 So let me begin with an overview of what I hope to cover. Unfortunately, I will probably have to erase this. But it can still be on the board for just a little while longer, maybe. So the plan for today is to give an overview of the cosmic censorship conjectures in general relativity. Can everybody see this, or should I write larger? It's okay? By the way, how many of you have heard of the cosmic censorship conjectures in general? Okay, so quite a few. Okay, very good. I will not assume that you will have, but in any case it's... ...repetition is the mother of learning. So after this overview, the next lecture, let's say, will be a discussion of spherically symmetric dynamical space times. So this is sort of a world which is very dear to my heart. And it's actually a world in which you can understand a lot of what's new and interesting in general relativity. And in particular, it's a world in which you can really understand what's going on concerning these conjectures. So in some sense my favorite way to really understand what the issues at stake are for these conjectures is really to understand this world. And another nice thing about this world is it's in a world where you can really make formal sense out of the notion of a Penrose diagram. So I'm going to add this here, and Penrose diagrams. So this is something that I'll use informally in today's lecture, and I know many of you have nodding familiarity with them. But in this context, we'll actually be able to sort of discuss these well-defined mathematical objects. And you can even state theorems just with pictures. So it's nice. Okay, so now I guess I'll actually have to rub out this nice outline starting. So the third, again, let's say, lecture, in some sense a continuation of this second topic. So in the world of spherical symmetry, in the simplest case, that's to say one of a self-gravitating scalar field, there's actually a complete proof of both cosmic censorship conjectures, the so-called weak and strong cosmic censorship conjectures, due to Sothulu. So I'm going to try to devote a whole lecture to at least explaining those statements and a little bit what goes into the proofs. So we'll see these acronyms a lot later, so I might as well initiate them already here. This is weak cosmic censorship and strong cosmic censorship for spherical symmetric self-gravitating. And this work really is sort of the place to start for anyone wanting to understand more generally these conjectures. This is what you have to understand. And well, the final lecture will be concerning sort of a result which is beyond spherical symmetry, in fact beyond any symmetry assumptions, and it actually sort of leads to a reformulation of strong cosmic censorship, or if you want sort of a disproof of the original version of strong cosmic censorship. So this is a recent work of myself and Jonathan Luke from Stanford University. And well, I'll just write the statement here and later it will be clear what is its relevance for strong cosmic censorship. The story is a bit complicated and it doesn't fit in a tweet or in this part of the board. So let me just write for now the C0 stability of the... So this is my plan. I think I'm giving four lectures and these are four topics. Although I have to warn you that so I'm from Greece and I'm not so good in planning ahead. And so it could be that the plan sort of disintegrates before our eyes. But in any case, this way you know what my plan was. Okay, so let me begin this overview. And by the way, feel free to interrupt me at any time. But also if you don't quite understand everything yet, that's fine. You shouldn't because the whole point in some sense of topic two is topic two will in particular explain formally in a certain world everything I said today. But in any case, nonetheless, do feel free to stop. So let's begin our overview cosmic censorship conjectures. Okay, what are these conjectures about? So these conjectures, the very big picture, whatever they're about, they're about the nature of singularities that form dynamically in general. At the end of the day, both of these conjectures you should think of them as being about the nature of singularities as they form dynamically. So I want to focus for a second on that last word dynamically because until you formulate these questions in terms of dynamics, you will not be able to rationally study them. So before you say anything about these conjectures, you have to understand the notion of dynamics in general. So let me sort of review that very briefly. Possibly this has already been discussed a bit last week and actually I know that Harvey Rayell, his lectures will be about the initial value problem in, well, not just in classical GR, but beyond classical GR. So in any case, this is something very important. Without dynamics, we could not talk rationally about it. So let me fully deserving of a bullet point. So dynamics. So what is dynamics? Well, let me first discuss the vacuum. So for the vacuum equations, Ricci curvature equals zero. We learned that there is some notion of initial data and the initial data is a three manifold which I've drawn here. So it's some three manifold with a Riemannian metric and with the second fundamental form or what will be the second fundamental form. And maybe you heard last week that these actually are not completely free. They have to satisfy these constraints and that's a nuisance in the theory various people study. So the fundamental theorem of dynamics, which is due of course