 You can all hear me. So let me review where we got at the end of last time. So for spherically symmetric Einstein mother systems, given spherically symmetric asymptotically flat initial data, so remember last time we talked about the case where the initial data had either one or two asymptotically flat ends. Today I will only talk about the one-ended case because it is the physical case. I will talk about this case. So Harvey at the end of his lecture talked about the maximal Cauchy development of general sort of initial data for the Einstein equations, more generally Einstein mother equations. So what we said was the following, but the maximal Cauchy development of spherically symmetric initial data is itself spherically symmetric, and moreover can be covered by a system of global double null coordinates. So coordinates u comma v, such that the metric takes this form here. Moreover, you can, without loss of generality, for the reasons that we said last time, you can assume that the range of the coordinates u and v is bounded. So you can think of the pair of coordinates as defining a bounded map of m into two dimensional Mankowski space. And it's precisely the image of such a map that we call Penrose diagram. So because by definition, the maximal Cauchy development is globally hyperbolic, admitting the initial data as a Cauchy hyper surface, then just that fact, global hyperbolicity upstairs, it translates into something downstairs about p. And in the one-ended case, what it translates to is the following. So there's this boundary sigma, which was just the initial data. And there's this boundary gamma, which corresponds to the fixed points of the SO3 action. And there has to be such a boundary since I've restricted to the one-ended case. So this function r is actually zero on that boundary. And it is strictly positive everywhere else in p. And p itself, as a subset of r1 plus 1, is not globally hyperbolic. But it has the property that any point in p here, if you follow back null curve in this direction, you'll hit either gamma or sigma. Whereas if you follow null curve in this direction, you'll hit sigma. So that's the analog of global hyperbolicity at the level of this quotient. And from that, it follows immediately that you can decompose the boundary of p. And now when I say the boundary, I mean the boundary except for this, the new boundary that you get because this, you're looking at it as a subset of r1 plus 1, you can decompose the boundary. And what does the boundary look like? It looks like it potentially has a null segment coming in from this point. I say potentially because this could be empty. It potentially has a null segment coming in from a future endpoint of this gamma. Again, could be empty. And then the rest of the boundary is generated by things I want to call first singularities. So these are points such that they are entirely preceded by a characteristic rectangle which is in p, except for the endpoint. So the whole rest of the boundary could just be first singularities, so that's exactly the case in Oppenheimer-Sneider. But alternatively, from any given first singularity, you can also have these null pieces. So I encourage you to think about why this is true. This is really a very simple exercise in the geometry of globally hyperbolic subsets of two-dimensional Minkowski space. So that is the claim. So what I want to do now, so far everything has really been basically trivial as far as what I said about p. So I want us to start putting in more meat into this. So now I want to start using the equations. But actually, we're not really going to be using the equations yet because we don't yet have equations. What do I mean by that? So for now, I'm really going to be in this world. So these are the Einstein equations for general energy momentum tensor. And of course, when you're in this world, you can think of the right-hand side as a definition. If I haven't yet told you what matter this is and what equations the matter satisfies, but nonetheless, let's first just do this to introduce some notation. So if you write the Einstein equations for general energy momentum tensor in this double null coordinate system, then the energy momentum tensor decomposes like this. And I claim to you that the Einstein equations, all the information the Einstein equations, is in these four equations. So already, you'll notice that this is a wave equation for R on, actually, you can think of it as a wave equation on that as a subset of Minkowski space. This is a wave equation for log omega. And these equations actually have a name in a much more general setting. These are the so-called Reichadoury equations for null hypersurfaces. So if you know more generally, this is a constant u hypersurface, d by du is this way. So this is a constant u hypersurface. And this is a constant v hypersurface. u equals constant, v equals constant. And secretly, these two equations are the Reichadoury equation. So this is only relevant if you know what that means in this null hypersurface, respectively. So we'll use these very, very shortly. So of course, without, if I don't introduce any other assumptions, then I'm not going to get anything. So let me now introduce the first assumption. And that is that our mother model, whatever it is, satisfies the null