So it's a pleasure to give a talk here. So thanks to the organizers of this conference,

and apologies to anyone who has heard some version of this talk. Please feel free to not hear this talk by leaving or paying attention to something else.

So this is the title of my talk, stability of the Kerr-Koshy horizon and the strong cosmic censorship conjecture in general relativity. Let me immediately give you an outline of this talk. So first and foremost, this talk is really about the inside of black holes and a

big conjecture in general relativity, which in some sense is motivated by the behavior of black hole interiors. So a few weeks ago, I gave a talk about aspects of the stability problem inside and outside of black holes. I ended up talking mostly about the outside, so

this talk will be only about the inside. So this talk is about the black hole interiors. So this problem for many years was understood from the point of view of sort of linearized and nonlinear toy models, which I'll run through. And the main result of this talk, which will be,

it's a joint theorem with Jonathan Luke from Cambridge, in some sense sort of resolves.

It's still there. It's still there on paper. Resolves this issue in some sense.

Okay, so I'll, so I say resolves this issue. There's still something very important left to be done and that will be the end of the talk. Okay, so this is my outline. So off we go into the interior of black holes and this strong cosmic censorship conjecture. So the story, if you want, starts with

the Schwarzschild family of solutions, as any good story about general relativity should. Cecile, in her talk this morning, already introduced the initial value problem for the Einstein vacuum equations, the equation Ricci curvature of a Lorentzian

form manifold equals zero. In some sense, the first non-trivial solution of these equations to be discovered is the celebrated Schwarzschild family. This was discovered actually already in December 1915 and published in January 1916. So we are really celebrating the

centennial of the publishing of this solution. So what I want you to know about the solution, one could certainly give a whole lecture just about the solution, but what I want you to know for the purpose of this talk is the following facts, all of which are written on this slide and which I'm going to go through. So first of all, you can think of this as a solution

of the initial value problem and it arises from sort of good initial data, in particular asymptotically flat complete initial data. The only funny thing about the initial data is that it has two asymptotically flat ends and I've depicted sort of the initial data in this sort

of schematic description, where actually every point here is a sphere, so this topologically is s2 cross r and this and this are the two asymptotically flat ends. So it's a solution that arises from regular initial data, nothing is wrong with initial data. On the other hand, the solution itself is geodesically incomplete. So in some sense, this is the first example of

why global existence does not hold for the Einstein vacuum equations. You already see it in this solution. So the solution is geodesically incomplete, but at the same time observers at infinity, they live forever. So how is that possible and what does that mean? So you should

think that observers at infinity is just a way of parameterizing the limit t plus r goes to infinity. So that limit is very important in general relativity because that's where gravitational radiation is detected as we found out last February.

So that sort of limit, you can think of it as an ideal boundary of space-time called future null-infinity. This is depicted here, it has two connected components because, as I said, the data has two asymptotically flat ends. And that boundary at infinity is

itself complete. So if you normalize time at null-infinity, then time goes on forever in both the forward and past directions. At the same time, if you ask yourself what is the past of null-infinity, what are the points in space-time that can communicate with these far away

observers, then that past has a non-empty complement. And that's what, if you learn to sort of read these space-time diagrams, that's what's depicted here. So in general, when the past of future null-infinity has a non-empty complement, we call that region the black hole region, basically. So Schwarzschild is also the most basic example of a solution of the Einstein vacuum equations with a black hole region. So it so happens that

any observer, so any time like geodesic, that enters the black hole only lives for finite time, but all the observers who refuse to enter the black hole, they live forever. So somehow you have geodesic incompleteness, but the incompleteness is hidden inside of

this black hole region. So that's sort of the second thing that I want you to take away from Schwarzschild. But then there's a third thing, which in some sense is the most interesting from the point of view of this talk. What happens to observers who enter this black hole? Well, I already told you that they only live for finite time. But why is that? So it turns

out that you can picture them as asymptoting to a singular boundary of space-time at which the curvature blows up. And this boundary is r equals zero. And there's something more you can say about the singular boundary than the fact that the

curvature blows up. So first of all, it is space-like. So again, that is manifest from this depiction if you know about space-time diagrams. But one can also relate it to the talk of Zag yesterday. So in the language of his talk, that's the statement that every point in the boundary is non-characteristic. So that's really sort of

equivalent to the statement. So this is a space-like singular boundary. And moreover, not only does the curvature blow up, which we know from the PDE theory, curvature blowing up per se, is not necessarily fatal. The metric itself

blows up in some sense. And well, the correct way to say that is that the metric is inextendable beyond r equals zero, even as a merely continuous Lorentzian metric. And actually, that statement was only recently proven in a nice paper of Jan Spierski.