to Yvon Chocabra from 1952, is that given sufficiently regular such initial data, you can solve these equations. So what does that mean? So there exists a Lorentzian four manifold that satisfies these equations and admits this three manifold as a hyper surface which is space like. And such that this is the induced first and this is the induced second fundamental form. Moreover, and this is actually important for being able to talk about dynamics. So the way I said what it means for a space time to solve this initial value problem, then solutions are severely non-unique. Okay, because you could always then take sort of a type of subset of this and this would also solve the initial value problem. Or maybe you could take another space time which is a bit bigger in one direction that also solves. So actually in 1969 Bob Garrosh and Yvon Chocabra, they were able to show that there exists a maximal Cauchy evolution. So there exists a maximal solution. So there exists a maximal Cauchy development of, okay. So this object is globally hyperbolic. So globally hyperbolic means that all inextendable causal curves in the space time, they intersect the initial sigma three once and only once, okay. And it is the largest globally hyperbolic solution of this initial value problem in the sense that any other solution embeds isometrically into this. So anyway, I can't emphasize more how from the point of view of language, this object is absolutely fundamental in rationally discussing all problems and dynamics in general relativity. Because you should think that any problem essentially in dynamics is a problem of connecting properties of initial data, okay, to properties of this object. Any question in general relativity essentially can be phrased in that way. So I can't resist saying a certain side story that actually the proof of this theorem by Bob Garrosh and Yvon Chocabra in 1969 actually appealed to Zorn's lemma. So if you know what that is, that's one of the equivalences of the axiom of choice. So that sort of already hint that it's non-trivial to construct this object. And the reason is that if you have two globally hyperbolic developments, then they're not sitting, they're not living on the same manifold. And it's not clear how to compare them a priori, which is bigger. And that sort of is essentially why one had to appeal to Zorn's lemma. For those of you who like the philosophy of physics, you don't have to identify yourselves. But for those of you, you might think that that's actually quite problematic. If indeed, you know, talking about dynamics in general relativity relies on the axiom of choice, you might be sort of worried. So actually a bright young graduate student in Cambridge a few years ago, Jan Zbierski, he succeeded in disornifying this proof. So actually this proof is now completely constructive. So I don't remember the year of that, so let me guess that it was 2015. So actually his paper on this is I think also the best reference to discuss sort of, you know, this proof from a modern point of view. Okay, so that's great. I fade lip service to dynamics, and we will certainly see in use this object later on. And we will see the significance of global hyperbolicity later on. So if you don't think you have a sense for what it means, don't worry. Sort of everything will become much clearer in lecture two. Okay, so that's the first thing we have to pay lip service to say anything. So we have a notion of dynamics. So the second thing that we have to remember, and again, we can only say these words because of dynamics, is that bad things happen to good data. Okay. Bad things happen. So, okay, so what do I mean? So this is data. Okay. So the first thing that, okay, you can know this in this framework is that there are space times where the data is impeccable. There's nothing wrong with the data. The data is complete. It's asymptotically flat. There's no singularities in the data. In fact, the data can be, you know, very nearly flat in some sense, very close to Euclidean space. And yet bad things happen. So happen always means the maximal crochet development of the data. The maximal crochet development of the data has, in quote, properties. Okay. So this, if you want, you already see this from the explicit solutions that we know in law. So Schwarzschild, something which I'll talk about later, Oppenheimer-Sneider, in which it's even more clear, in some sense, the, you might quibble that the data in Schwarzschild or Kerl, let me also add Kerl. Okay. It's complete, asymptotically flat, but it has a funny topology. So maybe you don't like that. So here in Oppenheimer-Sneider, which I'll show you very soon, you can't quibble about anything. So bad things happen to good data. And again, ah, means it's the biggest. It's the biggest. So it's the biggest solution with this property. And biggest because, again, you can't compare a priori. So biggest means any other one isometrically embeds to this one. So any other one. So any m tilde. Okay. G tilde, which is globally hyperbolic. Okay. And admits sigma t bar k. Okay. As sort of hyper-surface with induced dot as a Cauchy surface. Any other one isometrically embeds into m comma g. Okay. So that this diagram commutes. All right. So to get back here, so bad things happen to good data. And again, keep in mind, especially with the discussion of, let's say, Schwarzschild. Okay. You see, in the anti-Diluvian discussion, and I mean before the relevance of dynamics, sort of, was clear, then people saw