energy condition. So that's to say that these components of the energy momentum tensor are non-negative. Of course, again, since I haven't yet discussed any particular mother, you can think of this as just an assumption on the Riemann curvature. It's just the assumption that the Riemann curvature satisfies, the Riemann Riemann curvature satisfies R u u is greater than or equal to 0. R v v is greater than or equal to 0. So sometimes this is called the null curvature condition. So why is this immediately relevant? Well, you see that this immediately introduces a certain monotonicity in this and this equation, which may be familiar to people from the proof of Penrose's incompleteness theorem, because that's exactly what comes in there. So under this assumption, we immediately have that these terms have a sign. So I'm going to introduce another assumption. Actually, the relevance of this assumption was first really highlighted by Christodulu, which is what I'll call the no anti-trapped surfaces assumption. So my other assumption that I want to make, maybe I'll write it here, is that du of R is less than 0 on sigma. So remember, d by du is this direction. Sigma is initial data. And you should think that I am free to impose assumptions on initial data. That's sort of the only place where I'm allowed to impose assumptions. So in particular, this tells you that no, so if you think of these points of sigma as spheres in the initial data, they cannot be what's known as anti-trapped or past-trapped surfaces. So why is this relevant? Well, we can see immediately. So let me give you a little lemma, which is sort of trivial, as you'll see. So again, let this be any constant u curve. And so I'm going to just let this be any constant v curve in p. So here is my claim. So let's look at this one first. dv of R is less than equal to 0. Some point here at p, then it's less than equal to 0 at q. Any point to the future of p along the constant u curve. And similarly here, so if du of R is less than equal to 0, let's say here, then du of R is less than equal to 0 here. And similarly, if I have strict inequality here, I have strict inequality here. And similarly, if I have strict inequality here, I have strict inequality here. So the proof is trivial, but you just have to know this one thing. So remember, double null coordinates, as we've said many times, are non-unique. Because you can always rescale u, and you can rescale v by arbitrary functions of the old u and the old v. As long as those functions are, let's say, monotonically increasing, in the sense that the derivative is strictly positive. So in particular, a nice exercise. Given any, let's say, this is a constant u curve, you can rescale the v coordinate here so that Omerra is identically 1 here. And it's clear that you can do this, if you think about it geometrically, because actually that's just saying I'm going to parametrize this affinely. I'll keep my u coordinate as it was, and I'll reparametrize this affinely. So well, if you change the v coordinate like this, you never change the sign of dvr. And the reason that you never change the sign of dvr is that, again, you're always rescaling by something which is monotonically increasing. So these rescalings preserve the sign. And that should be clear because, of course, the sign of dvr and dur is something geometric. It's only depending on the direction, not on how you're scaling the coordinates. All right, so you go to this coordinate where Omerra equals 1, and well, now this equation is just saying that d squared u, I'm sorry, well, if I'm doing this one, I should look at this one. This equation here is telling you that dv squared r is less than or equal to 0, from which both these relations hold with respect to that v coordinate. OK, but again, now you go back to your favorite v coordinate that you had because the sign of dvr is independent of the parametrization. So similarly here. So let me give you two corollaries of this. So the first corollary, and this is sort of the motivation for this. So since we assume du of r is less than 0 initially, and since every point in the spacetime is accessible to initial data by going backwards in this direction, and this is a constant v curve, it follows from this relation with a strict inequality that du of r is strictly less than 0 in all of p. So that's interesting. You can almost, it's tempting to, this is sort of some. What this is telling you is that the notion of anti-trapped surface is non-evolutionary. That's to say, if you do not initially have an anti-trapped surface, you cannot form one in evolution. And this is sort of different from the notion of trapped surface, which do form in evolution. Anyway, so this will be actually quite convenient. There's another corollary, which is that we can now define something to be called null infinity, and we can say a little thing about it. So corollary 2. So I'm going to, well, OK, let me, I mean, there are various, yeah, so let me just say it like this. So consider, sort of tedious to write this in words, but maybe I will just once. Consider the subset of the boundary p in this ambient one plus one dimensional Minkowski space. So this, remember, this was this boundary. So the boundary is this. With the property, such that, so consider the subset of the