So to summarize, Schwarzschild, it emerges from perfectly fine initial data. But the space-time is not fine. It's geodesically incomplete. On the other hand, faraway observers, they live forever. And the incompleteness is in a black hole.

And if you are so silly as to enter the black hole, then you only live for finite time. And moreover, you will be torn apart by infinite tidal deformations. And in some sense, being torn apart by infinite tidal deformations,

you should think about as being related to the fact that it's the metric itself that breaks down, not just the curvature. So it's a very strong singularity. Moreover, the singularity is space-like, which means nearby points on the singularity do not communicate with each other. Or again, in the language of Zag, the points of the singularity

are non-characteristic. So this is Schwarzschild. But let me already say a big conjecture in general relativity that I'm not going to talk about, but is motivated already by Schwarzschild. And this is the so-called weak cosmic transition.

So when Schwarzschild was first understood geometrically, people thought that all these behaviors were bad and pathological. And there was a hope that all these behaviors at the end of the day were a result of Schwarzschild being very symmetric. And in particular, there was a hope that if you perturbed the initial data leading to Schwarzschild,

then space-time would be geodesically complete, you wouldn't have a black hole, etc. And it's important to remember that that hope was spectacularly falsified in a very short seminal paper of Penrose from 1965, when he proved his celebrated incompleteness theorem.

And a corollary of this theorem is the statement that if you perturb Schwarzschild's initial data and solve the Einstein vacuum equations, you're still geodesically complete. So when you take a second look at Schwarzschild, then Schwarzschild isn't that bad, because at least this incompleteness is hidden in black holes. So this gives rise to a conjecture known

as weak cosmic censorship, which says that for generic, and I'll get back to this word generic, asymptotically flat vacuum initial data, you always have a complete null infinity. So you should really think of this as the statement that very far away observers in the

radiation zone, they live forever. So this is really, if you want, the global existence conjecture in general relativity, which is still compatible with Penrose's incompleteness theorem. So why generic? Why not for all initial data? Well, actually, our only understanding of this conjecture comes from a toy model studied

back in the 90s by Dimitri Shostadoulou, namely Spherically Symmetric Einstein scalar field system. And he proved the analog of this conjecture restricted to spherical symmetry, but he also at the same time gave examples of spacetimes for which complete for which future null infinity is not complete. So such spacetimes are said to possess

a naked singularity. So he proved that they were non-generic, but he also proved that they exist. So this is something to keep in mind. I'm actually giving you this conjecture also so that the name of the main conjecture I'm going to talk about makes more sense. But anyway,

it's a nice conjecture that we should all know if we want to study relativity. All right, back to Schwarzschild. Well, Schwarzschild, it turns out, does not come alone as a solution of the Einstein vacuum equations. It's embedded in a larger two-parameter family of solutions, which are much more subtle and were discovered much, much later

and are known as the Kerr family of solutions. So if I could spend a whole lecture on Schwarzschild, I could spend maybe a week of lectures on the geometry of the Kerr solution. I don't have time for that and you certainly don't want to listen to that. So let me just tell you what

we need to know about the Kerr solution for the purpose of this talk. So first of all, just like Schwarzschild, these Kerr solutions are again geodesically incomplete. And they again have a non-trivial black hole region. In fact, qualitatively speaking, the initial data looks much like Schwarzschild. It's sitting on the same topology. It's again

asymptotically flat with two ends. And again, there's a future null infinity with two connected components. Again, future null infinity is complete, so far away observers live forever. This is not a counter example to weak cosmic censorship. But if you look at the past of future null infinity, that's this region here and this region here, that has a non-trivial

complement in the spacetime, the black hole region. So all that is just like in Schwarzschild. But now there's a difference when we look at the interior of the black hole region. It turns out that the interior of the black hole region is not bounded by a sort of a

space-like singular boundary where all observers approaching are torn apart. On the contrary, the interior of the black hole region, or at least the interior of the region, which is uniquely determined by initial data, is bounded by a null boundary on which the

solution is everywhere regular. Moreover, that means that you can smoothly extend the solution beyond this boundary, and we call the boundary a Cauchy horizon. So what's happening here? You see, we can extend the solution beyond this boundary,

but these extensions will no longer be globally hyperbolic. That's to say, they are no longer uniquely determined by initial data. So here is a situation that does not happen in the model problems that we heard about in the talk of Zag. Here you see that the boundary of the Cauchy development

can be everywhere characteristic. And moreover, nowhere on the boundary does anything blow up. So this is something very, very strange from the point of view of the model problem that we saw about. And it's because you don't necessarily have a quote of first singular time.

