I want to sank 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'll 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 the 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 Riemani in geometry, the notion of curvature play the 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 know, you have Lorentzian matrix, Lorentzian geometry is at the heart of general activity. So, that means you have many faults, maybe I should take some chalks and also put something on the blackboard. Is there a chalk somewhere? Below. Oh, here, yeah. Ok, so, obviously the starting point of Lorentzian 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 Riemani in geometry. The only difference is that the metric of interest is not Riemani in metric, but rather Lorentzian, 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 that 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, right? 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 this 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. And this corresponds to observers. Observers in general 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, geodesic, again, the concept of geodesic is exactly the same as in Divanian geometry, and null simply means geodesic, which is such that the tangent at every point gives you g xx equal to zero, or x is a tangent to the point. Proper time, which is nothing else, so the proper time along a curve is nothing else than the alpha in parameter. Just like in Divanian geometry. Tidal forces in, which have 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 Divanian geometry. Given the metric g, you can associate the so-called Riemann curvature tensor. So, R-I-J-K-L, 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, 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 else but asymptotic flatness.  And typically, the manifolds of interest, particularly if you talk about isolated system, is the ones which are, say, defomorphic, at least outside a sufficiently compact space. They look like r1 plus n from a topological point of view. And you want the metric also to look Minkowski in, if you are sufficiently far away from a desisolated system. All right. So, 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 imanian geometry, you can associate the connection, which is the so-called Levy-Civita connection. So, exactly the same properties as in imanian 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 move forward in time and backward in time. Curves, which are time-lark 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. Causal, which means that you allow the tangent also to be null. So, future and past sets. Well, let me go here. Let me move here. Future and past sets. Again, a very simple concept, but it's quite important in Lorentzian geometry and, of course, in generality 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 a null cone in Minkowski space, then the future of this point, so you distinguish between time-like future and causal future, so the 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, hypersurfaces. 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 hypersurfaces of interest. The ones which are space-like. So, space-like hypersurfaces is that if you look at the unit normal at every point, it's time-like. So, unit normal is time-like at every point, which means that if I look at the Lorentzian metric and I restrict it to the space-like hypersurfaces, 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 hypersurfaces of space-like hypersurfaces to charnal. So, this is a null hypersurface. So, as you see, it's ruled by geodesics to charnal or curves to charnal, 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 do not intersect the hypersurface, right? So, anyway, it's a very important concept and we'll talk more about this. All right, so, the next topic... We have an 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, 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 the Emanian metric, typically a frame is a set of vector fields, say, u1en on g, which is such that the metric G, E, I, eJ is equal to delta ij. So, for all i and j. And this is a Kronecker symbol. In Lorentzian geometry, of course, you can define the same thing, so you can have frames which are called orthonormal. So, for example, if you are in Minkowski space, you can take a time-direction, say, e0, And then e1, en minus 1, or en, let's say, if I am in a plus one dimension, will be space-like curves, space-like vector fields, which are perpendicular to e0, and the zero 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 minus 1, if I am in n plus one dimensions. So these two are null, so gl l is gl l bar l bar is equal to zero. You also normalize them. Of course, you cannot do anything like in the money geometry, where you take one of them to have lengths one, because they are null. But I can normalize l and l bar, so that this is equal, say, minus 2. So, for example, if I am again in Nikovski space, I can take vector fields, so this is a null cone, at the point, 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 2. 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 money 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. So, for example, here is an example of a null frame in Nikovski 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. 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 carton formalize, well, again, this is exactly like in money in 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, so I have say, in the money in geometry e1en, 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, the covariant derivative. So it's induced by the Levy-Civita connections. And then the carton formalize tells you that somehow, derivatives of gamma plus gamma, so you get equations, which are the carton equations, which are equal to curvature. So on the right hand side, you have the Riemann curvature tensor. So I'm writing it just very formally. We'll have, whenever we need, I'll write it more explicitly. But these are the fundamental equations in Riemann 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 carton formalize, I'll be using null frames. Folijations, so another thing, which is extremely important in problems in general activity, is to have folijations. So one kind of folijations will be given by in Mikowski space. For example, it's just the 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 folijation by space-like hypersurfaces. So this will be on an arbitrary Lorentzian manifold. Another kind of folijation, 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 sorry, optical function is a function u, which verifies the ikonal equation. So gi, j, di, u, dju is equal to zero. In fact, actually, to distinguish between Riemani 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 zero. So this means nothing else that if I take the exterior derivative of u, so if I take the differential of u, then gi, du, du is equal to zero. In other words, if I look at u, it called constant, that 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. It's null. So in other words, if I solve the ikonal equation, which is, of course, an equation extremely important in all sorts of branches of mathematics and physics, if I solve the ikonal equation, I get the solution u, the level surfaces of u would give me a foliation by null cons. So, for example, again in Nikovski space, I can have a foliation given by, for example, like this, where all these are null cons. So I can foliate part of the spacetime by null cons. 