Schwarzschild. They saw that it was singular. Actually, there was a load of confusion of what was singular, et cetera. They even lost over that. They saw that it was singular. And many people's reactions, including Einstein, was that, okay, that just means we should throw this solution out. And, okay, it's sort of a reasonable thing to do in theories which are not governed by dynamics. But the point is that you're not allowed to throw, you cannot sort of throw out the maximal Koshy development just because you don't like one of its properties. You can only throw out the initial data. That's to say, you know, judgment calls can only be made on the level of initial data. Okay. And the initial data is flawless. Okay. And it's not singular. So somehow, the understanding that dynamics is the important problem, it fundamentally sort of changed the point of view of these solutions. Okay. Anyway, all this will become clear later, but it's worth saying this sort of philosophy at the beginning because, you know, anyway. Okay. Bad things happen to good data, but the story gets worse. And the reason that the story gets worse is that, again, these here, so Kerr, of course, was discovered later than Schwarzschild and this Oppenheimer-Sneider that I'll talk about. So these solutions, of course, they were discovered rather early, this one already in actually in December 1915, this in 1939. We'll see in just a second. And okay, from the point of view of dynamics, they exhibit this principle. But of course, these are just some explicit solutions. You might hope that the reason that bad things happened was because these solutions are very symmetric. Okay. So even though, indeed, bad things happen to some very particular good data, at this point, you could still be hoping that for generic data, bad things don't happen. Okay. Now this was very much the hope in the early 1960s and the hope was completely destroyed in 1965 by Roger Penrose. Okay. So Penrose, what he showed in 1965 is the following. So these three solutions, they have various bad properties. They have, in this case, they have singularities, whatever. They have various types of bad properties. Okay. Or what were in any case considered to be bad properties. Okay. But one thing that they have in common is that they are geodesically incomplete. Okay. And what Penrose showed in 1965 is that geodesic incompleteness is stable to preservation. Okay. If something is true, and this something is actually true in all these examples, then geodesic incompleteness is stable. So to perturbation of what? To perturbation of initial data. Because again, to make sense of all these things, you have to always refer to dynamics. It's stable to perturbation of initial data. So again, to go back to this picture, what is the claim? So all these, okay, you can think of them as solutions of the problem of dynamics with some special initial data. Okay. So the result of Penrose says that if you wiggle ever so slightly the initial data, okay, and you solve the Einstein vacuum equations, or while in this case, more complicated system of Einstein-Matter equations that we'll see in just a second. So you consider the maximal Cauchy development of wiggled initial data. Okay. Then the wiggled initial data will still be geodesically incomplete. So this theorem is traditionally in the literature known as Penrose's singularity theorem. But it is time that that terminology be retired, and that it be called what it is, namely Penrose's incompleteness theorem. Okay. Because that's actually what it says. So, and it will be clear later on why, okay, this is just semantics, but why this is a much more useful semantics. Okay. So to recap where we are so far after these three bullet points, the fundamental problem is one of dynamics. Bad things happen in dynamics to good data. Those bad things, or some bad thing, namely incompleteness, is stable to perturbation. So it wasn't because the data was symmetric. It wasn't because the data was fine-tuned. Okay. These bad things, they actually happen sort of generically, or they happen sort of with non-zero quote probability. So maybe I should say it like that. Okay. So where does that leave us? Well, this is now where enter the cosmic censorship conjectures. The cosmic censorship conjectures, they enter to make the best out of a bad situation. Okay. So cosmic censorship, what is it all about at the end of the day? It's about making the best of a bad situation. And if you want, so let me make a quote to carry on from here, and we'll understand in stages this quote as we go along. So the statement basically is given the above situation. Okay. Classical GR is as good as it gets. So let me say it like that. But there's one little caveat, and that caveat I'll write it here, generically. Okay. So with probability one, let's say it like that for now. So that's what cosmic censorship is supposed to be about. And let me immediately sort of try to give a first... Okay. This is not even a formulation. This is what is... This is, okay, a quote connecting to both cosmic censorship conjectures. Because as we will see, there are two sort of cosmic censorship conjectures. And then as time goes on, we will put meaning to these quotes. So, okay. So maybe I'll move back here now. Say again. Yeah. So there was not... So that's a good point. But yeah. So the question was that if you take 1965 and you subtract 1969, you get the number which is negative. And so did he use close time like curves? I think this was the question. So