in connection with where it started in general activity,  but in many other applications of problems in Riemannian geometry.  And they have to do somehow with this, with the geometry of these initial data sets together  with those constraint equations.  Once you have set up the initial conditions, you can talk about solving the equations in  the same way as you solve it here.  Of course, this is a linear problem.  That one is nonlinear.  It's much harder, but nevertheless it turns out that there is a very general theory of  nonlinear hyperbolic equations, nonlinear wave equations, which allows you to solve the  Einstein equations in the wave coordinate.  So you have to use a wave coordinate conditions.  To start with initial data, you construct a spacetime at least for some time.  It will have to be local to start with, because typical nonlinear problems can develop singularities.  But you can solve it for some time.  You can also talk about the maximal future development.  In other words, this would be something similar to what happens for simple ordinary differential  equations.  So x dot is equal to x squared.  If I want to solve it, some initial data, x of 0 is given.  Then obviously I can only solve it for a local amount of time, because you can have singularities.  But you can talk about the maximal time of existence of solution of this equation.  And the analogous of this notion, and this, by the way, is not just for this equation,  for any nonlinear ordinary differential equations, you can talk about maximal time of existence.  And the same concept can be discussed in the context of these initial data sets.  So in other words, I can start with an initial data set, and I can talk about developments,  and I can talk about the maximal development.  And then somehow the entire subject of general activity, you could say, is really nothing  else but the discussion of the character of this maximal development.  So the theorem, which was proved by Ivonsho K. Brown in 1952 and complemented by Gerroch  in 1969, is that to any initial data set, you can associate a maximal future global  hyperbolic development.  Global hyperbolic development, what does it mean?  It simply means that the space time that you have constructed, by the way, all this concept  came in the French school, by the way, I should say.  Then the formulations of the initial value problem as a hyperbolic problem was really  not well understood until Loré, Schokebouin, and many other people in the French school.  So this is really a good place to talk about it, because it's a...  So anyway, so this was...  The hyperbolic character of the equations was definitely an important part of the work  of Loré and certainly Ivonsho K. Brua, and then Bruha Gerroch.  You associate with initial data, you can associate this maximal future hyperbolic development.  That simply means global hyperbolic development.  By the way, global hyperbolicity was defined by Loré, in fact.  But for us, it simply means global hyperbolic, it simply means that the space like you are  starting with, when you construct your space time, this three-dimensional, limanian surface  together with a first and second fundamental form of this, is what you want to say.  So I want you to say that relative to the space time that you have constructed, this  is a Cauchy hyper surface.  So global hyperbolicity is something very simple.  When you solve the Einstein equations using PDE techniques, you automatically construct  a globally hyperbolic development.  So that's part of the construction itself.  So again, you have this maximal global hyperbolic development, you can associate with any initial  data and therefore what's left is to discuss the character of this maximal future hyperbolic  development.  It sounds simple, but certainly not.  So let's talk now about the next step after this.  So the next thing that needs to be said before going into real problems is to talk about  what kind of solutions you could have in general activity.  Of course, solving the equations in general is very complicated and it's really the role  of mathematicians to solve the equations without necessarily producing explicit solutions.  But the physicists who started first to work on this, they wanted to look at explicit solutions.  So that's natural and we don't blame them.  In fact, it's extremely important.  So how do you look for specific solutions?  You have to look for symmetries.  You have to assume that somehow you have all sorts of additional symmetries.  So what kind of symmetries you can have?  Well, the symmetries in geometry are given typically by vector fields, which are key  link or conformal key link.  In other words, they generate, so these are vector fields, which generate one parameter  group of deformal fields, which correspond to isometries or conformal isometries.  So that's how you look.  So when you look for symmetries in general activity, you follow the same idea.  You look for solutions which have additional key link or conformal key link vector fields.  The simplest case is actually when you have a lot more, not just one, it's spherical symmetry.  So you look for solutions with spherical symmetry.  So in that case, you have an action.  It's more than just one or two key link vector fields.  It's an action of the SON group.  I'm not going to say much more about it.  There is a very simple definition of spherical symmetry, in which case once you impose spherical  symmetry, the Einstein equations become much simpler, and you can actually solve the next  velocity in certain cases.  So I'll say more about it in a second.  But of course, you can also look for solutions which have just one symmetry or two symmetries.  So say x1, x1, and then x1, x2, and so on and so forth, and you get more and more simplifications  as you go.  So if you impose enough symmetries, in fact, you reduce the Einstein equations, which are  partial differential equations, you reduce it to just ordinary differential equations,  for example.  But you can also, if you have one, for example, typically, you have to think about having one  such key link or conformal key link, typically key link, having one such key link vector field  reduces the dimension.  So suppose you are in one plus one dimension, and you have one symmetry, you reduce it to  n dimension.  So you reduce the problem from n dimension to 1 plus n dimension.  Maybe I should say 1 plus n minus 1, for example.  So you simplify the equations quite a bit, but they could still be very complicated.  That symmetry makes life much easier.  And in the case of the Einstein equations, of course, the simplest solution is the Minkowski  space itself.  The Minkowski space certainly verifies this equation.  So Minkowski space is a spatial solution of the Einstein equations in vacuum.  But then there are more such solutions.  The most interesting class of such solution was discovered by Schwarzschild.  So here is an example.  So this is Minkowski metric in polar coordinates.  So you see the difference between this and Schwarzschild is that you add these terms here,  where m is some constant, and you want to take it positive.  So m can be also 0.  The case equal to 0, it's exactly the case of Minkowski space.  But you see that the moment you introduce a norm 0 m, the solution becomes somewhat  more complicated.  The character of the solution was actually, even though the solution was discovered by  Schwarzschild in 1915, immediately after Einstein, it took more than 50 years to really understand  the character of the solution.  But the next, so this was discovered in 1915.  The next one was discovered much later in 1963, which is a famous care family of solutions.  So let me now go to here.  So this is how the care solution looks like in coordinates, which are again like polar  coordinates, T, RT times 5.  These are called the Boyer-Linckes coordinates.  And you see the metric is more complicated now.  But it's still amazingly explicit.  This is one of the, probably the most remarkable solution.  I mean, we should discuss, but I think it's one of the most remarkable solution in all  of physics, because it's clearly a solution of a nonlinear problem, highly complicated  nonlinear problem.  It's explicit, right?  Because all these coefficients are calculated by this very simply trigonometry formula.  So it's explicit, and it has huge impact, right?  I mean, basically, whatever we think about black holes today are essentially based on  these solutions, right?  So showing that this is a solution is not so easy at all.  I mean, if you try, you probably spend at least a few days to try to calculate and show  that to indeed the solution.  Of course, if you use a calculator, it's much smarter.  You will probably get it in a few hours.  Sorry?  You use the fracture forms.  It makes it easier, certainly, if you introduce correct.  All right.  Anyway, so this is the family.  Let me make some remarks about it.  First of all, you can see that this is asymptotically flat.  You see that asymptotically flat from the character of these coefficients here.  