And this really has to do with the fact that if you want, time is relative in general. So you might say this is much better than Schwarzschild, it's much better to not blow up than to blow up, right? And it's certainly much better from the point of view of this

observer here. This observer sails past this Cauchy horizon, presumably into some extension. We just don't know what the extension is. But from the point of view of the theory, this situation is actually thought to be worse. The reason that it's worse is that here we see a breakdown of determinism of the ability of the theory to predict the future.

And there's nothing that manifestly says that you have left the validity of the theory. There is no blow up. Nowhere. So that's very, very strange. So in some sense, the situation in Schwarzschild where everyone is accounted for, observers who don't enter the

black hole live forever, observers who enter the black hole are torn apart. This sort of is more satisfying from the point of view of determinism. I have a question. Is there any sense where an m tilde can be taken to be maximal? Do you have a choice of m tildes? Well, you can try to sort of take a bigger and bigger m tilde, but that's certainly not

a unique object. I mean, you can try to sort of look at extensions which are themselves inextendable in various senses. So there isn't that notion of maximal, but not a maximal. So in fact, this behavior is deemed to be so pathological and so sort of worrisome that one

stance to take is that maybe this is all the fluke, maybe it will go away upon perturbation of initial data. And this happy thought

motivated Roger Penrose back in 1972 to conjecture the following. For generic asymptotically flat initial data for the Einstein vacuum equations, then the solution space time, which is determined by initial data, so if you want the analog of the darker shaded region here, cannot be extended as a suitably regular Lorentzian manifold.

So of course, here it's clear why you have to say generic. The Kerr solution itself does not satisfy the predicate of this conjecture. So this says that in particular Kerr inside the black hole has to be unstable, but more generally, any generic initial data should have the property that the Cauchy development

is inextended. So as is written here, you can really think of this conjecture as the statement of global uniqueness in general relativity, just like the weak cosmic censorship is the statement of global existence. So in fact, those would be much better words

or names for these two conjectures, because there is nothing about this conjecture which is stronger than this conjecture. They're really two different statements. Of course, needless to say. So the only sense that this is stronger than this is that the

Kerr family is a sort of a counterexample for strong cosmic censorship, or a generic, whereas it is not for weak cosmic censorship. But in general, these statements, one is not

this is the conjecture. So far, I've only motivated this conjecture by philosophy. And actually, I just came back from a conference on philosophy of science. And believe me, after attending that conference, you don't want your conjectures to be motivated by philosophy.

So it turns out that there's an honest reason to hope that this conjecture might be true. And this was also first put forward by Penrose, and it's the so-called blue shift instability.

So what he observed is the following. Imagine you have two observers, observer A and observer B, where observer A enters the black hole and observer B does not. So these two observers are depicted here. And imagine that observer B sends a light signal to observer A at a finite rate,

as measured by observer B. So every two seconds, whatever. Now remember, observer B not entering the black hole means that observer B lives forever. So observer B sends infinitely many

signals over the course of his or her life to observer A. On the other hand, observer A reaches the Cauchy horizon in finite time. So all these infinite signals, observer A receives them in finite time, which means that the frequency of the signal as the observer A's

time goes to this time value goes to infinity. So it's infinitely shifted to the blue in the electromagnetic spectrum. So this sort of geometric optics type instability, Penrose argued, would cause solutions of the wave equation on this background to blow up

in some way at the Cauchy horizon. And then you can think of this wave equation, this is just the covariant wave equation, scalar wave equation on this background, as some naive model for the linearized Einstein equations. So maybe that means that, at least in linear theory, we see that you have some sort of blow up of something associated

to the Cauchy horizon. So this was actually his heuristic argument and this, in some sense, this was the heuristics behind making this much more ambitious conjunction. But actually, people took this argument one step further. So of course, if you look at a linear wave equation on a globally hyperbolic spacetime, then you can only blow up at the

boundary. You cannot blow up inside the spacetime, just as a matter of principle. On the other hand, if you now take into account the full non-linear theory, you might think that once linear perturbations start getting big, then non-linearities will

take over. And that would cause the spacetime to blow up before the boundary of the original spacetime that you were perturbing. That's to say, you might expect that you develop a space-like singularity before being able to reach this sort of null boundary.