boundary with a property that, so I'm looking at all x, so of all x in the boundary. So here's x, such that x is the endpoint of an outgoing null curve, so a constant u null curve, such that x, the endpoint of null curve, so I'll call these out. So since I'm in the one-ended case, it's sort of natural, I call these outgoing. So of an outgoing null curve, such that the curve is completely contained, first of all, in p, except, of course, for its endpoint, which by definition isn't, and such that r along this curve goes to infinity. So what I want to do is I want to define a piece of the subset, a piece of the boundary of p, and I want to identify that with null infinity. So let me already give it this name. So let me consider this subset, which I'm going to denote like this, of all x with this property that r goes to infinity as you approach x along a null curve, which is completely contained in p. So what can I tell you already about this subset? So actually, if you want, you might say, OK, what does it mean for r to go to infinity? What do I actually mean by that? I'm willing to even make it sort of weaker. Let me even just say the supremum of r is equal to infinity, because a priori maybe r has very weird behavior, oscillates, et cetera. So let's immediately make some remarks. So first of all, what I just changed is basically irrelevant, because in reality, by what we showed, by this relation here, if r is to go to infinity in any way, so if the supremum of r is going to be infinity on this null curve, then r has to be monotonically increasing. Because once the v derivative of r becomes 0 or negative, then it will stay negative, and r, its supremum, will not. So in reality, I could have said this in the first place. Doesn't matter. If the supremum is infinity, r will actually increase monotonically. Great. But moreover, so here's the claim. i plus is a subset of the null curve coming from here, from this point. So i plus is actually a subset of this. But moreover, I claim it's a connected interval. So i plus, in fact, i plus is a connected interval, which may or may not, well, first of all, it may be empty. No one says that it's not empty. And it may or may not include its endpoint. But it's a connected interval. OK, so let's see. So this is, as we'll see, it's, again, it's a trivial consequence of the monotonicity in Rai Shaduri. So first of all, why can this point x not be on i plus? It's clear. So let's suppose this point x was on i plus and draw these null curves. We know from this corollary that du of r is less than 0. So that's saying that r is always decreasing in this direction. So the supremum of r everywhere here is always less to the supremum of r here. But this is contained in some compact subset of initial data, so whatever that is, it's fine. So immediately, this no-antitrapped surfaces condition tells you that null infinity, as I'm defining it, can only be a subset of the part of the boundary, which is coming from here, if there is such a part. So now, why can it not look like some funny subset? So what I'm going to show you is, so if x, what's x? There's x. So if x is in i plus, then so is all of this. So again, this follows for the same reason. If x is in i plus, then what does that tell me? By definition, that tells me that this null segment is in p. And r goes to infinity along here. So of course, here was sigma. And r went to infinity along sigma, which is great. Now suppose y over here, suppose y, label here, suppose y was not in i plus. So what does it mean to not be in i plus? It means that, again, if I draw this, this all now has to be in the spacetime because of the geometry. So it's in the spacetime. r does not go to infinity. But again, the only way that r cannot go to infinity is that the supremum of r is bounded. So r is less than equal to some capital R everywhere there. So well, I told you that r goes to infinity as you go this direction. So go very far out, such that over here, let's say, r is bigger than capital R. And now follow this. Well, r is bigger than capital R. r is less than equal to capital R here. Then what does that tell you? That tells you that somewhere in between, well, OK. I don't even claim. I don't even need that. Let's just start here. Why say superfluous things? r is less than equal to capital R here. r supposedly is going to infinity here. So I look at this null curve. So d ur has to be positive somewhere. d ur has to be positive. I can choose a point here, such that r is bigger than capital R. If r goes to infinity as I go here, I can choose a point close enough to x, such that r is bigger than capital R. So I just look at this segment. Here, r is less than equal to capital R. Here, r is bigger than capital R. So there must have been some point here where d ur is positive. d ur is negative. So that's sort of nice. So that means that we can define this piece of boundary, which I'll call null infinity. Whatever it is, it's a connected interval with this as an endpoint if it's not empty. And it has the property that r goes to infinity as you approach any point on null infinity. And those are the only points in the boundary with that property. So by labeling a piece of the boundary, future null infinity, I am telling you that r does go to infinity as I go there. r does not go to infinity