You see that when r goes to infinity, right, that r goes to infinity, you get closer and  closer to Mikovsky space.  So that's just standard to see immediately.  Yes?  So these equations are homogenous.  Excuse me?  Homogenous.  Homogeneity.  Sorry.  All those in units.  I'm not looking at units now, right?  I mean, in terms of physical units, you mean?  Yes.  No, no.  I'm certainly, I'm writing it as purely a mathematical solution, right?  At this stage.  You can always put units.  But I'm not going to do it.  All right.  So anyway, so the first observation is that, again, this is asymptotically flat.  So it's an asymptotically flat solution.  It has symmetries because you can see immediately that since these coefficients do not depend  on time, right, you can see it here.  They don't depend on t.  It means that d over dt in this coordinate, so the vector field d over dt in this coordinate,  is killing, right?  OK?  So this gives you that these solutions are in fact stationary.  Stationary, by the way, I didn't say.  I should have said it earlier.  These solutions means, I guess I didn't.  Yeah, OK.  Anyway, I'll discuss about stationarity later on.  The important thing is that this vector field, which is killing, is also time-like.  At least as you go far away in the r direction, as r goes to infinity, it becomes, certainly  it's time-like.  We'll discuss about the actual global behavior later on, but for the moment, it's stationary  because this d over dt is time-like.  The second observation is that it's also axisymmetric in the sense, again, that if I look at all  these coefficients, they don't depend on, the coefficients don't depend on phi, which  means that d over dt phi is a killing vector field, right?  OK?  So this corresponds to a rotation.  So this corresponds to a rotation.  So these are, we say, are black holes which are rotating.  You see that there is an extra parameter instead of m that you have now two parameters.  You have a and m, and in fact, this set of solutions are supposed to be physical only  in the regime a is strictly less than m.  Actually, sorry, less or equal to m.  Equal to m is called the extremal case, right?  All right.  So again, you can see that if a is equal to 0, you are reduced to Schwarzschild, and of  course, if m equal to 0, you are reduced to Mikołski space, right?  So this is the famous two parameter family of solutions of the Einstein-Vakum equations,  which verify all the interesting assumptions that you want.  It's asymptotically flat, and it's stationary, and it's also axisymmetric.  All right.  So now, come much time.  Maybe I should take a few more minutes, and then we'll take maybe a break.  So here I want to talk about the D major conjecture in general activity.  It's probably, I mean, there are some others, but I would say that this is sort of the  dream conjecture that you'd like to solve.  So what is this conjecture?  So it's called the finite set conjecture.  So it says the following thing.  It says that generic means not all, there could be some exceptions to this conjecture,  right?  So I say generically, of course, you don't know what that means at this stage.  Of course, whenever you solve a major conjecture, you also have to define everything.  In particular, you have to define what you mean by generic.  But leave aside generic, any asymptotically flat initial data set that you take, right?  So it has to be asymptotically flat, has to be smooth, of course, but I make no other  assumptions.  So this is global future development.  So in other words, there is this global maximal development that I talked about before, right?  So you go as far as you can go.  And here you are saying that this maximal future avivory developments, which are maximally  extended solutions of the n-stand vacuum equations, so that's the same thing, look asymptotically  in any finite region or space as a care solution, right?  So let me make it a bit more graphic.  So you take an initial data set.  You take a very general sigma 0, g0, k0.  You take a very general initial data set, which is, of course, asymptotically flat and  smooth in whatever sense.  And you look at the maximum future hyperbolic development.  So this tells you that the maximum future hyperbolic development is not finite.  It doesn't terminate in finite time.  It goes all the way to infinity, right?  So this is the first major statement, which is included in here.  The notion of maximum hyperbolic development, when we talked earlier about maximum future  hyperbolic development, it could have been that if I look at a time like curve, the time  like curve terminates in finite time.  In other words, observers will die after finite time, right?  So that means from a physical point of view, it's not a very interesting solution, right?  So this is not an interesting scenario.  So here, it tells you that in reality, what happens, you can go for infinity time.  But it's even more interesting is that what you see asymptotically will be just a finite  number of care solutions, right?  So asymptotically in time, whatever that means, of course, you have to define the two.  But say intuitively, it's pretty clear.  So asymptotically in time, you are only going to see a finite number of care solutions,  which are black holes.  And they are going away from each other, right?  So because in principle, they could have interacted before, when they interact, they  form only one black hole.  So if you have two that come together to form only one black hole, so asymptotically, you  will see only a finite number.  In any compact region, in other words, you see only one care solution at most, right?  And they could rotate.  So that's why care, that's why the final states will be care.  So what is?  Rotate two black holes rotating around each other.  Well, the conjecture says that this is not possible, right?  But I'm not saying that the conjecture is true, of course.  And generically, right?  Yeah.  Because then they will evaporate, they will emit traditional waves in the area.  Correct.  Yeah.  Correct.  So that genericity is also very important in this business.  But of course, you also can have gravitational waves, which goes to infinity.  Right.  So what is the basis of this conjecture?  Intuitively, it's very clear what it says.  Somehow, any solution would radiate energy at infinity.  So these gravitational waves would be generated at infinity.  And therefore, you will be left to stationary solutions.  But the only stationary solutions, this is another conjecture we'll discuss in a moment,  any stationary solution has to be a care solution.  That's another belief that people have.  Right.  So anyway, this is a conjecture.  Now, I want to show you a little bit, before taking the break, I want to show you a little  bit what is behind, what are...  I mean, if you really try to understand this conjecture, you will see that it consists  on many, many things, which by themselves are extremely interesting.  So for example, the first thing that you will have to connect it to this is that if the data  is sufficiently small, in other...  Of course, if I am in Minkowski space, the Minkowski space obviously has no black holes.  You don't see any care solution at the end of the day.  Right.  It's just flat Minkowski space.  So you...  In order to solve this, the first thing you like to do is to see that if I have small  data, in other words, if I make a small perturbation of Minkowski space.  So by the way, the initial data of Minkowski space will be...  The metric here will be the Euclidean metric.  So maybe I'll write it Euclidean.  And this will be zero.  Right.  So I start with Euclidean, and the second fundamental form is zero.  Right.  And then I want to make a small perturbation of this.  The fact that a small perturbation leads to...  It doesn't lead to any black hole is what it's called the stability of Minkowski space.  Right.  So it's a statement which in itself is highly non-trivial.  The second statement that is hidden inside that conjecture is that large data, in other  words, if I take the data now sufficiently large, it may concentrate.  So it actually can lead to a black hole.  Right.  So it can lead to a stationary solution.  So you could produce stationary solutions if the initial data is sufficiently large.  So this is called the problem of collapse.  Right.  So for example, from a physical point of view, you know that if you have enough matter, then  stars can at some point create neutron star.  And if you have enough energy, you can create a black hole.  Right.  So that's the problem of collapse.  You can see that it can be formulated in pure mathematical terms, in a sense, and without  matter.  So this is just in vacuum.  Right.  But these sort of things can happen, and we'll discuss it.  The next thing which is hidden in that conjecture is that all stationary states are care solutions.  Right.  So you see, the care solution is just a family of solutions which happen to be stationary.  But how do you know that all other stationary states are care solutions?  Right.  So the conjecture tells you that you see at the end of the evolution, you see only care  solutions, which means that somehow all stationary solutions will have to be care.  