Or again, to use the terminology of ZAG, that might make you expect that the boundary should now be non-characteristic and singular. So somehow the working hypothesis that most people

subscribe to was that the generic dynamic solutions of the Eigen equation would look causally, the causal picture would look like Schwarzschild. This is sort of funny because, of course, the Schwarzschild family is non-generic within the Kerr family, but the claim was that if you looked more generally within all dynamic solutions, then the generic

case would look like Schwarzschild. And this is something that has been discussed in the physics literature for a very long time, and many people have written about this. So let me isolate this statement as what I'll call very strong cosmic censorship. And again,

because unfortunately strong cosmic censorship is not called the global uniqueness conjecture, I cannot call this strong global uniqueness, I have to call it very strong. Okay, so very strong cosmic censorship says essentially that for generic vacuum asymptotically

flat initial data, the part of the solution space-time determined uniquely by initial data cannot be extended to the Lorentzian manifold, and now I'll tell you exactly in what way, even with a metric assumed only continuous. Even with a metric assumed only continuous.

That's to say in exactly the same way that Schwarzschild could not be extended. And moreover, I'll throw in for good measure the statement that the singularity that can be naturally thought of as space-like. So this boundary is non-characteristic in terminology. So this is the very strong cosmic censorship,

and this is in fact the sort of formulation that was sort of widely believed. Can one always make this comment that whenever you have a space-like singularity, the metric will happen? No. In fact, as the paper of Zbierski shows, it's a very

tricky business to show that a metric cannot be extended even just as a continuous metric, exactly because there are no point-wise invariants that capture that. So it's tricky. And in fact, it was Zbierski who first proved that Minkowski space is not extendable

as a Lorentzian metric with continuous metric. So even that was not known. All right. So let me give you a little bit of the prehistory of the problem, even though some of the prehistory is very, very recent as these things go.

So actually, let me start with Penrose's heuristic argument. It's actually very, very easy to show that... So I'll draw for good measure. So this is this picture of the

Kerr metric. This is the event horizon, the boundary of the black hole region. This is the black hole region. This is null infinity, and this is the Cauchy horizon. It's very easy to show that if you have initial data, you can find initial data for the wave equation

in the energy class. So with finite energy, such that the solution here, so if I look at the hypersurface transverse to the Cauchy horizon, will have infinite energy here. And you can do this exactly by the geometric optics argument.

In fact, there's a proof of that by Zbierski using Gaussian beams. So that's very easy. On the other hand, of course, we all know even from the stability of Minkowski space that when we think of perturbing Kerr, we're not perturbing in the energy class, we're perturbing in some sort of weighted energy class. And now you might somehow worry

that the decay that that generates will compete with this blue shift instability effect. So actually, you can show that the blue shift instability wins. And this is a theorem both on sub-extremal Reister Nordstrom, which is a poor man's sort of version of Kerr,

but also on Kerr itself in the full sub-extremal range. And essentially, the statement is that generic solutions of the wave equation, which are localized, in fact, they can decay very, very fast to a space like infinity, will fail to have a finite local energy

everywhere on the Cauchy horizon. So indeed, in linear theory, there is some truth to what Penrose was saying. So anyway, there are all sorts of other comments that one can make. So in particular, you can actually relate specifically a lower bound

for this sort of tail of a solution of the wave equation on the event horizon to the blow up of the local L2 norm of its gradient at the Cauchy horizon. So there's something, in fact, you can say like that, which is very important,

particularly for the future. All right, so this is the blue shift effect. So it's there. But at the same time, it turns out that this blow up is weak. So I told you that

local energy of the solution, the generic solution of the wave equation, is infinite. But it turns out that the amplitude of the solution, C itself, remains uniformly bounded in the black hole interior, all the way up to the Cauchy horizon. And in fact,

not only is it uniformly bounded, it turns out that you can continuously extend it to the Cauchy horizon and thus to some sort of larger region. So this was a theorem of Anne Franzen. And while there are also various sort of extensions and generalizations, maybe let me not talk about them here.