as I go there. So this is very nice, because in this world of spherical symmetry, we can define a notion of null infinity only using the monotonicity properties. So there are no precise decay rates, sort of compactification, et cetera, nothing. Everything is done just with monotonicity. So let me introduce an assumption. So my assumption is that this is non-empty. So strictly speaking, I should not be in the business of introducing assumptions, which are not assumptions on initial data. So in practice, the claim is that you can retrieve this assumption from assumptions on initial data. And of course, there's a trivial way to retrieve this assumption, and that is to consider initial data such that the matter has compact support. Because if the matter has compact support in spherical symmetry, then outside, you are schwarzschild. Maybe more generally, if you have charge, you want to say that the sort of the matter, except for the Maxwell part, has compact support. And then outside, you'll be rice or Nordstrom. And well, null infinity is non-empty in those cases, and this property is inherited. So this is very nice, because in particular, we have defined some notion of future null infinity. Let me write this on the board. So this is my future null infinity. So now, for instance, you can immediately define a black hole region. So black hole, if you want, is just the complement of the past of future null infinity, where this is taken in the sense of the ambient geometry of 1 plus 1 dimensional L'Kofsky space. So in particular, you can already define for you what it means for the spacetime to have a black hole. So if this is the picture, everything here is in a black hole. OK. But remember that weak cosmic censorship, as we've defined it, is not the statement that singularities are hidden in black holes. It is the statement that future null infinity is complete. So let me tell you what that means, because we can immediately sort of define that. So in actually this, it's a much more general definition, but let me not talk about it in generality. Let me just sort of specialize it to this setting of spherical symmetry where null infinity is meant in the sense that I've defined. OK. So suppose here is your P, OK, and here is null infinity. I'm just assuming that it's not non-empty. So what does it mean for null infinity to be complete? So of course, if this is non-empty, then, OK, there is at least one null. There's a lot of them because of the connectivity statement. So here, there is in particular one outgoing null curve that goes to null infinity, along which, of course, R goes to infinity. OK. So the statement is done the following, that I parallel translate an ingoing null curve. OK. And again, this is canonical, it's sort of an ingoing radial null curve, OK, along this. And I look at now the ingoing null curves. So these should be straight lines in this picture. Generated by this factor, OK, so these are, of course, just the constant V curves, OK. For as long as they remain in the past of future null infinity. OK, so suppose future null infinity ends here. And remember, with this definition, future null infinity a priori could include its future endpoint, or it might not, OK. The way I've defined it, it's not clear. So consider these curves. And consider the affine length of these curves as measured by this vector. So remember, see, I want to measure the length of null curves. But of course, null curves have zero length no matter how long they are. Of course, you can talk about their affine length. But to normalize their affine length, you have to choose sort of the affine parameterization. So I'm using this vector to choose the affine parameterization. So if sort of this affine length goes to infinity, OK, then I will say that null infinity is future complete. And if this affine length goes to infinity, I will say that null infinity is past complete. So if affine length, this whole affine length, I mean, in both this direction and this direction diverges, OK, as this point goes here, we say, OK. So this is the definition of what it means for future null infinity. So I claim that if you think a little bit about null coordinates, OK, let's call this curve, I don't know, u equals u0, So this affine length, the way I've defined it, I claim to you, is just the integral du of omega squared u comma v over omega squared u0 comma v, OK, where the integral is from u0 to whatever. Let's say this is u equals u1, and this is u equals u minus 1. So du plus or minus 1. So it's explicit if you want. So that's, so if this goes to infinity as v goes to, so let this be v equals v0, then I declare future null infinity to be complete, OK. So in practice, the past completeness, OK, sort of follows easily from assumptions on initial data. So in particular, if indeed my initial data, the matter, is of compact support, OK, then I claim the past completeness of null infinity follows from the past completeness of the null infinity of Schwarzschild, OK. And if you think about it, that just follows from the fact this is a good exercise, that in sort of bondy u coordinates, OK, so bondy u coordinates is trying to normalize this so that it be 1, OK. So if you normalize this so that it be 1 at v equals v0, OK, then the claim is that the coordinate u takes the range