Merken

# The stability of the Kerr Cauchy horizon...

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at a time and you and so it's there it's a pleasure to give it to a clear and so thanks to the organizers of this at the conference and and apologies to anyone who has heard some a version of this talk please feel free to not here is stalled by leaving Gore was paying attention to something else so that this is the title of my talks stability of the current coffee Verizon and the strong cosmic censorship conjecture In general activity let me immediately give you an outline of his talk so 1st and foremost this book is really about the inside of black holes and area the conjecture general relativity which in some sense is motivated by the behavior of black hole interiors so there's a few weeks ago I gave a talk about aspects of the stability problems inside and outside black holes I ended up talking mostly about the outside this will be only about being so there is still concern is about the lack of interior so this problem for many years was understood from the point of view of linear I and nonunion toy models which run through and that the main the result of this talk which will be it's a joint appearance with Jonathan

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look up from and in some sense sort of results use is still there still there complacent and at at at and that results from this this fish some again so I'll alum so I say resolves this issue there's still something very important left to be done about stuff that would be me and of talk so this is my outlined areas off we go into the interior of black holes and the strongest of contract so and so the story if you want starts with the 4 children families solutions as any good story about general wanted show some and sincerely her talk this morning already introduced the initial value problem for the nation's vital questions questions curvature of the ransom for was 0 the some since the 1st nontrivial solution of these equations to be discovered is the celebrated Shawshank this was discovered actually already in December 1915 and published in in January 1969 so we are really celebrating the centennial of the of the publishing of this so what I want you to know that I know about the solution and 1 1 could certainly give the whole election just about solution what I want you to know for the purpose of stalkers the following a fax all of which are written on this flight and which aren't going to go so 1st of all you can think of this as solution of the issue problem and it arises from 2 good initial garden particularly as told the flat complete sure Dr. the only funny thing about the initial rather is that it has to a simple the flat aunts and I'm depicted a sort of the initial doctor at this sort of schematic description of where actually every point here is a sphere that this topologically says to cross art and this and this are the 2 most father or so it's a solution that arises from regular about nothing nothing is wrong with from the other hand the solution itself is genetically incomplete so in some sense this is the 1st example of why global existence is not hopefully for the Aschenbach during this year In this 1 it was almost as you musically incomplete but at the same time observers of infinity they live forever so how is that possible and what does that mean so we should think that observers at infinity it's just the sort of a way of saying know parameterize advising the limit T plus are goes to infinity of so that is very important generosity because that's where gravitational radiation is detected as we found out that last February so so what sort of limits you can think of it as an ideal boundary of space-time called traditional inherited this is depicted here it has to connected components because there's as I said the doctor has to us simple compliance and those sort of boundary and Infinity's itself complete interview normalize time acts knowledge affinity than time goes on forever both past at the same time if you ask yourself what is the pastor of knowledge in the world .period in space and that can communicate with these followers observers than that past has a nonempty complement and that's where he learned to swim to read the space and diagrams that's what's depicted here so in general that when the prosecution Finley has amounted to complement recalled region the black hole which search for shield is also the most basic example of a solution the reaction back inquiry with a black so it so happens that any observer so any time I duties that enters the black home only lives for finite time but all the observers who refused to enter the bloc holding for somehow you have incompleteness but the incompleteness is hidden inside this block it's about sort of the 2nd thing that I wanted to take away from from 4 but then there's a fair which is sometimes the most interest from the point of view this talk what what happens to observers who entered this black hole while I already told you that they only live for finding time the wiser so it turns out that a you can picture them as asymptotic to a senior boundary of space at which the courage of those and this boundary is articles and there's something more you can say about this singular boundaries and the fact that the courage of blows so 1st of all it is space-like so again but is manifest from this depiction of you know about space and paragraphs but 1 can also related to the talk design that yesterday at so the language of the of his talk was the statement that every point in the boundary there is known characters so that's really the state so this is a space like Cingular boundary moreover not only does the curvature of which we're sort of we know from as PDE theory curvature going up for sale is not necessarily fatal and those the metric itself blows up in some sense as well the correct way to say that it is done the magic is extendable beyond articles 0 even as soon as a nearly continuous actually that the statement was only recently approved a nice paper against so to summarize 4 chilled it emerges from a perfectly fine initial doctor but the space-time is not find its Judas Akley incomplete I'm not far away observers they live forever and incompleteness is black hole and if you or so silly as to enter the black hole but then you only liffe finite time and moreover you'll be torn apart by infinite titled information the ants and in some sense being torn apart by by infinite titled the formations to should think about as being related to the fact that it's the metric itself the brakes it's a very