Right.  So this is by itself a highly non-trivial statement.  There is no reason not to expect other solutions.  So this has to be also studied as a mathematical problem.  Care solutions are stable.  So this is another major thing, is that if the conjecture were to be true, then care  solutions themselves will have to be stable.  Because if I make a small perturbation of a care solution, I can't get something crazy.  Right.  So the stability of the care solution is also part of this conjecture.  And this is just the soliton resolution conjecture.  Exactly.  Exactly.  It's a soliton resolution conjecture in the class.  For relativity.  But with the difference, I mean relative to the one that you guys are talking about, there  is a small difference.  But I'll talk about it.  I mean stability.  Stability is a major difference, because you do have stability of all these states.  So the solitons are all stable.  Right.  So that could be a, anyway, we can discuss it.  So, okay.  So other things that are included in the final state conjecture, which are huge conjecture  in their own right, which is the cosmic censorship conjecture.  So if this final state conjecture were to be true, then you should also verify the cosmic  censorship conjecture.  Now the cosmic censorship conjecture says something which again can be made more precise,  but I'll do it intuitively.  You see what happens in the, when you look at the, say, Schwarzschild metric.  The Schwarzschild metric has, it can be drawn by a diagram, which is called the Penrose  diagram.  I'm not going to talk, I mean there is something here at infinity, but I don't care about.  There is a black hole region.  So this is the separation.  So this is black hole.  And we'll talk more about it in a second.  So this is the part outside the black hole.  So this is the exterior region.  This is a black hole.  And once you are in the black hole, you reach a singularity.  So this is something that the character of the solution that we wrote down will tell  you immediately that once I start up as an observer, in other words, I started with a  time like curve in the black hole, I will necessarily have to hit a singularity in finite  time.  Finite time means relative to the affine parameter of the corresponding time like curve.  So in finite time, I hit a singularity.  So in other words, it's something very objective that you can, it doesn't make sense.  It's a proper time of the observer.  On the other hand, if I'm here, and of course if I'm stupid enough to move in this direction,  I will also hit a singularity.  But I have a chance to escape because I can go this way.  And I don't hit any singularities.  So because of this fact that you can see that somehow the expectation is that outside black  holes, you can't have singularities.  So the only singularities are hidden by horizons, which are the boundaries of these black holes.  And so this, of course, happens in the particular case of Schwarzschild solution.  The generalization of it is called the cosmic censorship conjecture.  So the generalization is that any initial data you take, again, generically, because  they are counter examples, but generically, any initial data you take, it will have to  have be such that you can define a black hole region where you can have black holes, and  therefore you can have singularities in those regions.  But outside the black holes, you can't have any singularities.  So this will also have to be part of that conjecture, of the finite state conjecture.  And by itself, it's as difficult as any other conjecture in mathematics.  And in fact, maybe even more so.  And probably more interesting than many other conjecture in mathematics.  Alex, you believe that the Riemann hypothesis is the most interesting conjecture in the  world?  You have to admit that there are some other, very hard and extremely interesting.  All right, anyway, the other thing that is also hidden here is a two-body problem, two  and maybe more, because in fact, actually, this corresponds to the LIGO experiment.  The recent LIGO experiments have to do with two black holes interacting, right?  So they get closer and closer to each other, they rotate around each other, and at some  point they coalesce, and they form a new black hole, right?  So there is no mathematical theory of such a thing at this stage.  There is a lot of numerical evidence and asymptotic expansions, but there is no mathematical theory,  right?  But this is clearly also part of the conjecture.  I mean, you will not be able to solve the conjecture if you don't understand how black holes interact,  right?  And of course, this has a clear and time, I mean, clearly clear physical significance  right now, right?  Because of the LIGO experiments.  All right, so there is a lot of evidence for the conjecture.  There is astrophysical conjecture, numerical, mathematical, but we are extremely far from  solving anything like this.  The mathematical evidence, I am not going to talk about these ones.  Of course, this we know from a lot of, I don't know, newspapers talk a lot about it,  right?  So we know that people believe that there are black holes all over the place, and therefore  somehow this conjecture has to be true.  I think the most montreal part of the conjecture is the absence of singularities.  Suppose that there are some singularities of mild type, which is the physical evidence  we have again.  Yeah, I agree.  I agree, but they are all connected in a sense.  But they are all connected in a sense, right?  I mean, this lack of singularities connected to stability, for example.  Stability is also, I mean, obviously, if you don't have stability, you don't have anything.  Right?  You don't.  Okay, well, we'll talk more about this.  Anyway, so there is simplifications.  Linear theory, one can do linear theory and one can be quite rigorous in linear theory.  But that's a huge simplification because the equations are nonlinear, highly nonlinear.  There is a symmetries.  You can take, look at various symmetries and you reduce the problems to something much  simpler.  And what I will talk about is the emergence of stronger mathematical techniques to deal  with this, at least to some of these issues.  Right, so then, I don't know, we should take maybe a three-minute break or?  Okay, so I talked about, so introduction and I mentioned this main problem in general  relativity, which as it stands, is much too difficult and it contains a lot of deep problems  in their own right.  And therefore, as mathematicians, the only way we go when we have a huge problem is to  divide it into smaller steps, right?  Smaller problems.  And so that's what I want to talk about now.  I want to pick, in particular, I want to pick three issues, three questions where I think  one can make progress.  By the way, here was something else that I should have done in the first hour, which  has to do with the picture, the standard picture of gravitational collapse, which is at large  energy concentrations, right?  May lead to the formation of a dynamical black hole, which settles down by gravitational  radiation to a care or schwarze in black holes.  So that's the kind of picture that you see here.  But anyway, I'll make some of these things more precise.  Anyway, the important, what I want to get out of the discussion in the first hour are  three questions which we believe are difficult, they require new mathematical techniques and  a lot of new ideas, but they are not impossible.  I mean, they could, in principle, be solved and they would be relevant in their own right.  So the first one is that the issue of collapse.  How can you show, can you actually produce this situation where you start with data which  are, as I said, not that small, so they're sufficiently large, but still regular in the  sense that there are no black holes in them to start with?  And how can I form a black hole later on?  So this is the problem of collapse and there are now a lot of interesting results of this  type, but we are very far away from a general understanding.  But at least we do understand in some very interesting examples how collapse can happen.  So that's the first question.  The question of rigidity, which I mentioned earlier, are all stationary states of the  Einstein-Vacuum equations, are they care black holes?  So this is the problem of rigidity and it's the easiest in a sense.  You'd think that it's the easiest problem.  There are lots of problems like this in elliptic, in Riemannian geometry.  And you'd think that at least this could be solved.  Unfortunately, it still isn't.  And then finally, there is a problem of stability, which is among them, I would say, the hardest.  And we made a lot of progress, I think, but we are still relatively far.  But in all these problems, you can see that there is an emergence of mathematical techniques  that dilutes them and fortunately, hopefully, many of these problems will be solved.  So this is, by the way, these three problems is what I call the reality of black holes.  