 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 Kosmic 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 and 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 with Jonathan Luke from Cambridge in some sense sort of resolves he's still there he's still there on paper and resolves this this issue in some sense okay so I'll I'll um so I say resolves this issue there's still something very important left to be done and that's that would 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 Kosmic Censorship Conjecture. So the story if you want starts with the Schwarzschild family of solutions as any good story about general relativity should. So Cecil in her talk this morning already introduced the initial value problem for the Einstein vacuum equations the equations Ricci curvature of the Lorentzian format 4 equals 0 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 in January 1916 so we are really celebrating the centennial of the of the publishing of this 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 okay where actually every point here is a sphere okay so this topologically is s2 cross r and this and this are the two asymptotically flat ends okay so it's a solution that arises from regular initial data nothing nothing is wrong with the initial data on the other hand the solution itself is geodesically incomplete okay so in some sense this is the first example of why global existence does not hold for the for the Einstein vacuum equation you already see it in in this solution so the solution is geodesically incomplete okay 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 the sort of a way of saying you know parametrizing the limit t plus r goes to infinity okay so that limits very important general relativity because that's where gravitational radiation is detected as we found out last February so um so now a sort of limit you can think of it as an ideal boundary of spacetime 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 those sort of that boundary at infinity is itself complete okay so if you normalize time at null infinity then time goes on forever in both the forward and past directions okay at the same time if you ask yourself what is the past of null infinity what are the points in spacetime 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 spacetime diagrams that's what's depicted here so in general when the past of future non-infinity has a non-empty complement we call that region the black hole regions basically so Schwarzschild is also the most basic example of a solution of the Einstein vacuum equations with a black hole region okay so it 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 okay so somehow you have geodesic incompleteness but the incompleteness is hidden inside of this black hole region okay so that's sort of the second thing that I want you to take away from 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 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 spacetime at which the curvature blows up okay and this boundary is r equals zero okay and there's something more you can say about this singular boundary than the fact that the curvature blows up so first of all it is spacelike so again that is manifest from this depiction if you know about spacetime 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 okay so that's really sort of equivalently the statement so this is a spacelike singular boundary and moreover not only does the curvature blow up which we're sort of we know from PD theory that curvature blowing up per se is not necessarily fatal the the metric itself blows up in some sense and well the correct way to say that is that the metric is inextensible beyond r equals zero even as a as a merely continuous Lorentzian metric and actually that statement was only recently proven nice paper of Jansbierski so to summarize Schwarzschild it emerges from perfectly fine initial data but the spacetime is not fine it's geodesically incomplete on the other hand faraway observers they live forever and the incompleteness is in the 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 and moreover the singularity is spacelike okay which means nearby points sort of on the singularity you know do not communicate with each other or again in the language of zag the points of the singularity are non-characteristic okay so this is Schwarzschild all right 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 censorship 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 you know spacetime would be geodesically complete you wouldn't have a black hole etc etc okay and it's important to remember that 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 perturbed Schwarzschild's initial data and solve the Einstein vacuum equations you're still geodesically complete okay so 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 okay 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 you know 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 the our only understanding of this conjecture comes from a toy model studied back in the 90s by Dimitri Stalulu namely the spherically symmetric Einstein scalar field system and he proved the analog of this conjecture restricted the spherical symmetry but he also at the same time gave examples of space times for which complete for which future null infinity is not complete so such space times are said to possess a naked singularity so he proved that they were non-genetic but he also proved that they exist okay 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 that's 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 it 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 the 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 judicically incomplete okay 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 all right but if you look at the past of future null infinity okay that's this region here and this region here that has a non-trivial complement in the spacetime the black hole region okay 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 turns out that the 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 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 so moreover that means that you can smoothly extend the solution beyond this boundary okay and we call the boundary a cushy horizon okay so what's what's happening here okay 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 dot so here's a situation that does not happen in the model problems that we heard about in the talk of zog here you see that the the boundary of the cushy development can be everywhere characteristic and moreover nowhere on the boundary does anything blow up okay so this is something very very strange from the point of view of the the model problem that we saw about and it's because you don't necessarily have a quote at first singular time all right and this really has to do with um well the sort of fact that if you want time is relative in general so um 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 cushy 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's no blow up nowhere so that's very very strange so in some sense uh 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 there an m tilde can be taken to be maximal you have a choice of m tilde well you can try to sort of take a bigger and bigger m tilde but that's certainly not a unique object right i mean you