minus infinity to infinity, OK. So the fact in this direction it goes to minus infinity is the past completeness, OK. So great, so that's, OK, we've made a little bit of progress. So now, in principle, we have a completely well-defined formulation of weak cosmic censorship that we can entertain its truth or falseness, namely for our favorite Einstein-Matter system that now we have to sort of consider for generic initial data, future null infinity, as I've defined it, be complete, OK. So everything I'm talking about is for spherical symmetry, everything. Now this definition, actually, if you're willing to drop the requirement that these stay in the past of future null infinity, you can make it without explicit reference to future null infinity itself. That's to say, you can define the notion of having a complete future null infinity without having a future null infinity. And actually, this definition has been made by Christodulu. And so it's a way of actually formulating weak cosmic censorship very generally without depending on the sort of, in particular, the old Penrose constructions of infinity that, unfortunately, don't hold generic. OK, so very good. So now we can start section three of the lecture series, namely, consider Christodulu's proof of weak and strong cosmic censorship for Einstein's scalar field restricted always the spherical symmetry. So let me keep those equations on the board. In fact, let me throw in also the space-like singularity conjecture, some version thereof, since we talked about it. OK, for a spherically symmetric Einstein scalar field system. OK, so let me write down what that system is. So this is the Einstein equations. Well, I've already written them down here. Where the energy momentum tensor is that of a massless, minimally coupled scalar field. So this is the energy momentum tensor. And if you want from the divergence of this, it follows that the phi itself satisfies the good old wave equation on the metric background g. So the system is star with this as energy momentum tensor. And you can even think of this as a consequence of that. So what do these equations look like in double null coordinates? Well, you can easily compute that the component Tuv actually vanishes. So bye-bye these terms. And this expression also vanishes, so goodbye to this. And now it turns out that Tuu is nothing but Tuv squared. And Tuv is nothing but dVv squared. So OK, so these are sort of the Einstein equation. And if you want, you can also write down explicitly the wave equation. Maybe I won't do that unless I need it. OK? Turns out that what's actually more useful than writing down the wave equation is extremely useful for general spherically symmetric spacetime. So maybe I'll still write it for general energy momentum tensor and then specialize to scalar field. So you can define something, a quantity called m. OK, so this is a function of u and v, which is simply the following. This, so it's 1 minus the Lorentzian gradient squared of r. But lest you think that that's positive, I'll write it out explicitly. So this is the Lorentzian gradient of r squared, this expression here. So this was first written down, well, in some other coordinates a long time ago. And in that context, it maybe should be called the Misner-Sharp mass. But this actually coincides with the Hawking mass of the sphere u comma v upstairs. So I'll refer to this as the Hawking mass. This is the Hawking mass that you may have heard of. So why is the Hawking mass great? Because it satisfies the following evolution equations with respect to u and v. And again, this is a general, I'll write the general case here. Get the factors right. Maybe I won't. OK, so in general, it looks like this. OK, so this is sort of dual. And of course, specialize to the Einstein scalar field system. Of course, these components don't exist. So we just get whatever. Let me write it like this. And in this case, I'm going to write it like this. OK, so why have I chosen to write it like this? Well, so OK, maybe I should make a note. So I told you that TUU and TVV are what I wrote here. So in particular, these are greater than or equal to 0. So this model does satisfy the null energy condition. In fact, it satisfies the dominant energy condition, which is just that these should be greater than or equal to 0. And so should TVV, which is 0 actually. So these are greater than or equal to 0. So in particular, everything that I've said so far, which remember, was using that assumption, it applies to this model model. So in particular, so given that I'm always assuming du of r less than 0 on data, I have that this quantity is always positive. So this is telling me that this is always greater than or equal to 0, DV of m. On the other hand, I claim to you that m is always 0 on gamma. OK, so I'll leave it as a little exercise for you because it's actually very easy. So it's very easy to show that m has to be greater than or equal to 0 everywhere on sigma. So exercise. And now, since every point of p is preceded, I mean, if I go in this direction, I either hit sigma or gamma. And this is a constant u curve. So this is d by dv. This tells me that this hooking mass is globally non-negative. But this will be useful. All right, so that's the first remark. Another second remark that I want to make, which