Because obviously, if any of these statements will be wrong, then physically black holes  will not exist.  So because they exist, we believe that all these results have to be correct.  In other words, we should have the ability to collapse, because after all, black holes  have to form somehow.  We have to have rigidity.  In other words, the black holes are really care black holes, or maybe care Newman if  you have Maxwell equations involved.  But right now, I'm doing everything vacuum.  Or in the stability, of course, if the case solution is unstable, then obviously it's  not an observable thing.  So you think that in the minimum, these three things should be true.  So first of all, what is a general black hole now?  So a general black hole has to be a stationary solution, stationary asymptotically flat Einstein  vacuum equations.  So you are solving the equations we are talking about.  And we are talking about the black hole, but the external part of the black hole.  Not so the visible part.  So black holes are not visible, but the part that communicates with the exterior.  And the definition is that the external black hole is an asymptotically flat globally hyperbolic  Lorentzian manifold with boundary.  Defo-morphic to the complement of a cylinder in R1 plus 3.  And so this is one very mild requirement.  The most important one is that the metric has to have an asymptotically time like killing  vector field T, namely vector field such that the leader of the vector field of the metric  G is equal to 0.  So that's the definition of a guinea vector field.  And it has to be asymptotically time like because as we shall see, the care solution  has such a stationary solution, but which actually tilts and becomes space like near  the black hole.  So it's only far away from the black hole that you can expect to have real time like.  And then there is another thing which is called completeness of null infinity, which means  basically that there are no singularities in the exterior of the black hole.  I am not going to use.  In the outside.  Sorry, in the outside.  There are no singularities in the outside.  So that's basically, so the only important definition is really this one.  Just understand, so you stop the black hole at the horizon.  Exactly.  So the external part stops at the horizon.  So you don't look inside the horizon.  You have a completely different behavior.  By the way, this conjecture that I mentioned, the final state conjecture involves only the  exterior of black holes.  Inside black holes, you have lots of other conjecture that are equally difficult, maybe.  But how do you understand that you might want to avoid it?  But how do you avoid this mathematically in terms of definition?  Because the fact that your space time is not complete because there are some, the basics  would go inside the horizon.  Yeah, but you have causality.  You see, you still have causality.  So when you solve partial differential equations, you are interested in causality.  I mean, you, you know, so it's up to the boundary, everything is causal.  Right?  So, you know, it's like here, right?  Right?  If I'm in a region like this for the wave equation, right?  This is not complete, of course, because I can go this way.  But from the point of view of solving the, I understand, solving the wave equation,  all you care is what happens here, right?  And you construct this.  I want to take a black hole and stop it even earlier before the horizon.  You know, you, you, you remove.  Actually usually stop beyond the horizon, actually.  Inside the definition of black hole.  Sorry?  If you, if you, if you stop before the horizon, well, it will be just this business of, of  this competence of null infinity if you want.  I mean, that will not be solved anymore.  It will not be satisfied, yeah.  All right?  Okay.  So this, this is, all right.  So now, a care family is, of course, an example, right?  So again, I'm looking outside the region where, of the horizon.  I'll say it in a moment where the horizon is.  Anyway, this is just to remind you how things are.  The important thing in the discussion is this delta, which is r plus a square minus 2mr.  Now, Schwarzschild's solution, first of all, this is Schwarzschild.  This is a complete picture of the Schwarzschild's solution, in the so-called conformal compactification,  the Penrose diagram.  So what the conformal compactification does for you is to allow to go to infinity with  r.  So, and r goes to infinity, but there is a way of conformally changing the metric so  that all the directions, all the important things, the causal structure is preserved.  But this way, you get something finite.  So r is finite here, right?  So, so this corresponds, in fact, to points at infinity in the physical, for the physical  thing.  This is also now hyper-safety.  By the way, everything at 45 degrees here is null, right?  So if you have an observer, observer, which is time-like or causal, will hit, will hit  a scribe, but null hyper-safety, null geodesics at 45 degrees will hit scribe.  The other ones, at time-like, will hit there at the point at i plus infinity, which corresponds  to another type of infinity.  But the important thing to see here is that there is a horizon.  So there is r equal to 2m.  You see r equal to 2m comes, of course, in the metric.  r equal to 2m is not a singularity, as you might think, because you see you have 1 minus  2m over r to the minus 1 dr squared.  It's not a singularity.  It's actually what is called a horizon.  Now this is what I mentioned earlier, that even though this metric was discovered very  early on in 1915, it took about 50 years to understand the character of that singularity.  We're extremely confused, in fact Einstein was really confused about it.  Apparently he had some discussions with Zad Amar when he was in Paris.  At Amar was bugging him exactly about that singularity, and Einstein was very upset because  he couldn't answer.  Anyway, so there was a lot of questions, apparently, in the mathematical community in Paris about  that singularity.  It took a long time.  It was understood by the sixties it was understood.  It is understood in that that singularity has only to do with the coordinates, is because  you are using a specific coordinate system, which is good away from this horizon.  It's used in this region.  By the way, it's also good in this region.  If you go for R less than 2M, you can also put this metric is still relevant in that  region, but of course it's not relevant, and they're equal to 2M.  But you can change, you can write down another system of coordinates in which that singularity  is not there anymore, and in fact this is completely irregular.  So actually the complete extension of the metric was found much later, and we now have  these beautiful pictures.  You have a part R larger than 2M, another part R larger than 2M here, and you have the region  R less than 2M, and the singularity at R equal to 0.  You can see again at the metric that the R equal to 0 is a problem, and indeed R equal  to 0 is a true singularity, you have that the Riemann curvature tends to actually become  infinite.  In fact, we got scalars quantities which are formed from curvature, which becomes infinite  at R equal to 0.  Yes?  Does the black hole rotate?  No, no, no, this is a Schwarzschild solution.  It doesn't rotate.  So this is the external part of the black hole, and you see that everything is organized around  R.  So R is less than 2M corresponds to the black hole.  R equal to 0 is a singularity.  R less than 2M is black hole.  R larger, event horizon is exactly R equal to 2M, so that's the boundary between the  exterior region and the black hole region.  If you are in black hole, you necessarily fall into a singularity.  In other words, any time by curve in the black hole, if you extend it towards the future,  it will go to that R equal to 0, which is singular, and therefore everybody dies in  the black hole.  If you are here and you are careful, you survive for all time, you go for infinite time, of  course nobody lives for infinite time, but that's for other reasons.  Anyway, so this is a picture, right?  So R larger than 2M.  By the way, R equal to Cm is also something very important.  There is something very important happening at R equal to Cm, but maybe that's not the  time to say it now.  Maybe I'll say it later.  What happens at R equal to Cm?  So in other words, it's another important value of R. And then of course, it is R equal  to infinity, which corresponds to these boundaries here and here and here and there.  The care solution is similar in the external part.  So the external parts are very similar.  You don't terminate at R equal to 2M.  You terminate instead at R plus, which is the root of the delta polynomial R plus A  square minus 2MR.  You see that there are two roots.  One is this R equal to R minus.  That's a smaller root.  And the higher root is R plus, which corresponds to the horizon.  Now inside the black hole, you have all sorts of new phenomena.  And there are all sorts of interesting new results in mathematics.  They have to do the character of this R equal to R minus.  It's called the Hoshi horizon.  