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 foliation, 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 would be Riemannian. So we call them, so if 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 n 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. So these are compact Riemannian manifolds. And they, 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 arise naturally as boundaries of future and past sets. 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 it's, if there are, so these are typically ruled by null geodesics, so the boundaries of future are ruled by null geodesics. 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 Poincare, who said, parmi ses action implicit, il est un qui sembler mérité quelque attention, parce que en l'abandonant, on peut construir une quatrième geometry, aussi cohérente que celle de euclide et de lobachesque de Riemann, je ne citereré qu'en de cette harem et je ne choisiré par le plus sangulier. 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 it's very different from what happens in the Manian geometry. And this leads to a lot of interesting mathematics, which has been somewhat ignored by geometers. I would say that Poincare is always his visionary, and he's understood that this could be very important. And indeed, these null hypersurface are extremely important in general relativity. OK, so with this, let me go now to the beyond Lorentzian geometry. What is the geometric framework of general relativity? Well, you have field equations. So this is now, obviously, you are looking now 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 relativity 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 2 tensor. And the equation looks like this, minus 1 over 2g, times the scalar curvature. So the scalar curvature, so this is Ric. 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 that 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. So 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 0, which is, of course, makes perfect sense. So this will be, if you are to make an analogy with Maxwell theory, is studying pure electromagnetic waves without sources. 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 reach has to be 0. So the equations of interest in my lectures would be this one. So you have reach equal to 0. Sorry? Sorry? What do you mean conversely? The top equation is equivalent to the reach. Yeah, correct. So if t is equal to 0, these two are equivalent indeed. OK, 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. 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 richy flat, which g is equal to 0. And the first thing 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, we have to take into account another important fact about this equation, which is that there are deformorphism 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 1 plus n, 1 plus n. So a solution is actually a class of equivalence of solution modulo, this huge group of deformorphism of the manifold. Because these are generally deformorphism. 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 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, was possible 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. 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. So this is defined exactly in 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 are called 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 Ivan Trokebrov, 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. OK, 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 involves the metric and first derivative of the metric, and say it's quadratic in the first derivative of the metric. So in any case, it's recognizable. The character of the equation is now recognizable. And 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 in these things. OK, so equations are hyperbolic. That's what we discover. We discover that if we use the gauge dependence of the Einstein equations, if we use 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, so the metric is then just a Minkowski metric, then, say, I want to solve just the inversion of phi is equal to 0. So the downversion being, again, this is minus dt squared plus Laplacian of phi is equal to 0. And typically, one solve this by choosing initial conditions, say, t equals 0. It can be any space like hypersurface. You can take initial conditions on any space. But in posimplicity, you just take t equals 0. And then you get phi of 0x is, say, f of x. And dt of phi of 0x is g of x. So this you prescribe. And then you show that there exists a unique solution that solves this equation. So this is for the simplest possible equation. And of course, you can do that for much more general classes of equations. Yes? Does the formalism of general relativity is the same that the formalism eminator used to demonstrate theorems about physics? There is, of course, a connection, obviously. But I'm not going to get into this now. But of course, there are conservation laws, and all sorts of things that you have to do eminator. 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 unique solutions for the wave equations you find if you prescribe these two initial conditions. 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 0 hypersurface, 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. So we are in four dimensions. So this will have to be three dimensional, starting in one dimension less. This will have to be a Riemanian metric, because I know that if I can solve the Einstein equations, I will get a Lorentzian manifold. And this original hypersurface sigma 0, embed it into it, will have to be spacelike. And therefore, this, as an input, will have to be a Riemanian metric. And this will correspond to the time derivative of the Lorentzian metric. This is called the second fundamental form of the original spacelike hypersurface. So typically I prescribe g0 and k0 and sigma 0, 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. So 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 as a 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 Riemanian geometry. This you could view, the constraint equation, initial data, constraint equations can be viewed as a subject in Riemanian geometry. 