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strong signal nor over the similarities space flight To means nearby points and so on singularity you don't communicate with each other on again in the language of staff at the points of similarity are known characteristics that so this is fortunate but a litany already on sale at bay conjecture in general to be not going to talk about that but is motivated already but fortunately this is so recourse and so when was 1st understood geometrically people thought that all these behaviors were bad from and there was a all but they're all these behaviors at the end of the day there were a result of should be very soon and in particular there was a halted Jupiter the initial Dr. leading to fortunate then no space that would be due the sickly complete you wouldn't have a black hole eccentrics and it's important to remember that that that hope was spectacularly falsified in a very short seminal paper at of panels from 1965 when he proved his celebrated ministerial and a corollary of this era is the statement that if you perturbed for children from Arkansas the action by but "quotation mark you're still duties so the stall when you take a 2nd look at Ford Field that's sure to this and that but because at least this incompleteness is hidden in bottles so and this gives rise to a conjecture known as we cosmic censorship which says that for generic and I'll get back to this word generic Hassan told me flat should not you always have a complete now infinite so you should really think of this as statement very far away observers in the radiation zone they live forever so this is really if you want to the global existence conjecture in general relativity which is still compatible with Penrose's so why a generic wine overall blacks are only understanding of this conjecture comes from a toy model started back in the nineties by means of a new enemy strictly symmetric eyes and skin fuel system and he proved the analog of this conjecture restrict from symmetry but he also the same time it gave the examples of space space-times for which a complete before traditional affinities is not complete such basement ascent to possess and make its singular J. proved that they were nonchalant but he also approved Texas so but this is something to keep in mind as I'm actually a giving you this conjecture but also so that the the name of the main conjecture going to talk about it makes more sense that any amendments that it's a nice conjectured that we should all know we want to stop the relative please I am not just for chilled Welch fortune turns out it is not come alone as a solution of the icing equations it's embedded in larger to brand the family of solutions and which are much more so were discovered much much later and analysts the co-founders solely if I could spend the whole lecture unfortunately could maybe in a week of lectures on the challenges the Curragh solutions that I don't have time for that incident don't listen to them so let me just tell you what we need to know about the current solution for the purpose of this stuff so 1st of all just like Sean chilled these curves solutions Oregon and you basically incomplete and they don't have a nontrivial vocal range In fact qualitatively speaking we initial got the looks much like for children sitting on the same topology again and told the flock to answer and the gathers future volatility with 2 connected components again fictional Infinity's complete itself far away observers live forever and this is not a counterexample two-week cosmic censorship and but if you look at the past diffusional infinity apparent this region here in this region here that has a nontrivial complement the space and the black hole so although is just like export but there's a difference that when we look at the interior of a black hole region the terms of the the In terms of the Bicol region is not bounded by by a sort of a space like Cingular boundary where all observers approaching our are torn apart there on the contrary the the interior of the black hole region or at least the interior of the region which is uniquely determined by national party is bounded by now 900 on which the solution is everywhere regular L so moreover that means that you can smoothly extend the solution Beyond this boundary In and we call the boundary because she her rights so what's what's happening here that you we can extend the solution beyond its boundaries but these extensions will no longer be globally comparable as to say there are no longer uniquely determined by so here is a situation that does not happen in the model problems that we heard about it in the top of soccer hear you say that the the boundary of the coffee development can be everywhere characteristic and Lorillard nowhere on the boundary does anything and so this is something very very strange from the point of view of the the model problem responsible and it's because you don't necessarily have a sequel to the 1st time and and this really has to do With what we need the fact that you want time is right to do so and so you might say this is much better than fourfold as much by the 2 not then to blow right and it's certainly much mother from the point of view of this observer here this observer sales past this caution presumably into some extension just on an extension but from the point of view of the theory this situation is actually thought to be worse the reason it's worse is that here we see a breakdown of the terminus of the ability of the theory to predict the and there is nothing that manifestly says that you have left the validity of the no there's no blocked no so that's very very strange so in some sense that the situation