can try to sort of look at extensions which are themselves inextendable in various senses so so there isn't that notion a maximal but but not a maximal i don't know right um so so in fact this behavior is deemed to be so pathological and so sort of worrisome that uh 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 uh motivated uh Roger plenrose back in 1972 to conjecture the following for generic asymptotically flat initial data for the ancient vacuum equations then the solution spacetime which is determined by initial data so if you want the analog of the darker shaded region here okay cannot be extended as a suitably regular lorenzian manifold so of course here it's clear why you have to say generic the curr solution itself does not satisfy the predicate of this conjecture okay so this this says that in particular curr inside the black hole has to be unstable okay but more generally sort of any generic initial data okay should have the property that the koshi 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 okay so in fact those would be much better words for or names for these two conjectures because there's nothing about this conjecture which is stronger than this conjecture they're really two different statements so the only some sense sounds better than existence of course needless to say so the only sense that this this is sort of stronger than this is that the the curr family is a sort of counter example for strong cosmic censorship you know we're a generic whereas it is not for weak cosmic okay but in general these statements are not one is not weaker stronger than the other okay so this is this is the the the conjecture so so far i've 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 uh so it turns out that there's a 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 so 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 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 okay so observer b sends infinitely many signals over the sort of course of his or her life to observer a other hand observer a reaches the cushy horizon in finite time okay so all these infinite signals observer a sort of receives them in finite time which means that the the frequency okay of the signal as the observer a's time goes to this time volume 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 cushy 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 okay so maybe that means that at least in linear theory we see that sort of you have some sort of blow-up of something associated to the cushy horizon okay 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 if you now take into account the full nonlinear theory you might think that once sort of linear perturbations start getting big then nonlinearities will take over and that would cause the spacetime to blow up before sort of you know the the boundary of the original spacetime that you were perturbing okay that's to say you might expect that you would 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 sort of the boundary should now be non-characteristic okay and singular so um so somehow the working hypothesis that most people sort of subscribe to was that the generic dynamic solutions of the Einstein 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 sort of the so all dynamic solutions then the generic case would be would look like Schwarzschild and this is something that has been discussed in in the physics literature sort of for a very long time and many people have have 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 the strong global uniqueness I have to call very strong okay so very strong cosmic censorship says essentially that that for generic vacuum asymptotically flat initial data the part of the solution spacetime determined uniquely by initial data cannot be extended the salarency and manifold and now I'll tell you exactly in what way okay even with a metric assumed only continuous okay even with a metric assumed only continuous all right 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 the singularity that can be naturally thought of a space like so this boundary is non characteristic terminology merlin's act so this is the very strong cosmic censorship and this is in fact the the sort of formulation that was sort of widely believed can one always make this comment that whenever you have a space like singularity you'll have the metric will have no no in fact it's sort of as the paper of zbirsky 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 you know point wise invariance that capture that so it's it's tricky and in fact it was zbirsky who first proved that minkowski space is not extendable as a metric with as a lorenzian 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 current metric this is the event horizon boundary of the black hole region this is the black hole region this is non-infinity and this is the koshi 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 okay so if I look at the hypersurface transverse to the koshi horizon we'll have infinite energy here okay and you can do this exactly by the sort of geometric optics argument in fact there's a proof of that by zbirsky 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 sort of in some sort of weighted energy class and now you might somehow worry that the the decay that that generates will compete with this sort of blue shift instability effect so actually you can show that the blue shift instability wins and this is a theorem both on sub extremal rise or north stroke which is a poor man's sort of version of kerr but also on on kerr itself in the 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 they can decay very very fast to a space like infinity will fail to have finite local energy everywhere on the koshi horizon okay so indeed in linear theory there is yeah there is some truth to what panos was saying okay so um 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 the sort of tail of a solution of the wave equation on the event horizon to the blow-up of the local l2 norm of of its gradient at the koshi horizon okay so there's something in fact you can you can say like that which is very important for particular for for the future all right so this is the this is the the blue shift effect so it's there but at the same time um it turns out that this blow-up is weak so i told you that the 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 koshi horizon and in fact not only is it uniformly bounded it turns out that you can continuously extend it to the to the koshi horizon and thus to some sort of larger region so this was a theorem of uh an fransen and while there are also various sort of extensions and generalizations maybe let me not talk about them here uh so let me just say a few words about the proof of this just to give you an idea uh so just to recall this is all consistent right the generic solution will blow up in sort of h1 lock if you want but will be sort of remain continuous okay 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 of a sub-extremal Kerr black hole in particular you know that let's say tangential derivatives of c to the event horizon okay 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 okay so this was uh sort of for sub-extremal Kerr was a theorem of of mine with Rudnianski and Schlapp and Tokrothman 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 so that's easy