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 Riemanian geometry, played a major role not just in connection where it started in journal activity, but in many other applications of problems in Riemanian geometry. And they have to do somehow with the geometry of these initial data sets together with the constraint equations. OK, 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 the 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 maximal future development. In other words, this will be something similar to what happens for simple ordinary differential equations. Suppose I have 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 time, 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 Ivonsha Kevrba in 1952, and complemented by Gerovch 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 spacetime that you have constructed, by the way, all this concept came in the French school, by the way, I should say. Even the formulations of the initial value problem as a hyperbolic problem was really not well understood until Le Reche, OK Buah, 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 Le Re and certainly Ivonsha Kevrba. And then, Bruha Gerovch, 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 Le Re, 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 spacetime, this surface, sorry, 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 spacetime that you have constructed, this is a Cauchy hypersurface, right? 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. All right, 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 it's certainly not, right? 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 possible, 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 solv the equations without necessarily producing explicit solutions. But the physicists who started first to work on this, they wanted to look at explicit solutions, right? So that's natural and we don't blame them. In fact, it's extremely important, right? 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 keeling or conformal keeling. In other words, they generate, so these are vector fields, which generate one parameter group of the thermomorphism, which correspond to isometries or conformal isometries, right? 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 keeling or conformal keeling 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, right? So in that case, you have an action, it's more than just one or two keeling 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, Einstein equations become much simpler and you can actually solve them in specific 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, right?  But you can also, if you have one, for example, typically, you have to think about having one such keeling or conformal keeling, typically keeling, having one such keeling 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 a problem from n dimension to one plus n dimension, maybe I should say one plus n minus one, right? For example, right? So you simplify the equations quite a bit, but they could still be very complicated. Spherical symmetry makes life much easier. And in the case of the Einstein equations, of course, the simplest solution is the Minkowski space itself, right? The Minkowski space certainly verifies this equation. So Minkowski space is a special solution of the Einstein equations in vacuum, but then there are more such solutions. The most interesting class of such solutions was discovered by Schwarzschild. So here is a sample. 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 zero. The case equal to zero, it's exactly the case of Minkowski space. But you see that the moment you introduce a norm zero 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, r, t time five, 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, because all these coefficients are calculated by this very simple trigonometry formulae. So it's explicit, and it has huge impact. I mean, basically, whatever we think about black holes today are essentially based on these solutions. 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 in the solution. Of course, if you use a calculator, it's much smarter, you'll probably get it in a few hours. Sorry? 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, that R goes to infinity, you get closer and closer to Mikowski space. So that's just standard to see, to see immediately, yes. How does this equations of homogenous... Excuse me? ...homogenous, homogeneity. Sorry, how does in units... I'm not looking at units now, in terms of physical units, you mean? No, no, I'm writing it as purely a mathematical solution 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, 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? 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. Stationary solutions means, I guess I didn't, yeah, okay. 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 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 d phi is a killing vector field, right? Okay, so this corresponds to a rotation. So this corresponds to a rotation. So these are, we say, our black holes, which are rotating. The, 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 a equal to 0, you are reduced to Minkowski space, right? So this is the famous two-parameter family of solutions of the Einstein-Vakume equations, which verify all the interesting assumptions that you want. It's asymptotically flat, 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 de-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 would like to solve, right? 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, all right? So I say generically, of course, we 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. Have maximal future development. So in other words, there is this global maximal development that I talked about before, all right? So you go as far as you can go. And here you are saying that this maximal future development, which are maximally extended solutions of the n-stand vacuum equations, so that's the same thing, look asymptotically in any finite region of space as a care solution, right? So, okay, so let me make it a bit more graphic. So you take an initial data set, you take a very general sigma zero, g zero, k zero, you take a very general initial data set, which is, of course, asymptotically flat and smooth in whatever sense, and you look at the maximal future hyperbolic development. Well, this tells you that the maximal 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 maximal, when you, when we talked earlier about maximal future hyperbolic development, it could have been that if I look at the time like curve, the time like curve terminates in finite time. In other words, observers will die after finite time, right? So that means it's not a very, 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'll see only a finite number. In any Kampag 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 finite states will be care. So what is? Rotate to 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 meet gravitational waves in the future. Correct, yeah, correct. So the genericity is also very important in this business. But of course you also can have gravitational waves, which goes to infinity. So what is the basis of these conjectures? 