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for chilled where everyone is accounted for by observers who don't enter the bloc will live forever observers went to the black holes are torn apart this set of is more satisfying from the point of view of and

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determines we where the antisense work there until everything the rest of your choice of what you can try to sort of take a bigger and bigger and told them but that's certainly not unique culture Megan tried sort look at extensions which are themselves an extendable in various not so there is not a notion of a maximal but not a maximal with them so so about this their behavior is deemed to be so pathological and sold so worrisome that what stance to take is that may be this is all a fluke maybe it will go away upon preservation showed up and this hobby felt that and motivated Roger Penrose by 72 to conjecture the falling for generic until the flat initial dollar for the Aschenbach questions than that the solutions based which is determined by initial got so if you want the emblem the doctor shaded region here that cannot be extended as a suitably regular Lawrence enough Of course here it's clear why you have to say generic the Constitution itself does not satisfy the Pentagon to discourage cancel this this says that in particular :colon inside the black hole has to be unstable but more generally sort of any generic initial bout of should have the perspective the cautioned development so as written here you can't really think of this conjecture the statement of probable uniqueness general relativity just like the week cosmic censorship there is a statement of global exist so that those would be much better words for or names for these 2 conjectures because there is nothing about this conjecture wishes stronger than this congestion relief to different statements so the only constant and of course that needless to say that always sensed that this this is sort of stronger than that this is the car from the isn't it a counterexample for strong cosmic censorship the were generic whereas it is not for the week was but the generally statements not 1 is not weak or strong so this is this is the number of those conjecture so so far I've I've only motivated this conjecture by philosophy and actually I just came back from conference and philosophy of science and believe me after attending the conference you don't worry you're conjectures be motivated by the loss of so it turns out that there is there is enormous reason to hope that this conjecture might be true and this was also the 1st know put forward by panels and it's In the so-called blue shift so and when he observed is the falling so and I imagine you have to observers observer and an observer B where observer a enters the black hole and observer B does not so these 2 observers are are depicted here and imagine that observer beat sends the right signal to observe of a finite rate as measured by of so every 2 seconds when I remember observer being not entering the black hole means that observer be lives forever look at so observations in the men's singles Over the some of the costs of his or her life to the ring have observer rate reaches the coffee horizons in finite soul is influence signals observer the receives final which means that the the frequency think of the signal as observer race time goes to this kind of goes infinity so it's infinitely shifted to the blue electromagnetic spectrum so at this sort of geometrical that instability and Penrose argued would cost solutions of the wave equation on this background to block in some way coffee horizon and then you can think of this wave equation statistical variant wave equation scalar wave equation on this spot as some naive model for the linear function equations so maybe that means that in eastern India theory we seemed to have some sort of block of something associated discussion so this was actually his heuristic argument in this some sense this was the heuristics behind making this much more ambitious country but actually people took