So let me just say a few words about the proof of this just to give you an idea. So just to recall, this is all consistent, right? The generic solution will blow up in sort of H1 log, if you want, but will be sort of remain continuous.

So to prove this theorem, you can make use of results that tell you everything you want to know about solutions of the wave equation on the exterior of a sub-extremal Kerr black hole. In particular, you know that, let's say,

tangential derivatives of C to the event horizon, they decay like V to some negative power. And the only thing I really need to know about that power is that it's bigger than one. That's the only thing I need to know. So this was sort of for sub-extremal

Kerr was a theorem of mine with Rodnyansky and Schliapintok Rothman. So starting with this information and using so-called redshift vector field, this type of decay for C propagates, it's very easy to show, into a little

region inside the black hole interior. So that's easy to show. And then it turns out that you can do a nice estimate in the rest of the region using a vector field. It's not so difficult to write down. It's of this form.

And the only thing that's funny is the coordinates. So for those of you who know about sort of standard coordinates in black hole theory, these are Eddington-Finkelstein-like coordinates in the black hole interior.

So this is V equals infinity, and this is U equals minus infinity. So it turns out that you can apply, and P again will be greater than one, you can apply this as a vector field multiplier to the wave equation.

And this assumption allows you to bound the initial energy terms. Okay, so then if you think about it, bounding, so what does that mean on a hypersurface like that? You're bounding something like this dv.

So if you think about this, then after further commutations, this type of bound, because P is bigger than one, allows you to show that the C remains bounded, in fact, extends continuously. So this is what she did. Okay, so that's Anna's theorem. So now let me make some comments.

So of course, if you naively extrapolate now this linear behavior for the linear wave equation to the Einstein vacuum equations, then what would you do? You would identify

C with the metric and derivatives of C with the Christoffel symbols. I just told you that C doesn't break down, in fact, extends continuously, but derivatives of C, they blow up here. In fact, they're not even locally L2.

So that would suggest, if you could just extrapolate, that when you perturb Kerr initial data, the Cauchy horizon survives as a null bifurcate hypersurface, and the metric remains close to the Kerr metric, just in L infinity. All right. On the other hand, higher derivatives of the metric should be blowing up.

So that would mean that the boundary is something that one could call a weak null singularity, even though for various reasons, the name is not so great. This blow up is such that you're no longer even a weak solution of the Einstein equations.

But the blow up is much weaker than in Schwarzschild. So hence the name. So anyway, there was some evidence for this, actually, which came out of a sort of analysis of some fully nonlinear spherically symmetric toy models. Again, there's a big literature on

that, and I contributed to that many years ago. On the other hand, most people did not believe that this extrapolation was correct. That's to say, in particular, if you believe this original intuition, then in the absence of symmetries, then the nonlinearities of the Einstein equation

should just take over once the perturbations become large enough, and you should get a space like a singularity before. So the question is, which of these two scenarios hold? Okay, so let's leave all this linearized and sort of toy models behind and go to generic dynamical black hole interiors. So if you're a fan of this scenario, I sort of was, I admit,

then the first question you have to ask is, can you even just locally construct just a piece of vacuum spacetime which has no boundary, which is singular in the way that

it would have to be by this extrapolation? Because there are no explicit solutions of vacuum equation that sort of exhibited this type of singularity. And this is one of the reasons that many people thought that you wouldn't have that. So can you even just locally construct this?

And this problem was resolved by Jonathan Luke in a remarkable paper of a few years back. And he constructed such examples by solving a characteristic initial value problem. So let me sort of draw his theorem. So he considered initial data for the Einstein vacuum

equations that were posed on, if you want, on what would be the future of a sphere. Okay, so this is a sphere. It's outgoing and ingoing light cone. Okay, so I'm going to draw these two light cones just like this. Okay, so this is the sphere and these are the two light cones.