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, okay? All 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 Mikowski space, then Mikowski space obviously has no black holes. You don't see any care solution at the end of the day, right? It's just flat Mikowski space. So 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 Mikowski space, so by the way, the initial data of Mikowski 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, leads to, doesn't lead to any black hole, is what it's called the stability of Mikowski space, right? So that'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 the 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? But this is by itself a highly nontrivial statement. There is no reason not to expect other solutions, right? 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 stability of the care solution is also part of this conjecture. And... But this is just the soliton resolution conjecture. Exactly, exactly. It's exactly, it's a solitonal resolution conjecture. In the class. In the probability. For relativity. But it was a difference. I mean, relative to the one that you guys are talking about, there is a small difference, but I'll talk about. 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, say, Schwarzschild metric. 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, right?  But I have a chance to escape because I can go this way, right? So 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, right? 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 there are count examples, but generically, any initial data you take, it will have to 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, okay? So this will also have to be part of that conjecture, right, of the finite state conjecture. And by itself is, it says difficult as any other conjecture in mathematics, in fact, maybe even more so, okay? And probably more interesting than any 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 others, 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 this 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. Numerical. I think the most monstrual part of the conjecture is the absence of singularities. Suppose there are some singularities of mild type, which is the physical evidence of the gay. Yeah. But 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 is 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 it. 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 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. All right, so then, I don't know, we should take maybe a three-minute break. Okay, so I talked about, so introduction and I mentioned this main problem in general activity, 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, smaller problems. And so that's what I want to talk about now. I want to pick, in particular, I want to pick 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, may lead to the formation of a dynamical black hole, which settles down by gravitational radiation to a care or schwarz in black hole. So that's a 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 and you actually produce this situation where you start with data, which are, as I said, not that small, so they are 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. Second, 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 diluzem. And fortunately, hopefully, many of these problems will be solved.  Because obviously, if any of these statements would be wrong, then physically black holes would not exist. So because they exist, we believe that they have to, 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.   Stationally 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. In other words, 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, whose boundary default 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 be, has to have an asymptotically time lag, killing vector field T, namely vector field, such that the lead derivative with respect to this vector field of the metric G is equal to 0. So that's the definition of a guine vector field. And it has to be asymptotically time lag, because as we shall see, the care solution has such a stationary solution, but which actually tilts and becomes space lag near the black hole. So it's only far away from the black hole that you can expect to have real time lag, that this vector field is time lag. 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. In the outside. Sorry, in the outside. There are no singularities in the outside. So that's basically, the only important definition is really this one. 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. Inside or outside, 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.  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 spacetime is not complete, because there are some basics which 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, in other words, up to the boundary, everything is causal. So it's like here. If I am in a region like this, for the wave equation, this is not complete, of course, because I can go this way. But from the point of view of solving the wave equation, all you care is what happens here. And you construct this. So I take a black hole and stop it even earlier before the horizon. You remove? Actually, usually stop beyond the horizon, actually. Inside the definition of black hole. Sorry? If you stop before the horizon, well, it will be just this business of this completeness of null infinity, if you want. I mean, that will not be solved anymore. We're not dissatisfied, yeah. OK, so this is, all right. So now the care family is, of course, an example. Again, I'm looking outside the region of the horizon. I'll say it in a moment what the horizon is. Anyway, this is just to remind you how things are. The important thing in the discussion is delta, which is r plus a squared 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 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 this corresponds, in fact, to points at infinity in the physical, for the physical thing. This is also null hypersurface. 