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this argument 1 step further so cost that if you look at a linear wave equation on globally hyperbolic space time then you can only block about you cannot block in cyberspace to smother Prince on the other hand as if if you know take into account the full nonlinear fearing you might think that the once sort of linear perturbations started getting bigger than nonlinear parties will take over and that would cause the space-time to blow up before sort of you know the the boundary of the original space and were protect let's say you might expect that you would develop a space like singularity before being able to reach this sort novel on began to use the terminology of of Xi s that might make you expect that where the boundary should not be the known characters and senior so so somehow the working had all this is that most people sought subscribe to was that the generic a dynamic solutions of the action equation would look closely the coastal picture would look like 40 come this sir the funny because of course the fortune families non-generic within the profoundly but the claim was that if you looked more generally within so all dynamic solutions that generic he's being would look like Schwartz and this is something that has been discussed in the physics of the ship center for a very long time and many people who have have written about so let me isolate this statement as what I'll call very strong cosmic censorship and again because unfortunately strongest exams ship is not called it the global uniqueness conjecture cannot call this strong "quotation mark of Americans strong closing censorship said essentially the that's a for generic VOC you must build financial mother at the part of the solution space-time determined uniquely by initial bother cannot be extended Lorenzino tell you exactly in what way even with a magic assumed only continue look even symmetric assumed only contains say in exactly the same way that forces could not be extended and moreover I'll throw in for good measure the statement that the singularity it can be naturally filled with respect 6 of his boundaries known knowledge the world so this is a very strong centered and this is in fact those sort of formulation that this would want get 1 of these companies that have the flexibility that I to have the people and not now In fact the incidents the paper of Spears shelves it's a very tricky business to show that the metric cannot be extended even just continues matter exactly because there are no no .period was environments that capture solids it's tricky and that in fact that it was basically who 1st proved that Minkowski space is not extendable as symmetric with residents in metal containers let's say even that was not all right so let me give you a little bit of prehistory of a problem with the moved some of the prehistory is behind the recent and does the things

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got so and so actually and let me start with Penrose's heuristic argument that this is actually very very easy to follow then that yeah it's all draw for good measure so this is this picture of the the parametric this is the event horizon the boundary of the Black origins but only to actresses and infinity and this is the question and it's very easy to show that if you held initial daughter you can find initial doctor for the wave equation in the energy costs so with finite energy such that the solution here so if I look at the hyper surface transverse to the coffee horizon will have infinite energy here and you can do this exactly mindless would geometrical optics on fire as a proof of the minds beers using Nelson so that's very easy other and of course we all know even from the stability of closely spaced out and when we think of perturbing card were not perturbing in the energy class perturbing sort of in some sort of wait and watch and now you might somehow worry that the the indicated that generates will compete with this sort of blue shift instability so actually you can show that the the blue-chip instability winds and this is a theorem both on some extreme will rise from Nordstrom which just 4 months were version cops but also on Encore itself and the fall of extreme range and as essentially the statement is that generic solutions of the wave equation which are localized impact they can begin decay very very fast too a space like infinity that will fail to house the finite local energy everywhere on the coast 2 so indeed in linear theory there is there is some truth to what weapons was b so and so they're all sorts of other comments that 1 can make better than some particularly can actually relate specifically a lower bound for this sort tale of a solution of the way the question on the event horizon 2 then the blowout of the local L 2 norms of of its gradient on the come to there's something that you can you can say like that which is very important for a particular for the all right so this is this is not the blue shifted effect so it's not but at the same time it turns out that is blocked this week so I told you that the local