sort of is a big story behind this, is the following, proposition. So this Einstein scalar field model is what I'll call tame. So this is actually a terminology invented by a former student of mine, Jonathan Komemi. So let me write it like this. So Einstein's scalar field is tame. And what this means is the following. So this is just a word to describe the phenomenon that I will tell you. But it's useful to have a word because this phenomenon is extremely general for spherically symmetric Einstein and other systems. So the statement is the following. All first singularities not arising from the center, and I'll remind you what that means. So remember, this is the boundary. And the first singularity was a point on the boundary such that it was preceded completely by a characteristic rectangle contained, except for that point, in p. That's the first singularity. So all first singularities not arising from the center. Not arising from the center is just, remember, here's the center. Well, if the center has sort of an endpoint like this, I also call this a first singularity because, well, it's preceded. It's not a rectangle because it can't be, but anyway. So any other first singularity has the property that the infimum of r in that rectangle, p0. So let me try to explain a little bit. So in general, why do I sort of distinguish these first singularities? Well, it's very clear that if this is indeed the maximal Cauchy development of an Einstein mother system, like this one, then in order for this to actually be a boundary point, these equations have to break down at that point. Because if these equations do not break down at that point, if these equations do not break down at that point, then you can go very close to that point and you can sort of erect initial data if there was no breakdown as you come close to that point. You should be able to apply local existence for these equations and continue beyond and falsify that this is on the boundary. So given a first singularity, you know that something goes wrong there with these equations. Now, a priori, lots of things could go wrong. For instance, phi could blow up. du phi could blow up. All sorts of things could happen. Without anything else, omega could blow up. Or omega could go to 0. That's also a degeneracy of these equations. r could blow up. Well, it can't. That's the first thing we showed. r does not blow up on first singularities. r can only blow up on what we define to be null infinity, and those points are not supersede. But anyway, r could go to 0. And what this proposition says is that if r does not go to 0, then none of those other bad things can happen. And thus, it cannot be a first singularity. So if it is a first singularity, r has to go to 0. So I won't give the proof of this. It's not so hard, although it's not completely trivial. It's much, much easier than the main difficulty of the proof, which, again, I won't give, but I'll try to talk about it a little bit later. But this is certainly the first thing to try to learn if you want to understand a little bit of the analysis, because it's reasonably elementary, but there's already content. And a good exercise is to try to prove this for a more complicated Einstein-Matter system. So that's a good exercise. So a reference for this notion is a paper that I wrote a while ago called Spherically Symmetric Space Times with a Trapped Surface, and also the paper of Jonathan Komemi. I don't remember the title, but you can look it up where he proves, in particular, this statement for a much more difficult example of the charged scalar field. But in any case. So I claim we can already run away with this fact, and we can already extract a corollary just from this proposition, which suddenly will simplify incredibly our picture of what the structure of space time can look like, and then we can go from there. So I have to erase something, so maybe I'll go here and try to fit what I want to say on this board. So corollary, so P can look like the following things. So case one, this was sigma, this is gamma, this is I plus. We know, by the way, an explicit solution that has this property, so Minkowski space is an explicit solution of our system with fee vanishing, and it's exactly like that. Turns out that one can show that small perturbations of Minkowski space also look like that. That's the stability of Minkowski space for this model in spherical symmetry, which is also a relatively easy thing to prove, but good exercise for someone wanting to learn about these things. So this is one case. Case, let me see how I numbered them, because I don't want to number them in some funny case. So case two is what we really don't want, namely I have a first singularity on the center from which comes sort of the boundary looks just this null cone, and this meets off null infinity, okay? And finally, case three. So case three is the following, that I have null infinity, and null infinity, it's past, okay? It's not the whole space time. There's more of its past, okay? So I claim in this case, I can already say the following. I might have, can you even see some people? I guess this was very unfortunate. Let me draw case three somewhere else, because I'm gonna draw, okay, this is sort of awful, but I'll draw case three here, because it's sort of, so