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 scry. But a null hypersurface, a null geodesic at 45 degrees will hit scry. The other ones at time like will hit there at the point at time 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. People were extremely confused. In fact, Einstein was really confused about it. But he had some discussions with Zad Amar when he was in Paris, where Zad 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 mathematical community in Paris about that singularity. It took a long time. Anyway, it was understood by the 60s it was understood. It is understood in that that singularity has only to do with the coordinates. It's 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, 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 R equal to 0 is a problem. And indeed, R equal to 0 is a true singularity. You have that the Riemann-Kerberge tens actually becomes infinite. In fact, in fact, we got scalars quantities which are formed from Kerbergev, which becomes infinite at R equal to 0. Yes. Does the black hole is rotated? 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 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, anytime black 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 any time. Of course, nobody lives for any time. But that's for other reasons. Anyway, so this is a picture. 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? 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. OK. The care solution is similar in the external parts. 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 2M R. 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 is a very interesting. There are all sorts of interesting new results in mathematics. They have to do the character of this R equal to R minus. That'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 these. But again, we are only interested in the external parts, 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 this initial data here and see the evolution. And we'll discuss about it later on. All right, so that's external part, again. This is a picture of the time-like stationary killing vector field that I mentioned we have in care. There is this d over dt, which looks like this. It does always look exactly like in 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, and you become more and more Minkowski. And therefore, this t also looks perpendicular, just exactly like in Minkowski space. But as you approach the horizon, you see that 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. All right, 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 killing vector fields, right? So this has to do not as theorem. The fact that this is space-like in this region means you can extract energy from the black hole, right? 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 with a time-like killing vector field, has cohesivity properties. And this cohesivity is unfortunately violated when t becomes space-like. So there are all sorts of issues about this. There are also these strabnal 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 by definition stationary. Stability is a care family stable, so one of the arbitrary small perturbation, right? And collapse can black holes form starting from reasonable initial configurations. So this has to do with formation of strab surfaces. So I hope to talk about all of them. I will not talk in details. 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.  Stability, again, is a problem that has to do with the initial data, because you can start with the initial data set of care, right?  Particularly, you don't want to have singularities. Slava, so this, you see, the problem of stability has also a problem of singularities, because even this could have singularities, right? Anyway, so, and then I will start now talking a little bit about each one of these problems, right? So let me talk about rigidity first. So rigidity is care family, exhaustor stationary asymptotically flat. All right, 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, a regular hypersurface on which T is orthogonal. This is sort of an integability condition, and it's always verified in Schwarzschild, but not in Keating. So it's not true in Keating, 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 actually 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. The point being that if you assume, not only rationality, but also axial symmetries, you take solutions of the Einstein-Bachme question, 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 pure Riemannian geometry problem of harmonic maps, rigidity of harmonic maps, and that's why it's true, right? Okay, 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 as analyticity, then you can reduce the general case, in other words, the stationary case without axisymmetry, you can reduce it to 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? Except, of course, this assumption of analyticity is kind of no justification for it whatsoever. What causes it to work? And why is it... No, you cannot assume analyticity. No, how can you use analyticity to reduce the axisymmetry? 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 actually is space-like, but it's tangent to the horizon. Once you have a kelling vector in the horizon, you can say that it generates a rotation on the horizon, okay? So you find a rotation exactly along the horizon, and now you use the Einstein equations in analyticity, in other words, Košek-Valevski, to extend this in the interior, and that's it, right? So in other words, you show the existence of a second kelling vector field, which is a rotation based on the fact that you can, by a soft argument, you can construct one exactly on the horizon, right? Okay, but the problem is, the problem is analyticity. Why are you allowed to assume analyticity, okay? So, well, okay, you can say, if I am away from their origin, this is a very reasonable assumption because stationary solution, so if I have the Einstein equations together with a time-like kelling vector field, then the equations become, 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 Laplacian of phi equal to zero, and if I know that phi is stationary, that doesn't depend on time, then I'm left to just Laplacian 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 reasonable 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 kine vector field, since t is not time-like, but space-like, the corresponding equations will be hyperbolic. No analyticity whatsoever. In fact, it's even worse, because there is a transition between elliptic and hyperbolic. This is, these are the hardest problems in partial differential equations, the ones where you have these transitions, right? In fact, they are not understood even in much, much simpler cases. So his argument is completely faulty. I mean, in fact, it's not entirely without merit because it definitely shows that there can be no as explicit solutions of