to show and then it turns out that you can do a 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 uh sort of standard coordinates in black hole theory these are Eddington Finkelstein like coordinates in the black hole interior okay so this is v equals infinity and this is u equals minus infinity okay 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 uh bounding so what does that mean on a on a hypersurface like that you're bounding something like this dv okay so if you think about this then after further commutations this type of a bound because p is bigger than one allows you to show that the c remains bounded in fact uh extends continuously so this is what she did um okay so uh so that's um honest 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 pc with the metric and derivatives of c with the crystallism so i just told you that pc doesn't break down in fact extends continuously but derivatives of c they 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 curr initial data okay the cushy horizon survives as in no bifurcate hypersurface and the metric remains close to the curr metric just in l infinity all right uh 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 uh this blow-up is such that you're no longer even a weak solution of the Einstein equations okay but the blow-up is much weaker than in schwarz okay so hence the name um so anyway there was some evidence for this actually which came out of a sort of analysis of some fully non-linear spherical symmetric toy models again there's a big literature on that and i contributed to that uh many years ago on the other hand uh most people did not believe that this extrapolation uh was correct that's to say uh in particular if you believe this original intuition then in the absence of symmetries then the non-linearities of the Einstein equation should just take over once they once the perturbations become large enough and you should get a space like a singularity before okay 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 and go to generic dynamical black hole interiors so if 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 a null boundary which is singular in in the way that it would have to be by this extrapolation because you know there are no explicit solutions of the 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 uh 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 he constructed such examples by solving a characteristic initial variable so let me sort of draw his theorem so um he considered initial data for the Einstein vacuum equations that were posed on uh if you want on on what would be the future of a sphere okay so this is a sphere it's outgoing an 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 um so how do you prescribe characteristic initial data for the Einstein vacuum equations well actually there's in certain senses it's 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 christel luke and kleinerman is typically denoted by a he hat and he bar hat this is the shear of this cone this is so shear just means that the traceless part of the second fundamental form okay so that's really the free data you you basically get to prescribe that um arbitrarily okay and then you're going to try to solve the Einstein vacuum equations in a double no 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 spacetime 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 uh i'm going to sort of you can think of these light cones as defining two null coordinate systems okay so my uh 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 us and these are constant these 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 crystalline symbols fail to be l2 okay so this is a crystalline symbol it's second fundamental form okay so what's your favorite function which is not an l2 but is an l1 why should it be an l1 i want the metric to be continuous and the metric is an integral of this you should think so my favorite function is what's written there v to the minus one uh log minus v to the minus p for p bigger than one okay so this is this is v equals zero this is uh sort of barely an l1 okay but it's certainly not an l2 okay so this is very singular initial dot okay so you might expect that even if you can prove well positeness 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 so i can't say much about this proof but let me just say a few words which are really geared to to to the experts um so when you write the sort of ishon equations in such a double null coordinate system you have the metric you have sort of christoffel symbols and you have curvature okay where these are written with respect to a sort of a null frame which is tailor made to to 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 uh we also have curvature and what um what jonathan did is the following so first of all it turns out that uh 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 drop these and you can redefine rho and sigma and you can write again a closed system of equations so the redefinition of rho and sigma essentially you replace rho with the 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 away to and and now sort of these quantities are less singular so this is actually something that arises from some earlier work of rodianski and luke so what what they actually showed is that if he had is assumed to be somehow if he had is assumed to be an infinity okay then you can um you know you can still sort of close estimates for this renormalized system okay and prove a local well poses okay but here uh he had is much uh less regular so it turns out that yeah yes but it's it's a yes and philosophically yes certainly philosophically yes okay so what what uh what he showed here is that you can prove weighted estimates okay for these quantities where you introduce a weight which sort of uh cancels this singular behavior and you can close estimates uh for these quantities now of course secretly what's going on is there's there's a very subtle null condition for these renormalized quantities that allows you to sort of control things okay so i don't really have uh time to say more uh about this but just sort of remember what's on the board and remember this funny expression here okay so the other thing that maybe is is good to sort of emphasize you can think of this as a low regularity well posed as theorem and when you think of it as such you see that it's much lower than the the sort of best general local well posed in this theorem which is the l2 curvature theorem of serju and uh Jeremy and Rognanski okay and the reason is that here the yeah the crystal full symbols are not in l2 okay so it's really much much more singular on the other hand that singularity is compensated by extra regularity in other quantities okay and if you want this sort of null condition plus this renormalization is what tells you that okay this is consistent and propagated okay 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 read read yeah change u equals 0 to here so this is u equals minus doesn't matter okay 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 okay so you have existence up up to if you want this sort of bifurcate sphere okay so this creates a little piece of spacetime okay such that the spacetime is has a weak null singularity here and here okay all right moreover you can you can extend continuously sort of