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energy of the solution the generic solution the wave equation isn't it but it turns out that the amplitude of the sort of see itself remains uniformly bounded In the black interior all the way up to cushion and in fact not only is it uniformly bounded and it turns out that you can continuously extended to the coffee growers in the summer sort of larger so this was a theorem of and Franzen and friends while there are also various sorts of extensions and generalizations maybe maybe not talk about them here and so let me just say a few words about the proof of this just to give you an idea and so just to satirical this is all consistent right the generic solutions I will blow up in sort of H 1 lock if you want but will be 2 remain continuous the so the promise syrup can make use of results that tell you everything you want to know about solutions of the wave equation on the exterior of of extreme curb on hold in particular you know that that's a tangential derivatives of C to the event horizon In fact they decay like Evita some negative power and the only thing I really need to know about the powers that is because this was so this was sort of force of extreme Carroll was inferior my name is pronounced the entrapment of across so starting with this information and using so-called red chips that the field at this type of decay for a seat at propagates it's very easy to show into a little region inside the black interior so so that's easy to shop and then it turns out that you can do that and base their estimate in the rest of the region using about the field it's not so difficult to right now instead of this form and the only thing that's funny is the court so for those of you who know about Standard coordinates black hole theory is are Eddington Finkelstein like audits in the black hole interior so this is at the request of civility and this is new people's minds so it turns out that there you can apply and the again will be rate greater than 1 you can apply this as vector field multiplied the way the question and that this assumption allows you to balance the initial energy targets so that if you think about it then bounding so what does that mean on the on the Cyprus service like that involved something more this the the but so if you think about this then after further commutations this type of abound because P is bigger than 1 allows you to shoulder the scene remains found in fact that extends Kentucky so this is what she did better focus so as to that honestly so let me make some comments so of course if you naively extrapolate now this linear behavior for the linear wave equation too the inviting equations than what would you do if you would identify C With symmetric and derivatives of staying with the crystals so I just told you that seat doesn't break down bucked extends continuously but after reviews of the they they blow up here in fact they're not even locally so that would suggest if you could just extrapolating that when you perturbed correlation mother the coffee Verizon survived Aston no by forgive hyper Service and the Magic remains close to the crematory just In Oh infinity the other had a higher dreams of metric should be going so that would mean that the boundaries something that 1 could call it a week now singularity and all for various reasons the names not so great that that this book is such that you're no longer even a weak solution the Irish equations but the love is much weaker Lynch fortune OK so hence the name the Senate where there was some evidence for this actually

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which came out of a sort of analysis of some fully nonlinear it's replacement .period mortals began big the dish on that I contributed to the death many years ago I had a most people did not believe that this extrapolation I was correct not to say that it particularly if you believe original intuition that in the absence of symmetries that the nominee artistes of the action equation should just take over once they once the perturbations become large enough and he should get a space like from Priscella question this week which of these 2 scenarios Bob so let's leave all this linear toy models behind them and go to generic dynamic about to so give if you're a fan of this scenario assumed was admitted that the 1st question you have to ask is can you even just locally and construct just the piece of Vikings space-time which has no boundaries which is singular it in the way that it would have to be by this extrapolation this year there are no explicit solutions of the vital question that sort of a exhibited this type of singularity and this is 1 of the reasons that many people thought you would so that so can you even just locally constructors and this problem was resolved by Johnson woke up in a remarkable paper used back and he he constructed such examples by solving a characteristic initial so that sort of draw his fear so that he could see the initial other for the Irish and watching equations that postpone there with you want on on what would be the future of the sphere of Prince of this is its outgoing inning going like all of that so I'm going to draw these 2 icons just like this because this is the sphere in each of the 2 like OK so so how do you prescribed characteristic initial doctor for the affiliation bottom equations election there is a certain senses is it's more easy than space like initial doubts because it's more easy to deal with the constraint equipped and it turns out that essentially the Free Gotta is given by the sheer all of this call in this column about plus some information coming from so this year is in honor of crystal environment that is typically denoted by then he had and he is the share of this column so sheer just means that the trace was part of the 2nd fundamental so that's really the free got you you basically get to prescribe them and arbitrarily and then you're going to try to solve the Irish equations in double data on that sort of Taylor very nicely to this sorts of let's say you're going to construct locally is based I'm just phoning in by in going I'm going all comments OK which are drawn like this and in this picture this is sort of please would be just like comments from and I'm going to so that you can think of these likens defining to knock audit systems so my bed so what sort of caught system will be the most constant and the other will be you equals costs so this all make a decrease 0 at and this year it's you at and so these are constant use and these are constant so what he says is OK I'm going to choose the initial doctor To be singular as you go could have more work to be singular us you go here are right in a way that would make the Christoval symbols failed to be altered so this is is a result it's 2nd fundamental form so what's your favorite function which is not enough to but is now 1 why should it be know what I want the metric To be continuous in the metric is integral all risk we should so my favorite function is what's written their vitamins 1 the load Wednesday after Monday's plea for peace can't so this is this is the equal 0 this is uh so barely an hour 1 again but it's certainly not enough to what's is very