the Einstein equations, right? Because analyticity, I mean, we know that the k-solution is analytic. So if you are looking for as explicit solutions, you are not going to find them, right? That's basically what Hawking tells you. But certainly it does not answer the problem. Okay, 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, okay, so forget. Forget about analyticity, 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 analyticity, it's infinity 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 result, and this is, we have many versions of this result. I mean, it was 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. So let me mention something about this trap null geodesics. So if you remember the picture that we had, this is Pandro's 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 equals 3M in the case of Schwarzschild. So, and this creates huge problems, right? So it creates huge problem because somehow all the intuition, they come from geometric optics, is wrong, right? And that leads to issues in stability, but also rigidity. 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, right? Now, you could say, where does it come from? Well, it comes from, well, 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, right? So this is something that you have to check. I mean, there is no particular reason. It just simply happens to be true, right? 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, right? But more importantly, the conjecture is that this should be between gel. If there are no T trap null geodesics, you can actually do it, right? 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, 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, OK? 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, right? OK. So they will have to be unstable. And maybe somebody should try to prove this conjecture. I think none of us at this point, none of the people who work. Why do you think they are unstable? 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, for which you have this tetrapnar geodesics, I look at the corresponding data on a space like hyper 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, OK. 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, if it's stable, it's even better, because it will be more interesting. It will definitely lead to something very interesting. And, of course, it will mean that the cosmic censorship conjecture is also wrong. No, sorry, not the cosmic censorship, but the final state will be wrong, right? Will be wrong. Yeah, OK. 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 difficult, but probably not impossible, given the techniques we have today. And then the other thing is to show that there are spacetimes, which are stationary and which have this T-trap, now geodesics. 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 believe the evidence. All right, OK. Anyway, let's go to the issue of collapse, because we don't have that much more time. So collapse, can black 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 no 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 spacetime. So in other words, you have to understand 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, in connection with his incompleteness theorem, which allows you to reduce the problem of formation of black holes. 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 exist trap surfaces, 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 you don't care about the global picture. So that's why it's very powerful. It's a concept introduced by Penrose. Here is what it is. It's very simple. Imagine a spacetime, and imagine a two surface in the spacetime. So if I am in Minkowski space and I have a two surface, then I can generate two types of null cones. 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. 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 in this direction. 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 outgoing direction, the other one incoming. So these areas 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 is called expansion. So you can actually write down some geometric quantities, which are called trezky and trezky bar. I'll talk more about these quantities later on, but for the moment I just mentioned 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 trezky. So this measure and this measure trezky bar. And if this is positive, at points where trezky is positive, the area is increasing. At points where trezky is negative, is decreasing. So here, trezky will decrease in this region, but will increase everywhere else. And now, with all this, a trap 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 the incoming and the outgoing direction. So this is obviously something that you cannot have in Minkowski space. But you can have, in fact, if you look at Schwarzschild, if you look at the picture of Schwarzschild, 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. And in fact, you can show that if you have a trap, so this is a singularity theorem. So let me mention all the singularity theorem. So Penrose theorem is the 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 is equal to zero, that's obviously true. 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 you 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 trap surface now, so this is the hard condition, if it has a trap 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. 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. But it's nevertheless a remarkable theorem because using very little you can prove something which is extremely interesting.  Right? Because the cosmic censorship means you cannot have singularities outside black holes, but if you have a trap surface would automatically have to mean that there is also a black hole because it has to be hidden under the black hole.  Right. So now, so the questions which are raised by this term, yeah. Have you stare in? Yes. So if you, you want to have a nice in short data, so. Yes. And you want to have a trap surface. What is this? It has to be B, yeah? Yeah, of course. You cannot be, 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 trap surfaces form in evolution in other words starting with things which don't have trap surface. So I start with the initial data free of trap surfaces. I want to develop one in the future. All 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. 