the theorem was not originally formulated with respect to the maximal Cauchy development. Because of course that notion didn't exist. But Penrose's theorem, one of its great strengths is that it's actually a purely geometric fact. So it's not even about a solution to a PD. In fact, it's about a solution to a partial differential inequality. But somehow in order to interpret the theorem, it turns out that the natural object to interpret it too is the maximal Cauchy development. Because if you are not interpreting it with the maximal Cauchy development, then all this is saying is that, well, maybe you should extend further. So somehow it is really the sort of, indeed, you should really think of this as a corollary of Penrose's incompleteness theorem. But this is actually the corollary that has content for classical Gauss. It's a good point. Okay, so this is sort of the quote. This is what the cosmic censorship conjectures are supposed to be about. So let me try to make this slightly more specific, referring to both these sort of separate statements. So the first cosmic censorship conjecture, so this was actually the original cosmic censorship, and it's now called weak cosmic censorship to differentiate it from something else, which is called strong cosmic censorship. But if you look back at the literature, this is what used to be just called cosmic censorship. So what does this conjecture try to say? So again, I'll give you a quote, quotations for the quote. So this conjecture says the following, despite the fact that the maximal Cauchy development may be incomplete, as Penrose's theorem tells us, far away observers, whatever that means, and we'll make that precise later, far away observers live forever. So far away observers, which you should think of as sort of geodesics that are far away, some sense of that word, they are complete. So in particular, they never see anything that deserves to be called a singularity in finite time. So this is, if you want, the original form of the conjecture or the original informal form of the conjecture. And one thing that actually really somehow first came out from Stolulu's work in spherical symmetry, because this really is an impeccable model problem, is that this, one doesn't expect this is true without adding the caveat generically. So, yes, you have incompleteness, okay? But generically, there's this class of observers, the so-called far away observers, and they live forever, okay? And they don't, in particular, they don't see any singularities in finite time. So as we'll see, this is a property which is shared by all of these examples, okay? So if you want, it's motivated by what we see in the black hole spacetimes that we know and love. So that's weak cosmic censorship. And what is strong cosmic censorship about? So this is sort of harder to explain in a quote, and it will hopefully become much, much clearer when we start looking at examples and drawing Penrose diagrams. But nonetheless, I guess I should write something. So one comment I should immediately make is that these names are actually quite unfortunate, because this is not a weaker statement than what I'm going to write here. In fact, what I'm going to write here, the way I'll write it, it will look as if it has nothing to do with this. Of course, there are relations between these statements and there are reasons why, the historic reasons why they have these two names. But if it were possible to retire these names, then maybe we would. And I could imagine better names for both of these constructions. On the other hand, these are very cool names that have stuck, so I wouldn't, I'm not supporting retiring this. Here it's different, the reason is that in fact, there's nothing wrong with an incompleteness theorem, after all, Godot proved an incompleteness theorem. He's a great guy, and I don't think anyone thinks less of him, because it was an incompleteness theorem and not a singularity theorem. So I don't, I mean, I really think this, this amount should stick. Here we can, okay, I will keep these words, but we should, maybe it is good to sort of remember that they're not maybe so great. So what is this conjecture? What does it say? So it says the following, so generically, again, and I'll tell you a little bit about history in this context of the word generically. Generically the, well, maybe I'll say it like this. All incompleteness is due to singularity. Let me say it like this. So this actually goes right to the heart of this distinction, okay? So Penrose's theorem, commonly thought of as a singularity theorem, is actually just telling you that spacetime is geodesically incomplete. And there can be many reasons why geodesically, a spacetime is geodesically incomplete. One of them being that it sort of has an edge which is everywhere singularity. But that's not the only way that spacetimes can be geodesically incomplete. And somehow what strong cosmic censorship is saying is that generically, that is why spacetime is geodesically incomplete. It's incomplete because of singularity, because its edge is singular. So you might think that it'd be very, very strange at this point that this be thought of as something good. But I claim to you, and this will only become more clear for those of you who don't know the story, when we really understand the ramifications of this not being true. So this is good, okay? And in fact, secretly, this is the statement that general relativity uniquely predicts the classical fate of classical observes. So