Okay, so how do you prescribe characteristic initial data for the Einstein vacuum equations? Well, actually, in certain senses, it's more easy than space-like initial data, because it's more easy to deal with the constraint equations. And it turns out that essentially the free data is given by the shear of this cone and this cone, okay, plus some

information coming from here. Okay, so the shear is in honor of Christos Othello and Kleinerman, is typically denoted by he hat and he bar hat. This is the shear of this cone. So shear just

means the traceless part of the second fundamental form. Okay, so that's really the free data. You basically get to prescribe that arbitrarily. Okay, and then you're going to try to solve the Einstein vacuum equations in a double null gauge. Okay, that's sort of tailored very

nicely to this sort of setup. That's to say you're going to construct locally a space-time which is foliated by ingoing and outgoing null cones. Okay, which are drawn like this. Okay, and in this picture, okay, this is sort of the, these would be those light cones. Okay,

and I'm going to sort of, you can think of these light cones as defining two null coordinate systems. Okay, so my, so one sort of coordinate system will be v equals constant and the other

will be u equals constant. Okay, so this I'll make it v equals zero. Okay, and this if you want is, I don't know, u equals zero. Okay, so these are constant u's and these are constant v's. So what he says is, okay, I'm going to choose the initial data, okay, to be singular as you go

here. Okay, and moreover to be singular as you go here, all right, in a way that would make the Christoffel symbols fail to be L2. Okay, so this is a Christoffel symbol. It's second fundamental

form. Okay, so what's your favorite function which is not in L2 but is in L1? Why should it be in L1? I want the metric to be continuous and the metric is an integral of this you should So my favorite function is what's written there, v to the minus one log minus v to the minus p

for p bigger than one. Okay, so this is v equals zero. This is sort of barely in L1, okay, but it's certainly not in L2. Okay, so this is very singular initial data.

Okay, so you might expect that even if you can prove well positiveness for the initial value problem, the solution will only exist up to some region like this. Okay, and what Jonathan Luke proved is that no, the solution exists all the way up to v equals zero, at least if I restrict

to small enough sort of time in this direction. That's what he proved. I can't say much about this proof, but let me just say a few words which are really geared

to the experts. When you write the sort of Eigen equations in such a double null coordinate system, you have the metric, you have sort of Christoffel symbols, and you have curvature,

where these are written with respect to a sort of a null frame, which is tailor made to these sort of double null coordinates. So anyway, I mean, there's some notation for the metric components. And again, this is really for people who know about this. Otherwise, you can

read a book for hopefully not more than two minutes. So these are examples of Christoffel symbols. And then we also have curvature. And

what Jonathan did is the following. So first of all, it turns out that if you look at the expected behavior of some of the curvature components, it's just too singular to do anything. So a very remarkable thing happens, you can renormalize the system, you can draw up these, and you can redefine rho and sigma, and you can write again

a closed system of equations. So the definition of rho and sigma, essentially you replace rho with the Gauss curvature of the spheres of intersection, and you replace sigma with something that looks like this. So this is called sigma hat, or sigma check, rather.

But this actually has a geometric interpretation that I won't get into. And now these quantities are less singular. So this is actually something that arises from some earlier work of Rodnianski and Luke. So what they actually showed is that

if he hat is assumed to be somehow, if he hat is assumed to be infinity, then you can you know, you can still sort of close estimates for this renormalized system,

okay, and prove a local well process. But here, he hat is much less regular. So it turns out that this is already appearing in Christodoro. The fact that you are dealing with something which is singular.

Yeah, yes, but it's yes, and philosophically, yes. Certainly, philosophically, yes. So what he showed here is that you can prove weighted estimates for these quantities,

where you introduce a weight, which sort of cancels this singular behavior, and you can close estimates for these quantities. Now, of course, secretly what's going on is there's a very subtle null condition for these renormalized quantities that allows you to sort of control things.

So I don't really have time to say more about this, but just sort of remember what's on the board and remember this funny expression here. So the other thing that maybe is good to emphasize, you can think of this as a low regularity well-posed theorem.

And when you think of it as such, you see that it's much lower than the sort of best general local well-posed theorem, which is the L2 curvature theorem of Sergio and Jérémy and Rognanski. And the reason is that here, the Christoffel symbols

are not in L2. So it's really much, much more singular. On the other hand, that singularity is compensated by extra regularity in other quantities. And if you want this sort of null condition plus this renormalization is what tells you that this is consistent and propagated.

So this is what Jonathan did in his really remarkable paper. And in fact, it gets even better because he showed that, well, not only can you sort of have one of these singular fronts, but you can choose the data so that here also, let me change u equals 0 to here.

So this is u equals minus. It doesn't matter. You also have a singular front here, and you can arrange the data so that you can solve in this whole neighborhood here, in all of it. So you have existence up to, if you want, this sort of bifurcate

sphere. So this creates a little piece of space time, such that the space time has a weak null singularity here and here. Moreover, you can extend continuously sort of the metric beyond. So that was Jonathan's work.