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singular initial got on so you might expect that even if you can prove well positives for the nationwide problem the solution will only exist up to some region like this and while Jonathan proved is that no the solution exists all the way up to the 0 at least if I restrict 2 small enough sort of time that's what people some of the I can't say much about this probe but let me just say a few words which are really geared to to to to the experts there so when you write the sort of Eischen equations in such a double no court system you have the magic you have Christoval symbols and you have courage can wear these are written with respect to in all frame which is tailor-made 2 these sort of double knock or so anyway I mean there's some notation for the matter components and again this is really for people who know about this we did talk hopefully no more than 2 minutes these are exactly Christoval symbols and then we also have curvature company and what what Johnson did this the 4th so 1st of all it turns out that if you look at the expected behavior of some of the curvature components it just to singular to so very remarkable thing happens you can we normalize the system you can drop these and you can redefine Roland C ,comma and you can write again closest quick so the redefinition of Sigma essentially you replace Raul with gas curvature of of the

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spheres of intersection and who replaced signal with something that looks like this this is a good signal how much signature is

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actually has a geometric condition and walked away the and sort of these so quantities are less so this is actually something that arises from some earlier work of rodents and look forward when they actually show it's not as if he had is assumed to be a somehow if he had is assumed to be 0 infinity upon that you can do that you know you you can still sort of close estimates for this we normalize system band proved local wobbles but here he had is much then less regular so it turns out that it is going to be the 1st to use held some the yet yes but it's With said yes and philosophically yes set philosophically so what was was he showed here is that you can prove weighted estimates for these quantities where you introduce away which sort of councils this singular behavior and you can close estimates for these costs now costs secretly what's going on is there is there is a very subtle no condition for for these we normalize quantities that allows you to sort of controls so I don't really have the time to say more at about this but just sort of remember what's on the board and remember this funny expression here so the other thing that maybe it is good to sort of emphasized you can think of this as a low regularity what was the CIA and when you think of it as such you see that it's much lower than the 2 best general local locals and steel which is the onto curvature of feeling of surgery and Jeremy and romance architect and the reason is that here the yet the Christoval symbols are not enough so it's really much much more singular other hands that singularities compensated by extra regularity in other parts after and if you want this sort of no condition plus this renormalization is what tells you that produces is consistent and property so this is what Johnson did in his really Vermont and factored it gets even better because he showed that while not only can you sort of have 1 of these singular France but you can't choose the doctor so that here also let me read freak the change was erected here and so this is doing this month because you also have a singular here and you can arrange the doctor so that you can solve this whole neighborhood here all so you have existence up up to if you want this sort of bifurcate the sphere "quotation mark so this creates a little piece of space such that this space it has a weak singularity here and here but moreover can you can extend continues the of so that was that was Johnson's work so this is really great this tells you that there's nothing wrong in principle With having recalcitrant parties but of course does not tell you that we know singularities occur occurred inside black hole interiors it just says that in principle that it's inconsistent but so how does 1 show that we can all singularities occurring in black hole interior world you have to start with some information so of course in the linear theory when I talked about others approved that solutions of linear wave equation that remain bounded she could start with the fact that we know all the solutions of the when equation they decay polynomial Inc and so to 0 at sufficiently fast now the analog of that statement in nonlinear theory it is not yet known it's None other than the conjecture that the exterior region of the car black hole state so at the time so for the purpose of this talk I'm going to assume this project to be true so then the fear and which is forthcoming