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 trap 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, if they collapse in all directions. So somehow the intuition was, people is that you have to pump energy from all directions in order to form a black hole because the trap surface obviously it's a condition, a large conditions in all possible directions. All right. So that's finally what is the nature of the singularity created by Pendros. This is an open question. It's a fundamental open question and very little is known. But I'm not going to talk about this. So here is a result. So this is one of the results which was proved 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 Chris Odulu showed that there exists the 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 in vacuum in other words you don't have any matter. So the the formation of a trap surface has to do with gravitational waves, in fact. 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 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 hypersurface I take data here and here. So these are now hypersurfaces. So you can see it in this picture there. There is this now hypersurface and this now hypersurface. So I think I want to take data here and here. Right. Which corresponds to here and here. What he did was to take Minkowski data here. Right. 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. OK. And then he checked of course the following things. Right. That you can construct data which are 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 surface is here. So initially there are no trap surface of course there are no trap surface is here because this is Minkowski and data. Right. And then he has to show this initial data you have to solve the full Einstein equations. By the way in the case of the result of Penrose is based purely on the so-called Richard Dury equation which is just one equation of many in the in the Einstein vacuum 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 not a local result it's not like in Monchoke bra it's much 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. So do you take in this integration do you take the advantage of the center of mass energy to show the existence of the final trap surface?  Ja, no, so this is this is actually simpler than that in a sense, right? Because that one that one has to do it's too black hole. No, it's not. I mean it's in a way this is actually simpler. What's remarkable here is that you can do in vacuum you don't need anything else. We should discuss it doesn't seem to me I never understood what was it's the first time I'm starting to understand OK. 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 nonhyperserfaces, 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 I have a parameter delta let's say and the size is of 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 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 so you can extend it only for a short time so here you have to extend it for a whole I mean up to one in other words so data is of size delta to the minus a half but you solve it up to time one in this direction so this is repeated system it's clear that you are taking that shape center of mass because why do you throw things from two sides if you just get a pulse coming from one direction then by itself it will not form a threat surface so here 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 in two and then they kind of they work together in some region 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? but here there is no pulse coming from here there are no pulses from here so this is trivial data so the only pulse is here is just one pulse just on the left one pulse on the left that's all but I say even more so this is the first result so just one pulse here but the pulse is large in all angular directions so you are still pushing things you are producing a lot of gravitational energy in all from all directions so that's important so so in other words it's uniform, a long null so anyway so this is a general result but then in addition in that class by the way this is a hard part of the proof you have to show that to show that you can control the solution for long, long time is where you have to use a lot of analysis but once you have that once you have produced a space time which you control then there is a second part which is 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 the 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 this quantitative 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 discovered 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 that you might find more interesting so this is a result that I have was Ronjanski and on look right so right so it's actually what's interesting is that it's based on the first part of Christodulus result Christodulus result was very long and it was simplified and you know 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 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 Minkowskian say right so in other words it's the same situation is that except that instead of taking a pulse which is strong in all directions I take a pulse which is now strong in only one day of course I have to take sufficiently strong pulse and I form a trap surface later 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 doublin alpha variation in other words you foliate the spacetime by a family of light cons going this way and light cons going this way so that gives you a foliation which is represented here in this picture by these surfaces and these ones so and somehow this doublin alpha variation plays a very important role in the proof of the first result that you 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 the of your spacetime here so the trap surface would be right here so this is the 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 so the the surface is of the foliation of the doublin alpha variations at this one at these regular surfaces here and you have to find actually by an argument which is deformation argument you have to find this one 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 already get you can already get the trap surface so if you have two of course you think that it's a little bit easier this one here but obviously you have to focus it in this direction I mean you have to produce a lot of energy in this but of course you don't have a trap surface originally I mean as it is obvious because everything is is Minkowski and outside that region right yes does the trap surface as the maybe formalize as a cal no no no so this is trapped trap surface is not yet a black hole it detects a black hole 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 it takes in time for the formation of the trap surface so again if you have a pulse of size say delta to the one delta whatever so if it's delta as you know existence will sorry delta to the minus one excuse me this is delta to the I mean sorry 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 also time one this time one yeah right yeah so okay so anyway so this is formation of trap surface now there are many 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 so anyway if there are any questions I'll yes