let me write it like this. So this secretly is somehow the statement that the classical general relativity in its regime is deterministic, okay? Anyway, why that is the case might not be clear yet, but it's hopefully will be as time goes on. So secretly if you want in a more, for those of you who are sort of more, ah, there's a question, yes. Well, so actually the, yeah, so the question was, why all the fuss about geodesic completes? So the answer is actually, you sort of, so as we shall see in some sense, the, okay, it is not, so geodesics are convenient because, okay, these are sort of unquestionably good objects. So, okay, they're a convenient test of various things. But certainly, we can talk about classical physics fields without ever mentioning geodesics, okay? And if you want the, sort of, the rigorous formulations of these statements, they don't sort of refer to geodesics manifest. So they actually find their proper formulation just as in the world of the Einstein equations coupled to a billion sort of fields. So everything is in the solution. To pick it, to say it another way, in classical physics, you don't, you know, you don't need things from outside the, you know, the universe in order to measure the universe. Everything is in the universe, okay? So in particular, in, in, in classical general relativity, okay? We study the vacuum equations, and more generally the vacuum equations coupled to a billion fields, okay? And everything is there. So the observers, they're also there. They're, you know, they are one of the fields, let's say, okay? And so, so actually the proper way to say all these questions are as properties of that system of PDE. So you, you fundamentally, you don't have this issue that, okay, maybe you have in quantum physics that, okay, you have to talk about what, what is an, an observation. In classical physics, it's all, it's all in the system of PDEs. So all the, the questions fundamentally are statements about the, the system of PDE. So when we appeal to observers in time like Geodesics, it's just some limit of that system. So it's just something to help us, okay? But the, the, the primal formulation of everything is in a pure physics of fields. Because that's, because this is classical physics. Well, I, I, by which theory? By, by, well, first of all, these, these are not theorems. These, these are conjectures. Well, again, if it can, but, you know, Minkowski space can also have sort of, you know, such curves. I mean, sort of, you know, the fact that such curves exist, it's just the fact of life, okay? And it has some, you know, it has some interpretation. I mean, that's not sort of, that, okay, that's not inconsistent or in contradiction with anything. Anyway, we'll, we'll, we'll see sort of a lot of these properties and a lot of these subtleties actually in, when I talk about actually, well, hopefully very soon. So, okay, let me, let me try to put a little more meat into this by giving you what really you should think about as the sort of main model for gravitational collapse, at least historically, that sort of, if you want to shed light on, you know, what, what, what these statements are really saying. And this is Oppenheimer's night, okay? So, let me introduce to you Oppenheimer Snyder. If I get something wrong, there's at least one member of the audience who can help me out. So, this was first written down in a, in a paper from 1939. It was not presented in any sense in the way I'm going to present it. Actually, this solution in some sense is already implicit in the work of Lemaitre from, from a few years earlier and was from that work that it sort of eventually, yes? Ooh, sorry. Okay, okay. Yeah, I mean, in some sense this, it is already in, in, in Lemaitre, the, so Lemaitre was the first person to understand that the, the event horizon of, of Schwarzschild was not singular. And in fact, the reason he was thinking about that was sort of, because he was thinking about dynamical space times, dynamical space times in cosmology, but there is this connection with, with gravitational collapse and he sort of, so he already describes sort of this solution. He doesn't sort of write, write it down explicitly. So somehow there is, that's somehow where, where it's actually originating from. Anyway, so, so what is this a solution of first of all? So this is not a vacuum. This is a solution of the Einstein equations coupled to, well you can think of it as a perfect fluid where there's no pressure, however. So we often call this a dust, okay? So that's, I guess the energy momentum tensor is equal to this, where, where rho is pressure. And moreover, it is a, it is a solution of the system which evolves from very special initial data, initial data that looks as follows. So, so the initial data is, is, is spherically symmetric first of all. So if you want, I can think that I have a, a coordinate R, okay? That goes from zero. This is what I think about as a radial coordinate R. Okay? Which goes from zero to infinity. So this is time equals zero. Okay? And so the initial data is spherically symmetric but moreover it's actually homogeneous in some finite ball. So I have some R equals capital R, okay? And in this ball, I have rho, the density, constant and outside this ball I have rho equals zero. So this is the initial data. So if you want this, I can think of this as the initial state of a star. So this is the data, okay? So how does it evolve, okay? So to draw the space time that evolves on what I'm actually drawing