So this is really great. This tells you that there's nothing wrong in principle with having weak null singularities. But of course, it does not tell you that weak null singularities occur inside black hole interiors. It just says that in principle, it's not inconsistent.

So how does one show that weak null singularities occur in black hole interior? Well, you have to start with some information. So of course, in linear theory, when I talked about Anna's proof that solutions of the linear wave equation remain bounded, she could

start with the fact that we know that solutions of the linear wave equation, they decay polynomially to 0 at a sufficiently fast rate. Now, the analog of that statement in nonlinear theory is not yet known. It's none other than the conjecture that the exterior region of the

Kerr black hole is stable. So for the purpose of this talk, I'm going to assume this conjecture to be true. So then the theorem, which is forthcoming and hopefully very soon will be out,

which is joined with Jonathan Luke, says the following. If indeed the Kerr solution is stable in its exterior, then the Penrose diagram of Kerr is globally stable. And moreover, the solution is extendable beyond as a continuous

metric, just like Kerr is. So in particular, if the exterior region of Kerr is stable, then very strong cosmic censorship is false. So let me just say very, very briefly,

just to say something about this proof. Maybe I'll say it here because it sort of goes better with what's written here. So what does it mean to assume the stability of Kerr? It means that you can start the problem here. That's to say you can start the problem,

if you want, from a bifurcate null hypersurface that would be the event horizon of these dynamic spacetimes. And the assumption, which is given to you by the stability of Kerr, essentially says that the shear of this cone, and also this one, but let me always talk about

this side, decays at a suitably fast polynomial rate. So something like this. So in fact, all we need is that we have initial data. So this initial data is complete. V goes to infinity. So all we need is that we have initial data, which is approaching Kerr

at this rate. And these rates are not thought to be sharp. So this is sort of a weaker version of what's going to be true. P has to be bigger than one. That's all. Just one.

So now, again, it's very easy to show that things propagate to the analog of an r equals constant space-like hypersurface, a little bit in the black hole. That's very easy. You do it with redshift techniques. And then what you want to do is you want to apply the analog of

this vector field to all these components. So the remarkable thing that happens is the following. So you can apply these vector fields to all components of this renormalized system. And for a large time, as you're approaching the

Cauchy horizon, you think you're essentially showing a global existence result with a null condition. But actually, this coordinate v is related to sort of local coordinates

at the Cauchy horizon. Let me call local coordinate capital V by the transformation e to the minus alpha v equals, let's say, minus capital V, where alpha is some positive constant. So what's remarkable about all this is that if you do this transformation,

so this behavior sort of gets transformed into this behavior with capital V. And so these weights mesh exactly with sort of the weights that come in this theorem. And it happens without even

thinking about it. Yeah. Yeah. So here, the bigger the p, the faster you decay. Yeah. This p makes it less singular, remember. This p makes this less singular.

Because remember, this is blowing up, the log. So let me just make a comment. This theorem is not telling you that the boundary is singular. And in fact, the boundary will not be singular for all initial data, particular curr data.

It's not singular. This is just telling you that if it's singular, it's not too singular. On the other hand, since we expect that it will be singular generically, the estimates have to be compatible with exactly the singularity that supposedly will have. All right. Let me just give a few more comments. I will not take more than two minutes.

So one minute. OK. So let me first give an aside. There's a cousin of this problem, sort of younger cousin. You can add a positive cosmological constant to the Einstein equation.

And well, it's hard to find this solution in a textbook, but there is a solution called the Kerr-Dissider solution, which would be the analog of the Kerr solution if you add this cosmological constant to the Einstein equation. Now, unfortunately, sort of in the regime where we have black holes, the cosmological constant is zero. So these don't really occur in physics

as we understand it. But they're a very nice mathematical model. And in particular, very, very recently, a week ago, Peter Hintz and Andres Vassi showed the stability of a certain region of this space time, namely the region bounded between the event and

cosmological horizons. So actually, if you look at this Kerr-Dissider space time in the region beyond the event horizon, then essentially adding lambda is not so important. And our theorem actually applies. So in particular, using this and our theorem,

you can show that the analog of strong cosmic censorship if you add this cosmological constant is unconditionally false. And we will write something about this. Actually, this space

time has another interesting region, namely a cosmologically expanding region, and its ongoing work of Schlu to show the stability of that region. So let me finish with what's left to be done very, very quickly. Open problem one will prove the stability of the Kerr exterior. Once this is proven, then indeed, very strong cosmic censorship

will be falsified as a corollary. And open problem two, as I said, I'm only showing the stability aspect. I'm only showing that the Cauchy horizon is still there. So open problem two is to show that for generic initial data, the solution is inextendable if you require that