It's not a singularity.  And the question is whether it's stable or not.  And there are all sorts of very interesting issues connected to this.  But again, we are only interested in the external part, so I'm not going to talk about it.  So the external parts, again, the boundaries are equal to R plus.  You see that this is a causal region.  And this will be a space like hypersurface.  And of course, you'd like to, for example, if you do stability, you'd like somehow to  start the zinitial data here and see the evolution.  And we'll discuss about it later on.  So that's external part again.  This is a picture of the time-like stationary Keelink vector field that I mentioned we have  in care.  There is this d over dt, which looks like this.  It now looks exactly like a Minkowski space.  This is the part of the external part of the black hole, which looks like Minkowski space.  So because of asymptotic flatness, as you move, we are going to infinity, you become  more and more Minkowski.  And therefore, this t also looks perpendicular, just exactly like Minkowski space.  But as you approach the horizon, you see the t tilts and becomes time-like.  In the Schwarz's solution, this doesn't quite happen.  T actually remains tangent to the horizon.  So in other words, it's not time-like anymore.  It's not strictly time-like.  It becomes null.  And that has consequences.  So this is the region where t becomes space-like.  It's called the Ergo region, and it has a lot of physical and mathematical significance,  particularly it means that somehow the energy associated to t, but we know that energies  are always associated to time-like Keelink vector fields.  So this has to do with no theorem.  The fact that this is space-like in this region means you can extract energy from the  black hole.  So this actually has huge significance, both from a physical point of view.  Mathematically, it creates huge problems because typically the energy which is associated  to a time-like Keelink vector field has coercivity properties.  And this coercivity is unfortunately violated when t becomes space-like.  So there are all sorts of issues about this.  There are also these Strapnal geodesics, which I'm not going to talk about it now.  It's connected with what was before r equals 3m, I mentioned earlier, but I will not talk  about it now.  OK, so these are the tests of reality.  Again, the rigidity, stability and collapse, rigidity does a care family exhaust all possible  vacuum black holes.  So again, stationary, because black holes are the definition stationary.  Stability is a care family stable and the arbitrary small perturbation.  Then collapse can black holes form starting from reasonable initial configurations.  So this has to do with formation of Strap surfaces.  So I hope to talk about all of them.  I will not talk in detail.  So my hope is that I will later on talk in details about each one of them.  But for the moment, I want to give you sort of a general view of each one of them.  So they all can be viewed from the point of view of the initial data formulation.  So we start with initial conditions.  For example, the problem of collapse, you can formulate it as a problem in which you  start up with initial data, which are very reasonable.  There are no black holes initially.  And then you want to produce black hole later on.  Stability again is a problem that has to do with the initial data, because you can start  with the initial data set of care and make a small perturbation, make a small change,  and you want to know what happens in the long range.  Particularly, you don't want to have singularities.  So this, you see, the problem of stability has also a problem of singularities, because  even this could have singularities.  Anyway, so and then I will start now talking a little bit about each one of these problems.  So let me talk about rigidity first.  So the rigidity is a care-family exhaust or stationary asymptotic-ready-flat.  So what do we know about this?  So first of all, there is a case which is completely understood, which is a static case.  The static means, so we have a stationary solution.  In other words, there is a Keating vector field, right?  So there is a stationary Keating vector field, which is T. So in other words, we have a solution  of the Einstein equations, which has a Keating vector field, which is time-like, at least  far away, right?  In other words, in the asymptotically-flat region or in the region that becomes more  and more flat.  Now if you make another assumption about T, which is that T is hypersurface orthogonal,  so it's orthogonal on hypersurface, it's at every point you can find a hypersurface,  regular hypersurface on which T is orthogonal.  This is sort of an integrability condition, and it's always very hard in Schwarzschild,  but not in care.  So it's not really in care, but it's true in Schwarzschild.  And with that condition, with that extra condition, you can actually show that the solutions have  to be Schwarzschild solutions.  So the only static solutions are Schwarzschild.  So this is well understood, right?  The next thing is axially-symmetric case.  There is a beautiful result of Carter Robinson.  It goes back a long time ago, and there have been many other developments, simplifications  and extensions, or that result.  In fact, it's a result in Riemannian geometry, right?  Point being that if you assume not only stationarity, but also axially-symmetric, you take solutions  of the Einstein-Vakumov equations, which are static and axisymmetric, right?  So of course, care is axisymmetric also.  Then you are in care, right?  So this reduces to a problem about harmonic maps, in fact.  So it's a purely Riemannian geometry problem of harmonic maps, rigidity of harmonic maps,  and that's why it's true, right?  Okay, and then there is another result, which is due to Hawking.  It's really not a result.  It's just an observation.  So this is an observation that if you, in addition, assume an analyticity, then you  can reduce the general case.  In other words, the stationary case without axisymmetry, you can reduce it with this case,  and therefore, because of Carter Robinson, you are in care, okay?  So that's his result.  So this was viewed by people in physics as being the definitive result on rigidity, because  once you have this, you are done, right?  And of course, this assumption of analyticity is kind of no justification for it whatsoever.  But how does it work?  Why is it elitist?  You cannot assume a elitist.  How can you use an elitist to reduce like this?  Okay, so the proof goes like this.  Let me tell you roughly how the proof goes.  So you take that T, so you have a stationary solution.  You have a T, you can define the horizon, right?  That can always be defined starting from infinity, right?  So remember that has something to do with completeness of scribe, right?  So this, anyway, so this is a very, very weak argument.  I mean, a very soft argument that gives you the horizon.  Okay, then you can also show, it's also a very soft argument to show that T has to be  tangent to the horizon.  So it's not null anymore, it's actually space-like, but it's tangent to the horizon.  Once you have a key effect of being on the horizon, you can say that it generates a rotation  on the horizon, okay?  So you find a rotation exactly along the horizon.  Now you use the Einstein equations and analyticity, in other words, Koshy Kowalewski, to extend  this in the interior, and that's it, right?  So in other words, you show the existence of a second Keeling vector field, which is  a rotation based on the fact that you can, by a soft argument, you construct one exactly  on the horizon, right?  Okay, but the problem is, the problem is analyticity.  Why are you allowed to assume analyticity?  So well, okay, you can say, if I am away from the Ergo region, this is a very reasonable  assumption because stationary solutions, so if I have the Einstein equations together  with a time-like Keeling vector field, then the equations become an, the equations become  elliptic.  So the Einstein equations can be reduced to elliptic equations in the region where T  is time-like.  I mean, very much like the wave equation, right?  If I have a solution of the wave equation, dt squared plus Laplace N of i, equal to zero,  and if I know that twice stationary, that doesn't depend on time, then I'm left with  just the Laplace equation.  So it becomes elliptic.  The same thing here, this becomes elliptic, and therefore we know that even nonlinear  elliptic, very general nonlinear elliptic equations have analytic solutions, right?  So this is a reason of argument.  In fact, it can be made perfectly correct mathematically.  But this argument only works in this region where T is time-like.  When it becomes space-like, it's not true anymore.  In fact, the equations become hyperbolic again.  So in that region, the Einstein equations under the assumption that there is a Keeling  vector field, since T is not time-like, but space-like, the corresponding equations would  be hyperbolic.  No, no analyticity whatsoever.  