is a Penrose diagram. Okay, of course these type of drawings did not exist at the time, but in any case. So, so I'll just use this now as a, as a, as a picture. And those of you who already, you know, have nodding acquaintance can nod. And while those of you who don't, well, this will be motivation for understanding tomorrow's lecture where in the context of spherical symmetry, I'll tell you exactly what these objects mean. So, so the evolution, it turns out that it looks like this. So the, if you want the surface of the star, okay? Follows a path like this in, in, in space time, okay? And eventually reaches a singularity where r equals zero. And of course outside of the support of the star by Birkhoff's theorem, okay? See, the outside of the support of the star, the solution is vacuum, but spherical symmetric. So it has to be Schwarzschild. And many of you have already seen the Penrose diagram of Schwarzschild, okay? So this is the boundary at infinity known as null infinity. Already you should be thinking this is related to far away observers in the radiation zone, okay? The past of future null infinity is bounded by a horizon known as the event horizon. So the, the compliment, right? Of the past of future null infinity is this region here. This is exactly the black hole region. And, and so we have, we have this picture, okay? So, so you should think that by the way maybe I'll label this that r equals infinity in some sense on this boundary, okay? So we'll understand exactly how and why whatever next lecture. All right, so this, this is the, the, the, the Penrose representation of, of Oppenheimer-Snoijder. And let me try to interpret now this picture, okay? In the terms that I talked about here, okay? So you'll understand sort of, so the claim is that this picture exhibits both this property here, okay? And this property here, okay? So Oppenheimer-Snoijder if you want is the model par excellence for sort of what we want to be true generically in evolution in general activity, okay? So let's see why, okay? So, so of course, again, just to make a sort of historic point. So very often when one talks about weak cosmic censorship, the way one would think about it in, again in slightly anti-Diluvian days was as follows. So this boundary here is a singularity, okay? We'll get back to that in a second. And this singularity is cloaked by a horizon. This is horizon, okay? Remember the horizon is just the past of, the boundary of the past of future null infinity. This is future null infinity. This is its past. The horizon is the boundary of the past, okay? And the singularity is sort of cloaked by the horizon in the sense that observers who are outside the horizon, okay? If they remain outside the horizon, they don't encounter the singular. But actually, it's sort of, and this insight in some sense goes back to Bob Garrosh and Gary Horowitz and Gary Horowitz's thesis is that you should, you know, to capture the property that we're interested in, you should actually focus on future null infinity itself. Because actually, you know, the statement after all that observers who remain outside the event horizon don't encounter a singularity is meaningless if all observers eventually cross the event horizon. So what you really want to say is that there is a class of observers, okay, who do not encounter the singularity. That's really what you want to say, okay? So, all right, one thing that you might want to, one way you might want to say it is talk about find certain classes of time like geodesics, et cetera, but then you run into this problem which is partly related to your question of, you know, which geodesics and why, whatever. So actually, it turned out that the cleanest way of thinking about this is thinking of the completeness as a statement that lives at null infinity, okay? So, I claim that you can make sense to the following words that future null infinity is complete, okay? So, this is a funny statement because, okay, whatever future null infinity is, it looks like it is asymptotically a null hyper surface. And of course you can't talk about the length of null curves because the length is zero no matter how long they are, okay? But nonetheless, I claim and we'll discuss it next time, that you can actually make sense of these words, okay? And, well, a very pedestrian way to understand it for the experts is that if you normalize in a geometric way a retarded time coordinate, okay, along null infinity, then the range of this coordinate will be from minus infinity to plus infinity, okay? But you have to normalize it geometrically, that's the point, okay? So, you should think of it as meaning that, you know, very far away observers, they live forever in both positive time here and negative time there. Another way of thinking about it, sort of related to the meaning of null infinity, so you should think that null infinity is actually where gravitational wave experiments take place. So, another way of thinking of this is that those experiments can go on for as long as they're funded, okay? All right, so this, so if you want, this is what weak cosmic censorship is telling you that for now generic initial data for the Einstein vacuum equations and for a good class of Einstein mother systems and which class we'll have to discuss later, future null infinity is something that one should be able to define and it should be complete, okay? So that's this statement, okay? So what about strong cosmic censorship? So, so first of all, although this is complete, this