Christoffel symbols are locally square integrable. And the motivation of this, so this you can think of as a weaker formulation of strong cosmic censorship. It's due to Christoffel. And the motivation, if you want for this, is that it tells you that, well, maybe you can extend some sort of metric. But those metrics are not even weak solutions

of the Einstein equations because the derivatives of the metric are not square integrable. So this would be at least some consolation. So corollary of this would be that the Christoduo formulation of strong cosmic censorship is true in a neighborhood of the Kerr family. Now, I warn you to show that a version of strong cosmic censorship is

false. Of course, it suffices to show that it's false in the neighborhood of some solution because it's conjecture about generic initial data. To show that it's true, you really have to show it something about the whole modular space of solutions. So a complete proof of this version of strong cosmic censorship appears to be something far away in the future.

OK, sorry to go over a little bit. Thanks.

So when are we going to see the paper? Hopefully very soon. I tell my collaborators not to have babies, but they don't listen.

Maybe another question, which is you're giving us criteria on the horizon, and that criteria is shifting a little bit. So you have, I don't want to say philosophy, but you have some sense of what physically would be a more desirable criterion. Here you have very strong, here you

have weak singularities but not L2. Now you talk about Christobel symbols and their singularity, their L2-ness. We've had a variety of singularities on horizons. Just tell us what you feel about it. Well, I mean, so it's not to get into philosophy, which I heard a lot about in

Vienna last weekend, but actually your question is in fact, so I didn't mention, but there's another interesting thing about the case where you have a positive cosmological constant, which is that now you have many parameters in the solution. You have the cosmological constant, the rotation parameter, and the mass. And it turns out that heuristically, as you sort of

move towards extremality in these parameters and you sort of do some heuristics, then this suggests that actually the singularity that you would expect on the Cauchy horizon gets weaker and weaker. So in particular, you expect that the closer you are to extremality,

then the higher sort of LP norm, LP regularity you have for the Christophisms. So in particular, if you are sufficiently close, then they will be in L2. And that's really troubling because that tells you that the Christodural formulation of strong cosmological

is probably not true if you have cosmological constant. So that could even be an argument that there should not be a cosmological constant. People want a philosophical argument for sort of why cosmological constant is bad, that could even be used as such.

But in any case, that's to say that unfortunately the final answer is not as clean as we would have liked it to be. And as a result, it does open the door to all sorts of speculation. What is the interpretation of this singularity? Is it the end of space-time

or is it not? And it's not as definitive as it would have been had very strong cosmic censorship been true. I guess some would allow a continuation and others would not allow continuation. Yeah, so the question is sort of what types of continuation are

physically admissible. And we just don't know. Is it so that you expected the data that's allowable on the horizon is everything of the form chi hat's more than equal minus p? Yeah. And like your hope that it will be all data, just a simple net coin like that.

Well, that's the thing. I mean, we start with this assumption and we solve it. But I was thinking of this coming from asymptotically flat data. It doesn't, nothing now. I mean, it's sort of whether this came from an asymptotically flat region is not relevant anymore. No, I understand for your theorem, but I was wondering, so basically you expect that just the data that's consistent with

the asymptotically flat condition is just chi hat's more than equal minus p. Yeah. Well, I mean, there are more info. I mean, there's also, I'm suppressing the fact that you're also, in some sense, this should be a Kerr sphere. I mean, this is a limiting sphere. That's to say that your solution is approaching Kerr.

But you can think that this is sort of what parameterizes the free data on the horizon. Okay? And what you really need is that the free data is decaying sufficiently fast, and it's not, you don't need what's thought to be shot. All data that is decayed sufficiently fast is coming from something that's asymptotically...

Well, you have to add more data. You can try to add data here. Of course, as we all know, sort of solving this gathering problem has its own difficulties. So one... Apologize, but, you know, I'm under pressure. So we can have discussions. Thanks again.

Thanks.