In fact, it's even worse because there is a transition between elliptic and hyperbolic.  This is the hardest problems in partial differential equations, the ones where you have these transitions.  In fact, they are not understood even in much, much simpler cases.  So his argument is completely faulty.  In fact, it's not entirely without merit because it definitely shows that there can  be no as explicit solutions of the Einstein equations.  Because we know that the Keeling solution is analytic.  So if you are looking for as explicit solutions, you are not going to find them.  That's basically what Hawking tells you.  But certainly it does not answer the problem.  So let me tell you what we know now.  So analyticity is not reasonable.  It's an assumption.  And therefore, you have to do something else.  And it turns out that, so what you have to do is, of course, forget about analyticity.  Yeah, by the way, there is another.  Yeah.  So forget about analyticity, then analyticity makes no sense.  So you want to, it turns out that the problem becomes much, much more difficult.  I mean, once you don't have an analyticity, it's infinitely more difficult.  And you have to use a full Einstein equation somehow.  You cannot anymore do these kind of soft arguments.  So there are a few results.  Actually, maybe I'll mention the most interesting result.  There are some results that I have in collaborations with Yonescu and also with Alexakis and Yonescu.  This one, maybe I'm not going to explain it now.  But this one can be easily explained.  So I'll do that.  So the results, and this is, we have many versions of this result.  I mean, with weaker assumptions, in a sense.  But roughly what the result says is that if you have a stationary solution, which is sufficiently  close to care, in a sense that you can make precise.  In fact, you make precise by using the so-called mass-simon tensor, which I will maybe talk  about next time.  So you can make it precise.  And so again, if you are close to care, and you can make it precise what it means by closeness,  then you must be in care.  So it's, perturbatively, you are in care.  All right.  So here is a conjecture.  So this is connected to a question raised by Slava.  So the conjecture we have.  So we have a lot of methods.  I mean, this required completely new methods in terms of the kind of things that are used  in general activity.  And maybe I'll talk about it next time.  But for the moment, I just want to say that at the end of our work on these things, we  came up with a conjecture, which we think is very reasonable.  So the conjecture says that rigidity conjecture is true, provided there are no T-trap null  geodesics.  All right.  So let me mention something about this trap null geodesic.  So a trap.  So if you remember the picture that we had, the Spendros diagram of Schwarzschild, let's  say.  So remember that I said that there is a region here, which corresponds to R. So this is R  equal to 2M.  But there is a region R equals 3M.  And then R larger than 3M.  All right.  So if I am in the region R larger than 3M, any null geodesic will have to move towards  infinity.  Or if I am here, they will have to move inside the black hole.  So in other words, if I forget about this region, any null geodesics either moves towards  infinity or it moves inside the horizon, therefore in the black hole.  Once it goes into a black hole, I don't see it anymore.  It's finished.  I don't care about it.  Once it goes to infinity, I also don't care about it.  But unfortunately, you can have null geodesics which stay arbitrarily close to R equal to  3M.  In fact, there are null geodesics, complete null geodesics, which actually rotate around  R equal, rotate exactly on R equal to 3M in the case of Schwarzschild.  So this creates huge problems.  So it creates huge problems because somehow all the intuition they come from geometric  optics is wrong.  And that leads to issues in stability but also rigidity.  OK.  So anyway, this presence of trap null geodesics is very basic.  Now what we say here in this conjecture is that we don't care about the trap, in general,  about trap null geodesics.  We only care about those trap null geodesics which are perpendicular to T. Now you could  say where does it come from?  Well, it comes from our intuition, mathematical intuition.  But more importantly, it comes from the fact that in care, there are plenty of trap null  geodesics but there are no trap null geodesics perpendicular to T. So this is something that  you have to check.  I mean, there is no particular reason.  It just simply happens to be true.  So in other words, this explains this result because if there are no T trap null geodesics  close to care, I mean, in care, there are also no T trap null geodesics close to care.  And therefore, it explains the result here.  But more importantly, the conjecture is that this should be true in general.  If there are no T trap null geodesics, you can actually do it.  So the question is, can there be T trap null geodesics?  I see absolutely no reason to exclude them.  I mean, in other words, if you just think in terms of the original formulation of the  rigidity problem, there is absolutely no reason to exclude such things.  And of course, if such a, if these things exist, it's probably true that the conjecture  is wrong.  So in other words, I conjecture that in fact there are solutions of the ancient question  such as which are not care.  But they will probably be very unstable.  They will probably be very unstable, yes.  But they are there.  I mean, if you want to solve the conjecture, you have to take that into account.  So they will have to be unstable.  And maybe, you know, somebody should try to prove this conjecture.  I think none of us at this point, none of the people who were working.  Sorry.  Yeah, I think they, yeah, correct.  Yeah.  So in other words, the issue of stability should be measured in terms of the initial value  problem.  You see the rigidity conjecture has nothing to do with the initial value problem in some  sense, right?  It's just a problem about all possible stationary states.  And but if I also connected somehow to the initial conditions, and if I know that there  is one stationary solutions which is, which is for you to have the state trapped in our  geodesics, I look at the corresponding data on a space like a surface and I make a small  perturbation that would be unstable.  Yeah, but I don't see why.  It seems to me that you're simply emotionally attached to the first conjecture that you  made, which is this maximal global.  You want to save that one, but here you are really...  Well, yeah, okay.  So give me a chance to respond.  Yeah, give me a chance to respond next time because you have to go through a little bit  more.  But I believe it's unstable.  Of course, you know, if it's stable, it's even better because it will be more interesting.  It will definitely lead to something very interesting.  But I...  And of course, it would mean that the cosmic censorship conjecture is also wrong.  No, sorry, not the cosmic censorship, but the final state will be wrong, right?  It will be wrong.  Yeah, okay.  So anyway, let's leave it as it is.  It's an interesting issue, both from the point of view of proving the conjecture.  I think that is probably...  It's difficult, but probably not impossible given the techniques we have today.  And then the other thing is to show that there are space times which are stationary and which  have this T-trap nanogenetics.  And then finally, to show that they are unstable, which will...  I think will make Slava happy, right?  You'd rather have it unstable.  No?  Yeah, well, it's hard for me to eliminate the evidence.  All right.  Okay.  So let's go to the issue of collapse because we don't have that much more time.  So collapse can block form starting from reasonably initial data configurations.  And this has to do with trap surfaces.  Namely, you cannot...  It's not so easy to show that there are known black holes because the concept of a black  hole is kind of a tautological concept that you can only understand once you have a global  picture of the space time.  So in other words, you have to understand that whatever you do, you'll have to have a global  theory in order to understand formation of black holes.  So that's complicated.  But there is a concept which was introduced by Penrose, which in connection with this  incompleteness theorem, which allows you to reduce the problem of formation of black holes.  And if you assume the weak cosmic censorship conjecture, which of course is a huge conjecture,  but if you assume it, then the problem reduces to the connection to the fact that there exists  trap surfaces, right?  The existence of trap surfaces.  And the notion of a trap surface, which I'm going to explain in a second, is a local notion.  So it doesn't...  You don't care about the global picture, right?  So that's why it's very powerful.  The concept introduced by Penrose.  Here is what it is.  It's very simple.  Imagine a space time and imagine a two surface in the space time, right?  