day's time is geodesically incomplete. In fact, it turns out that there is, ah, there is colored chalk. Turns out that all, let's say time like geodesics which cross the event horizon, okay? They will be incomplete, okay? And in fact, in finite time, they are going to reach this boundary where r equals zero. So this is geodesically incomplete and in fact, one way to show that this is geodesically incomplete is of course to do this computation, which is not so difficult, it's probably the easiest, but another way is to notice that any sphere here, so if you think these are r equals, you know, these are sort of spheres of some r value, okay, so this is a sphere in Schwarzschild actually, because this is outside the support of the mother. And well, if you know your Schwarzschild metric, then you'll realize that this sphere is something called a trapped surface and that's exactly what you need to apply panerosis in completeness theorem, okay? So you sort of, from the fact that this is a trapped surface, you know, and the fact that this is a globally hyperbolic sort of evolution, okay? It has to be because I told you that this was the maximal Cauchy development, so I can apply panerosis theorem, this must be incomplete. So somehow, panerosis theorem already told you that this was it and I remarked that if you want for a reason. Okay, so let me just, maybe, since I bothered to say it, this is a trapped surface, trapped surface, so panerosis already tells you, okay? So, okay, but anyway, so this is an incomplete geodesic, okay? And what happens, as I just told you, is that all incomplete geodesics, so I just drew a bunch of them, they end up at r equals zero. Now, what's written in all of the textbooks is that r equals zero is a curvature singularity, that's what's often said. So the curvature blows up. Now, for instance, just so that that's unambiguous, you can look at the Kretschman scalar, which is a quadratic expression in the Riemann curvature tensor, which is scalar, okay? And that blows up as r approaches zero, okay? So that's great, of course, but the fact that the curvature blows up is not too justice to the nature of this singularity, because in reality, the singularity in, so by the way, this I should mention is true, both of these geodesics, okay, who sort of approach r equals zero in the Schwarzschild region, and these geodesics who approach r equals zero, you know, always inside the star, okay? So the curvature blows up, but this does not begin to do justice to how singular this boundary is. So in reality, not only does the curvature blow up, but observers torn apart these observers by infinite tidal deformations. So what does this mean? This means that if I'm a, so let me say it first in this language of observer, so if I'm an observer, so I'm following one of these geodesics, but I also have hands and legs, okay? And if you want, you can think of the hands and legs as Yacoby fields, you know, your differential geometry, because the ends of my hands and of my legs, they also follow geodesics, okay, which are infinitesimally close to sort of where my heart is, okay? So if you look at the hands and the legs, then depending on their orientation, they are infinitely stretched or infinitely compressed as I approach r equals zero. So I'm killed as an observer. This is very different from just the statement that curvature blows up. That's an instantaneous force, okay? That's just an instantaneous force, which is infinite. So if I just know this, okay, then no one says that I am killed as a classical observer, okay? But this statement here tells me that I'm killed. So, but again, in the spirit of that question, can I say these two statements without referring manifestly to observers just in the language of the field, okay? So can I somehow capture these statements just in the language? Yes, question? Yes, of course, yeah, and we will see this later on. Okay, so actually this, it's sort of funny. So I think the first person to claim this statement was Roger Penrose, but he actually doesn't show it and the first calculations sort of show this in some sense are actually in Misner Thornin Wheeler. So there's a section of Misner Thornin and Wheeler about this. Okay, so how can you say this without referring to observers just in terms of the metric? Well, the statement that the curvature blows up as observers approach R equals zero, that's very easy to say in terms of the metric. This is just the statement that the metric, though the curvature blows up, so this is just saying that, let's say it like this, the Lorentzian manifold, M comma G, okay, is inextendable as a manifold with C2. So C2 means twice continuously differential, okay? So that's very easy to infer because of course, suppose you could extend, so here's a little bit of an extension, okay? Then very elementary geometry says there has to be some time like geodesic that passes into this extension, okay? But the curvature over here was blowing up as you approach there, okay? And that contradicts that this is a C2 metric, so locally nothing should go wrong. Okay, so that's what curvature blow up means. It means that the metric is inextendable as a C2 metric. So it turns out that this statement with observers being torn apart, well it's sort of difficult to directly translate because then you sort of get into this issue of which observers, maybe there are some special observers that are oriented in some way, such that. But the claim is that the correct purely field theoretic