the metric beyond okay so that was 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 okay it just says that in principle that it's not inconsistent but so how does one show that weak null singularities occur in 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 sort of to zero at a sufficiently fast rate now the analog of that statement in non-linear theory is not yet known it's none other than the conjecture that the exterior region of the curved black hole is stable so so for the purpose of this talk i'm going to assume this conjecture to be true okay 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 curve solution is stable in its exterior then the Penrose diagram of curve is globally stable and moreover the sort of solution is extendable beyond okay as a continuous metric just like currents so in particular if the exterior region of current stable then a 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 with what's written here so what what does it mean to assume the stability of curve 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 space times and the assumption which is given to you by the stability of curve essentially says that the shear of this cone and also this one but let me always talk about sort of this side okay decays at a suitably fast polynomial rate so something like this okay so in fact all we need is that we have initial data so this initial data is complete so v goes to infinity okay so all we need is that we have initial data which is approaching curve at this rate okay and these rates are not thought to be sharp okay so this is sort of a weaker you need sort of a weaker version of what's going to be true he has to be bigger than one that's all just one so so now again it's it's very easy to show that things propagate to a sort of 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 okay and 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 this behavior sort of gets transformed into this behavior with capital V and so these weights mesh with 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 or less i because remember this is blowing up the log okay so 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 core data it's not singular okay 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 okay all right let me just give a few more comments i will not take more than two minutes so one minute okay 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-Desider solution which would be the analog of the Kerr solution in if sort of 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 hints and andros wasi showed the stability of a certain region of this spacetime namely the region bounded between the event and cosmological horizons okay so actually sort of if you look at this Kerr-Desider spacetime 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 so so and we will we will write something about this actually this spacetime has another interesting region namely a cosmologically expanding region and its ongoing work of slu 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 koshy horizon is still there so open problem two is to show that for generic initial data the solution is inextendable if you require that the crystal symbols are locally square intubable and the motivation of this so this you can think of as a weaker formulation of strong cosmic censorship it's due to the crystal zulu and the motivation if you want for this is that it tells you that well maybe you can extend to some sort of metric but those metrics are not even weak solutions of the ancient equations because sort of the derivatives of the metric are not square integral so this would be at least some some consolation so corollary of this would be that the crystal zulu formulation of strong cosmic censorship is true in a neighborhood of the kerf 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 okay sorry to go over a little bit thanks so when are we going to see the paper hopefully hopefully very soon i mean i tell my collaborators not to have babies but they don't listen yes maybe another another question which is you're giving us a criteria on the horizon and that criteria is shifting a little bit so you have a 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 cristobal symbols and their singularity but their el tunas we have we've had a variety of singularities on horizons and just tell us what you feel about well i mean what what 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 i i didn't mention but there's another interesting thing about the case where you have positive cosmological constant which is that now you have many parameters in the solution you have the cosmological constant the the rotation parameter in the mass and it turns out that heuristically as you sort of move towards extremality in this in these parameters and you sort of do some heuristics then this suggests that actually the singularity that you would expect on the cushy horizon gets weaker and weaker so in particular you you expect that the closer you are to extremality than the higher sort of lp norm you lp regularity you have for the cristophosomes so in particular if you are sufficiently close then they will be in l2 and and that's really troubling because that tells you that the um cristidulu formulation strong cosmic censorship is probably not true uh if 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 uh for sort of why cosmological constant is bad that that could even be used as such but in any case that's that's to say that the the you know unfortunately the answer the final answer is not as clean as we would have liked it to be and uh you know as a result sort of it does open the door to all sorts of speculation what what is the interpretation of this singularity should we you know is it the end of spacetime or is it not and the sort of 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 yeah what types of continuation are physically admissible and we we just don't know okay is this something you expected the data that's allowable on the horizon is everything of the form chi had smaller than v minus p yeah and like your hope that that would be all all data just a simple net point like that with the that's the thing i mean we we start we start with this assumption and we solve the but there is thinking of that's coming from uh from uh selective fed data it doesn't nothing now i mean it's sort of whether this came from an asymptotically flawed region is not relevant anymore no i understand for your theorem but i was wondering so basically you expected just the data that's consistent with the asymptotically flat condition is just chi had smaller than v minus p yeah well i mean there are more info i mean there's also i'm i'm suppressing the fact that you're also some sense this should be a curse fear i mean this is a limiting sphere that's to say that your your your solution is approaching current okay 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 short all that that is the case officially fastest coming from something that's uh that's an asymptote well you have to add more data you can try to add data here of course uh as we all know sort of solving the scattering problem has its own difficulties so one part of it's so slow anyway so yeah they apologize but they you know i am under pressure can you say so we can have discussions uh thanks again thank you