I'll explain to you though. So case three, let me draw like this. So this is gamma, okay? So in case three, I am allowing a possibly empty null piece, emanating from the top of gamma. I'm allowing a possibly empty null piece, emanating from here, and everywhere else, my claim is that r extends continuously to zero. So the corollary I claim is that once you know this, okay? Then the only possible Penrose diagrams are one, two, and three, okay? So it's either this, this, or this, and moreover, let me tell you something else, okay? So in cases one and two, one and three, sorry, then null infinity is complete. And you may have noticed something else. So I've drawn this as a circle, which is sort of an open circle, no, the closed circle. So I mean to say, by doing that, that this point is not on null infinity in this case. And one way I can depict that is by telling you that r on this null curve, which I'm calling H plus, okay? It asymptotes to some limit, which I'll call r plus, which is less than infinity. And in fact, I can tell you something more. Not only is r plus less than infinity, but it's less than two mf, where this is the so-called final bond mass. And very, very quickly, using this monothonicity, and the fact that in the past of null infinity, dv of r better be greater than or equal to zero, quick lemma. So in the past of null infinity, I have the du of m is less than equal to zero. So if you think about it, I can define a limiting value of m here. So my assumption on initial data, okay, is that the supremum of m, initially, is finite, okay? So that's if you want the only version of asymptotic flatness that I need, okay? So from that, it follows that you can define something you can call the bond mass. I'll call it capital M of U, by just looking at the limit of little m as you go there. And that's a monothonic limit, so it exists. It's just the supremum, yes. Monotonically, sort of, it has limit monotonic, so r is non-decreasing on this null curve, which I call the horizon, and it attends the limit, which I call r plus, which is finite, and I'm actually asserting that r plus is less than equal to this quantity, which, let me say what this quantity is, for just a second. So by monothonicity, you can define mU, and by this, you can convince yourself that mU is non-increasing as a function of U at null infinity. So mF is just the infimum of mU. You can call that the final bond mass. So this type of equation is sometimes called a Penrose inequality. But this is the actual Penrose inequality that has not yet been proven outside the circle suit. That's to say, I'm telling you that the area of the event horizon is bounded by the final bond amounts. Okay, so this is the strongest type of statement. So all these are reasonably elementary things to prove. So unfortunately, I won't prove this here, but this is all proven in the paper that I mentioned before, a spherically symmetric space times with a trapped surface. And let me emphasize that that paper is not about this system. It is about any system that satisfies that proposition. So that's why if you want this proposition, this sort of property deserves a name that just from this sort of name, just from this property, you can say all this. And finally, let me just make one other remark. A sufficient condition for three. So I'll keep the equations, but I'll maybe erase this. So a sufficient condition for three is that there exists, and hence the name of that paper, at least one trapped or marginally trapped surface. So there exists a U comma V such that dV, I'm sorry, of R at U comma V is less than equal to zero. So given that dU of R is always strictly negative, then if at a point P, if at a point P, dV of R is less than equal to zero, then that point upstairs is a trapped surface. So I should say if it's strictly less than zero, it's what I'll call a trapped surface. If it's less than equal to zero, it's what I'll call a marginally trapped surface. So why is this true? Well, it's clear because by the process of elimination, you see by our monotonicity that we've used several times now, it cannot be the case that dV of R at this point or at this point is less than equal to zero. Because if dV of R is less than equal to zero, then it has to be less than equal to zero there. Similarly here, but it cannot be because R supposedly goes to infinity. So the only Penrose diagram that allows it is that one. So you can read about this in that paper, but it's actually, I mean, that paper of mine is not very deep. And I think it's even better to try to just prove this, prove what I said from this on your own, okay? Because literally you're just using monotonicity property. So though I should say the last part is maybe slightly more subtle with the Penrose. Okay. So now in the remaining time, I'll give an ever so slightly sort of more elaborate proposition which will be then the key to the discussion of weak cosmic censorship and strong cosmic censorship in this model. So unfortunately I have to erase this, but I'll replace it with another proposition. So here's the, here's the, here's the claim, proposition. And now this is really for the Einstein scalar field system, okay? So for Einstein scalar field and under all the assumptions that I've been making on the data that we've talked about, okay? So