So if I am in Minkowski space and I have a two surface, then I can generate two types  of null cones, right?  So I can look at null geodesic perpendicular to the surface moving in this direction, in  other words, moving out, and the ones moving in, right?  So you get light cones this way and that way.  So if I look at the corresponding area, in other words, if I change, if I move a little  bit in this direction, the surfaces, and I look at how the area changes, I see that the  area, of course, is decreasing in this direction and increasing this direction, right?  So that's a typical situation in...  Now, of course, you can have a more complicated situation where the two surface looks like  this, which in that case, if you look at the corresponding light cones, you'll see that  the ones here, in the region where the surface looks like this, then the areas will also  decrease in this region, but they cannot...  In other words, in the outgoing direction.  One is the outgoing direction, the other one incoming.  So this area can decrease locally in the outgoing direction, but globally, you can never make  something which is outgoing decreasing at all points.  And this can be measured by something which are called expansions.  So you can actually write down some geometric quantities, which are called trace-ky and  trace-ky bar.  I'll talk more about these quantities later on, but for the moment, I'll just mention  there are some very simple geometric concepts which measure exactly the change of areas  in these directions or in the outgoing direction.  So this is the incoming direction and this is the outgoing direction.  So here, again, it's in the outgoing direction for trace-ky.  So this measure and this measure trace-ky bar.  And if this is positive, at points where trace-ky is positive, the area is increasing.  At points where trace-ky is negative, it's decreasing.  So here, trace-ky would decrease in this region, but will increase everywhere else.  And now, with all this, a trapped surface is very simply is defined as being a surface,  but if you look at the expansion, these geometric expansions which can be defined in general,  both geometric expansions are negative.  So in other words, the area is decreasing in both directions, both in the incoming and  the outgoing direction.  So this is obviously something that you cannot have in Minkowski's space.  But you can have, in fact, if you look at Schwarzschild's, if you look at the picture  of Schwarzschild's, if you look at the point, so points inside the black hole region are  two surfaces, in fact.  All those two surfaces are, in fact, trapped.  So you see that in a black hole, you have trapped.  On the other hand, if I'm outside, every surface outside is not trapped.  So somehow, the trapping has something to do with the black hole.  So it kind of, and in fact, you can show that if you have a trap, yeah.  So this is a singularity theorem.  So let me mention all the singularity theorem.  So Penderos theorem is following.  If you have, say, Ricci in any null direction, Ricci is a two tensor, and I apply it to any  null direction L. If Ricci is positive, in particular, if Ricci equal to zero, that's  obviously true.  OK?  So in flat case, this condition is trivial.  If in addition, M contains a non-compact or she-hypersurface, this is typically our case,  because we talk about asymptotic flatness.  So we have something which is non-compact, as you go to infinity.  So this also is a very simple condition to generate.  And if it has a trapped surface now, so this is the hard condition, if it has a trapped  surface, then it must have a, not a singularity, it must be incomplete.  In other words, there will be, in that case, null geodesics, which will have to terminate  in finite time, in finite affine parameter time.  Right?  So it doesn't tell you anything about the nature of the singularity.  It doesn't tell you that the curvature blows up.  It doesn't, it's a very, very weak statement, but it's an incompleteness, we call it an  incompleteness statement, exactly for the reason that it only says something about the existence  of some null geodesics which are incomplete.  Right?  But it's nevertheless a remarkable theorem, because using very little, you can prove something  which is extremely interesting.  So in particular, if you have a trapped surface and the cosmic censorship is true, then you  must have a black hole, right?  Because the cosmic censorship means you cannot have singularities outside black holes, but  if you have a trapped surface, it automatically has to mean that there is also a black hole,  because it has to be hidden under the black hole.  So somehow, a trapped surface is detect black holes.  Right?  So now, so the questions which are raised by this term, yeah.  Have you stare in?  Yes.  So if you want to have a nice initial data, so.  Yes.  And you want to have a trapped surface.  What is this?  It has to be B, yeah?  Yeah, yeah.  So of course, you cannot be close to Minkowski space.  Right?  You cannot have close to Minkowski space.  Right?  But this will be exactly what I'm going to talk about.  Right?  All right.  So these are the questions.  Can trapped surfaces form in evolution, in other words, starting with things which don't  have trapped surfaces, or starts with an initial data free of trapped surfaces?  I want to develop one in the future, right?  This is sort of the analogous of saying that there are no blacks also originally, but later  in evolution, you form a black hole, right?  All right.  So can they form from non-isotropic initial condition?  So this is very interesting because essentially all the intuition that people have, including  the one of Pendros, is that you have to produce energy from all directions in order to form  a trapped surface, in order to have collapse.  Right?  So for example, if you have a collapse of a star, of a neutron star, to a black hole,  the collapse in all directions, right?  So somehow the intuition was, or is to many people, is that you have to pump energy from  all directions in order to form a black hole because the trapped surface, obviously, it's  a condition, a large condition in all possible directions.  All right.  So that's finally what is the nature of the singularity of Pendros.  This is an open question.  It's a fundamental question, very little is known.  But I'm not going to talk about this.  So OK.  So here is a result.  So this is one of the results which was proved by Christodulu in 2008.  Right?  Most of the results were proved in recent years, right?  So all these developments, many of these developments happened in the last 20 years, let's say.  So Christodulu showed that there exists an open set of regular vacuum, right?  So in other words, this is for the vacuum equations.  By the way, again, a lot of the intuition that physicists have is that you have to have  matter in order to produce a black hole, right?  Here this is actually happened in vacuum.  You have in vacuum, in other words, you don't have any matter.  So the formation of a trapped surface has to do with gravitational waves, in fact.  All right.  So this is a picture.  The initial data here are not taken on a space like hypersurface, which is a little bit more  complicated, to simplify, you take the initial data on two null hypersurfaces.  So this is a characteristic initial value problem in partial differential equations,  which is very well understood also.  So it's not more difficult than the initial value problem.  In other words, I can, instead of taking data on a space like hypersurface, I take data  here and here.  On two null, so these are null hypersurfaces.  So you can see it in this picture there.  There's this null hypersurface and this null hypersurface.  So I want to take data here and here, which corresponds to here and here.  So what he did was to take Minkowski data here.  So these are trivial data.  And on this side, in other words, on this side, you take what he calls a short pulse.  So in other words, you take initial data, which give somehow a pulse in this direction,  a strong pulse in this direction.  And then he checked, of course, the following things, that you can construct data which is  sufficiently large.  So the pulse has to be sufficiently large to form a trap surface.  So it has to be sufficiently large, but free of trap surfaces here.  So initially, there are no trap surfaces.  Of course, there are no trap surfaces here because this is Minkowski and data.  And then he has to show with this initial data, you have to solve the Einstein equation,  the full Einstein equations.  By the way, in the case of Penrose, you don't have to solve the Einstein equations at all.  I mean, the result of Penrose is based purely on the Sokol-Rachaduri equation, which is just  one equation of many in the Einstein-Valkomine equation.  So it's actually a very, very soft result.  The result of Penrose is very, very soft, but it's remarkable, nevertheless.  So here you have to solve the full Einstein equations and show that you have a solution  sufficiently long time so that the trap surface can form.  So this is a semi-local result.  