interpretation of this, okay, is the statement that the metric M4G is inextendable as a manifold with continuous metric, okay? So if you want, this is a, so from the point of view of differentiability, this statement lives at the same level as this one. I claim that these are not equivalent statements but in the spirit of what I told you, this is sort of the purely field theoretic description that the singularity R equals zero of Schwarzschild or Oppenheimer-Sneider is extremely strong, okay? Okay, so it's sort of of the strength that destroys observers, destroys. All right, so actually I should maybe even put an asterisk on this statement. So this statement is true about Schwarzschild but it was only very recently shown again by Zbierski. Okay, let me say, I don't know, 2018. And I don't know it to have been shown actually for the interior of the star. So any bright young student in the audience might want to read Zbierski's paper and see if you can actually extend this C zero, so continuous sometimes we C zero, C zero and extendability to the interior of the star. Okay, because this is really the statement that sort of, the singularity is strong. Okay, so in particular, let me try to now interpret sort of this statement here, okay? So that we see in the, let's say, the first part of the statement, okay, that indeed all incompleteness, all incomplete due to these six, okay, they go to a singularity. All right, that sort of this informal, version of the statement, okay? But the more correct version of the statement should be that there should be some obstruction to extending the spacetime. So spacetime may be incomplete, okay? But there should be some obstruction to extending it, okay? And what we see is that there is first of all the naive statement that indeed the spacetime is an extensible as a manifold with C two metric, okay? But there's actually a much, much stronger statement secretly there, okay? Which is that the spacetime is inextensible as a manifold with continuous metric. So let me now spell it out for you and try to interpret what I said here. So let me first say it in the language of observers and then I'll say it in the pure field theoretic language. So why is this telling us that GR uniquely predicts the classical fate of classical observers and why am I focusing so much on this destruction of observers as opposed to infinite curvature? So what's going on in the spacetime? Well, there are two classes of observers, right? They're the observers who don't cross the event horizon. Okay? And it turns out that these observers they all live forever. This is related of course to this completeness that I claim is the way we in the modern world the formula we cosmic censorship. So these observers live forever and of course the classical spacetime is telling me everything about this observer, okay? And then we have these yellow observers who only live for finite time and reach R equals zero. And what classical theory is telling me is that these observers as classical objects are actually torn apart, okay? So even if you believe, and certainly most people do that sort of maybe very, very close to the end we have to go beyond classical physics to understand sort of what happens in physics as far as this observer is concerned they have already been torn apart. They no longer exist as a classical object they're now some quantum soup and okay you can then use your favorite quantum theory of I don't know what to speculate what happens to that quantum soup. But as far as the classical physics is concerned classical physics has given up this classical observer to whatever is beyond classical physics as a non-classical object. So classical physics is in this sense self-contained. It tells you what happens to all classical objects, okay? So from the PDE point of view and let me just end with this comment, okay? So if I don't want to talk about observers, all right? Then I should go back to sort of just the initial value problem, okay? For my spacetime. So again, given initial data there is a unique object this maximal Cauchy development, M comma G. But the M comma G in what sense is it maximal? It is maximal as a globally hyperbolic Cauchy development. So no one says that that cannot be extended further just not as a globally hyperbolic spacetime. It is only maximal as a globally hyperbolic Cauchy development. We'll see exactly, I'm slightly behind so we'll see at the beginning of next time exactly in what way this can fail for instance, okay? So what we see here in this property, okay? Is that this object, the maximal Cauchy development is not just maximal as a Cauchy development it is maximal in any reasonable sense of the word. There is no possible way to extend this, okay? So in the sort of pure PD language again this is telling you that this unique object unique just by fiat is really unique in any sense it is the unique sort of classical object that we can associate to initial data, okay? So it is in this sense that this property represents determinism in general ultimately. Okay, so let me stop because I'm already two minutes over time and we'll sort of maybe in the beginning of the next lecture I'll explain so I'll maybe draw this picture again and I'll explain the many ways in which these nice properties could have failed to happen, okay? And then we'll start in earnest discussing spherically symmetric spacetimes where we'll be able to make formal sense of this picture and sort of really sort of show what you can prove, okay? Thanks.