for Einstein scalar field, suppose this is gamma and suppose that gamma ends in a first singularity, okay? Which will be the case in either case two or case three. In case one, there's nothing to do after all. In case one, future null infinity is complete. And well, maybe it's a good exercise to show that this is inextendable. So suppose gamma ends in a first singularity, okay? And suppose moreover, that there exists a sequence of trapped surfaces. In fact, all I need is marginally trapped. Let me write it like this. So there exists a trapped surfaces. So tending to this point. Suppose there exists such a sequence. So remember, what is a trapped surface? It's just a point such that since we know du of r is negative everywhere, such that dvr is less than zero. So suppose there exists such a sequence, okay? In my spacetime. So then, I'll tell you, I'll tell you what the Penrose diagram of the spacetime is. So then p is the following. So p looks like this, okay? r equals zero everywhere here, okay? This is space-like, and this is all for singularities. There are no null components here. It's all for singularities, okay? It connects this point to this point here, okay? And in view of that, so if this is the case, then everything is true, okay? So then, so then the, let's say, so i plus is complete. So the predicate of weak cosmic censorship, strong cosmic censorship, and even space-like singularity conjecture understood just the statement that the spherically symmetric Penrose diagram is this. I don't want to talk about what this boundary looks like upstairs because I don't want to define it. But just, if you define that conjecture to mean that this is space-like in the spherically symmetric Penrose diagram, then that's true too. So everything is true, okay? So then the predicate of weak cosmic censorship, strong cosmic censorship, space-like singularity conjecture are true. So everything, everything, everything, everything reduces to showing that for generic initial data, if I have, you know, if I'm not one, okay? Then I'll have a sequence of trap surfaces that go there, okay? So this proposition is actually also very, very simple. And in some sense what's key to this proposition, so you shouldn't think the difficulty in the proof of the cosmic censorship conjectures, this is, you know, really the achievement of Stodulu, is showing that indeed this happens generically, okay? But it's already very interesting, but somehow everything, everything, everything reduces to a completely local question about this point, okay? So I want to draw your attention to that. And actually let me draw your attention to one other fact. So this sort of statement that there exists such a sequence is sort of you could, you could also call this a conjecture, you could call this the trap surfaces conjecture, which sort of says that, you know, every first singularity, okay, is associated with a trap surface and by saying associated with a trap surface, I mean, you know, actually there exists the sequence that gets closer. So there is some much more general formulation of that conjecture, not in spherical symmetry, but in complete generality, it uses the language of tips, TIPs, if you know what that is. And that, it can be shown that that conjecture would imply weak cosmic cells. So that's interesting. Of course, in note, in view of what's already written on the board, in this model and in any tamed model, okay? If I just want to know weak cosmic censorship, I just need to know that there exists one of these, okay? Because of what's written in the remark, okay? If there is a single trap surface, then I know I'm in case three and future null infinity is complete, okay? So having the sequence, it turns out it is important so as to rule out a null component like this, okay? Which could be an obstruction for strong cosmic censorship because a null component like this could admit a smooth, let's say, extension through there. Okay, so I'll end in one minute. Let me just already wet your appetite with how you show this proposition. So I claim that the key to showing this proposition is the following. So if you look at this equation here, okay? I haven't written in a very useful way but I can rewrite it in view of my definition of M and well, let me get the factor, right? So I claim that I can rewrite this equation as du dvr equals minus M over two r squared omega squared. Okay? On the other hand, I've just told you that M is greater than or equal to zero everywhere. So it's sort of funny, this particular model and here it's really, really important that duv vanishes. In addition to Rai Shaduri monotonicity, it has an extra monotonicity, namely this monotonicity, okay? So you might want to see, let this be an exercise, how much of the proposition you can already prove just using this extra monotonicity, okay? So it's actually a good exercise. How much of that proposition you can prove just by using this, okay? So we'll continue with this, I guess, on Friday, in the last lecture and then I'll hopefully have enough time for an abridged version of lecture four, which will be about the stability of the Rice and Nordstrom and Cauchy horizon, a phenomenon that is not seen at all in this model, in fact, precisely because of this proposition that I've just formulated. Okay.