I mean, semi-global, sorry.  It's a semi-global result.  It's not a local result.  It's not like in Montchocke-Brua.  It's much, much more complicated.  In fact, the proof is very much, is quite similar to the proof of stability of Minkowski space  about which I'll talk next time.  Yes?  Do you take the disadvantage of the center of mass energy to show the existence of the  final trap surface?  There are similar setups in discussing physics where you collide particles and then a big  black hole forms which can.  Yeah.  No.  So this is actually simpler than that, in a sense, right?  Because that one has to do, it's too black hole.  No, it's not.  I mean, in a way, this is actually simpler.  What's remarkable here is that you can do it vacuum.  You don't need anything else.  We should just, it doesn't seem to me, yeah, I never understood this, because it's the  first time I'm trying to understand what the equation should be.  Okay.  All right.  Well, so please interrupt me.  I mean, there is no reason to go fast.  What I want to say is that, again, you could put the initial data here and here on two  non-hypersurfaces, right?  Minkowski and here, so it's flat.  You don't have to do anything interesting here.  But you choose a set of initial conditions here, which is sufficiently large so that  you can prove a semi-global result.  Semi-global means, for example, I have a parameter delta, let's say, and the size is, let's say,  delta to the minus a half or something like that.  So the size of the data here is delta to the minus a half, right?  Which means typically in the local existence, for people who know how to prove local existence,  you will get a solution only up to delta to the one half in time, right?  So you can extend it only for a short time, right?  So here you have to extend it for a whole, I mean, up to one, in other words, right?  So data is of delta to the minus a half, but you solve it up to time one in this direction,  right?  So this is a...  The physics, I mean, it's clear that you're taking that shape, central mass, because why  do you throw things from two sides?  If you just had a pulse coming from one direction, then by itself it would not form a flat surface.  So clearly you have to have two pulses which come in from two directions, one curve space  in one direction, by itself it would not be sufficient to fully on the other side start  curving the...  So actually they are intuitionists.  And then they kind of...  I should...  They work together.  Okay, so...  In some regions where they come close to each other.  So the two pulses propagate and they come close.  But you are talking about the pulse from here and one from here, right?  From that pulse to the right.  Right, but here it's...  There's no pulse coming from here.  There are no pulses.  There are no pulses from here, right?  So this is...  This is trivial data.  So the only pulse is here.  Right?  It's just one pulse.  Just one pulse, just on the left.  One pulse on the left, that's all.  But I'll say even more.  Okay, so this is the first result.  So just one pulse here, but the pulse is large in all directions, all angular directions,  right?  So you are still pushing things, you are producing a lot of gravitational energy from all directions.  Right?  Okay.  So that's important.  So in other words, it's uniform, a long null...  Yeah, so anyway, so this is a general result, right?  But then in addition, in that class...  By the way, this is a hard part of the proof.  You have to show that you can control the solution for long, long time is where you  have to use a lot of analysis, right?  But once you have that, once you have produced a space time which you control, then there  is a second part which says that among all those solutions, all those initial, class  of initial data which I control by my existence result, I pick up a special condition, an  additional condition, which is in fact a condition of uniformity along directions.  Then trap surface must form.  And at the same time, you show that the data is such that you don't have a trap surface  to start with.  So you have to take this quantitative, right, relative to this parameter delta.  So this parameter delta is a parameter that measures the strength of the pulse.  But why does it have to be uniform?  I mean, isn't this enough to...  I mean, in physics, there are such situations, this kind of...  You just throw in stuff from two directions, it comes close to each other, and you go trap  surface.  So let me show this as a result.  Okay?  That you might find more interesting.  So this is a result that I have with Ronianski and on look.  So it's actually...  What's interesting is that it's based on the first part of Christodulus result.  Well, Christodulus result was very long and it was simplified and we have much better  understanding.  Instead of 600 pages as it was originally, it's not only about 50 pages, let's say.  This is one of the cases where long proofs have been simplified quite a lot.  So anyway, but that result is still used here, except now the difference is that I take a  pulse which is concentrated only in one direction.  And everywhere else, I can make it to be Minkowskiian, say.  Right?  So in other words, it's the same situation, except that instead of taking a pulse which  is strong in all directions, I take a pulse which is now strong in only one direction.  Of course, I have to take a sufficiently strong pulse.  And I form a trap surface later on.  So the theorem is that it gives you conditions on the initial data here so that you form  a trap surface later on.  Now what's interesting is that this trap surface, you see, in the previous result, the trap  surface, the proof of the result is based on what is called the Dublin alpha radiation.  In other words, you foliate the spacetime by a family of light-cons going this way and  light-cons going this way.  OK?  So that gives you a foliation which is represented here in this picture by these surfaces and  these ones.  Right?  So and somehow this Dublin alpha radiation plays a very important role in the proof of  the first result, right, that you have this one here.  So the actual trap surface is actually in this foliation.  So you find the trap surface at the end of your spacetime here.  So the trap surface would be right here.  So this is a trap surface.  Right?  In our result, that's not the case anymore, you have to find a deformation.  So you see, you actually have to deform the two sides.  So the surfaces of the double-nile foliation are this one, these regular surfaces here,  and you have to find actually by an argument, which is a deformation argument, you have  to find this one.  Right?  So this is the case when you can show that actually all you have to do, you don't need  two pulses, as you said, you need just one pulse in one direction and you can already  get the trap surface.  So if you have two, of course, you'd think that it's a little bit easier, but...  Does it have to be focusing with pulse?  This one here?  Yeah.  Well, obviously you have to focus it in this direction.  I mean, you have to produce a lot of energy in this area.  But of course, you don't have a trap surface originally.  I mean, it's obvious because everything is Minkowski and outside that region.  Right?  Yes?  So the trap surface as maybe form like as a cow?  No, no, no.  So this is trapped.  Trap surface is not yet a black hole.  It detects a black hole.  So you know, if you have a trap surface, you can say that the black form will form later  once you have the trap surface, a black hole form.  But yes?  How long do you take in time to store the formation of the trap surface?  Yeah.  So again, if you have a pulse of size, say delta to the one half, delta, whatever.  So if it's delta, as you know, existence will go...  Sorry, delta to the minus one, excuse me.  This is delta to the...  I mean, sorry, the size of this is delta.  And the strength of the pulse is delta to the minus a half.  Delta to the minus a half, right?  And then you can... this falls at time one.  This time one, yeah.  So anyway, so this is formation of trap surface.  Now, there are many other results of interesting, but I guess I'll have to stop.  I wanted also to talk about stability, which is maybe the main focus of what I'll talk  about here.  But I guess I'll leave it for next time.  Right?  And if there are any questions, I'll...  Yes.  Okay.  So...  Okay.  Okay.  Yeah.  Okay.  So I guess I'll have to stop.  So I guess this is the end of my video.  Thanks for watching.  I hope you have enjoyed watching.  I would like to thank you.  I hope you enjoyed watching as well.  I'm going to try it.  Thank you.  Thank you.  If there are any questions, I'll...  Yes.  Thank you.  You're welcome.  Thank you.  Thank you very much.  Thank you.  Thank you.  Thank you very much.
Sergiu_Klainerman_-_14_On_the_Mathematical_Theory_of_Black_Holes_0_59.png Sergiu_Klainerman_-_14_On_the_Mathematical_Theory_of_Black_Holes_60_107.png Sergiu_Klainerman_-_14_On_the_Mathematical_Theory_of_Black_Holes_492_743.png Sergiu_Klainerman_-_14_On_the_Mathematical_Theory_of_Black_Holes_176616_178499.png