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hopefully very soon will be out of bed which is jointly Jonathan look says the following if indeed the current solution is stable in its exterior then the panel's diagram of color is going and moreover that the sort of solution is extended well beyond as it continues much just like come so particular about it the exterior region of current stable than that very strong closing should this fall so let me just say very very briefly just to say something about this problem but maybe I'll stay here because it's a goes better with With what's written here what was agreed to assume the stability of care it means that you can start the formal here are necessary stop the problems if you want From a bifurcated now hyper service that would be the event horizon of this dynamic space and the assumption which is given to you by the stability of cops essentially says that the Shia all this call and also this 1 but lately always talk In this side of the case added suitably that fast bowling on something like that so far all we need is that we have initial so this initial not those complete story goes to infinity so we need is that we we have initial model which is approaching but this race look and these rates are not thought to be shot so this is sort of a weaker you need instead of a weaker version of what's that was going to be true for currently has that's what just so long as so no ever again it's very easy to show all the things probably added to a sort of the analog of imparting his constant space like persists little bit in the black hole it's very easy to with reach and then what you want to do all this you want to apply the analog of this vector field to all these components and so on the remarkable thing that happens is the following so you can apply these vector fields to all components of this re-normalised system and for a large time as you're approaching the pushy Verizon you think you're essentially showing a mobile existence resolved with all condition but I actually this corner in the is related to sort of local coordinates article she horizon would become more important capital by the transformation eager the miners off of the equals considering the capital the wealth listen closely so what's remarkable about all this is that if you do this transformation that this behavior that sort of gets transformed into this capital and so those these weights match With exactly with the the weights of new and happens without even thinking about it yet it is yes so here we are in the festival here yet this speed makes it less singular remember this be makes this lesson yeah on site yeah because remember that this is blowing up the lock so so let me just make a comment this film is not telling you that the boundaries In fact the bombing will not be singular for all initial doubt particularly occurred that this is just telling you that if its singular it's not to sink on the other hand since we expect that it will be singular generically the estimates have to be compatible with exactly the singularity supposedly will have but they let me just give a few more comments I will not take more than 2 minutes or so what 1 minute on so they may 1st given aside that there's a cousin of this problem as a the younger cousins that you cannot oppose the cosmological

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constant Johnson equation and as well it's hard to find the solution in a textbook but there is a solution called occurred see the solution which would be the analog of the curse solution in it I sort of you add this cosmological constant attention now unfortunately sort of in the regime where we have black holes the cosmological constant 0 so these don't really occur In physics as we understand it but they're very nice mathematical model and in particular very very recently a week ago and Peter Hintze an on-the-spot show the stability of a certain region of this space namely the region that bounded between the event and cosmological authorized on so actually them sort of them you look at this as a courtesy the spacetime in the region beyond the event horizon than essentially adding land that is not so important and actually appliances some particular that using this and other fear you can show that the analog of strong censorship and if you add this cosmological constant is unconditionally falls so and so and we will we will write something about this they're actually this space that has another interesting region namely a mostly expanding region and its ongoing work of shoe to show the stability of the that's already finished with what's left to be done very very quickly that open problem 1 will approve the stability of the car exterior that once this is proven then indeed very strong closing censorship will be falsified as a corollary and open problem till as I said I'm only showing the stability aspect I'm only showing that the coffee horizons it is still there and so will over the problem to is to show for generic initial thought that the solutions in extendable if you require that the Christoval symbols on locally the square inch and the motivation of this so this even think of as a weaker formulation strong was closed due to the will and the motivation if you want for this is that tells you that well maybe you can extend some sort of metric but those metrics are not even weak solutions of the ancient nations because of the derivatives of the matter pronounce so this would be at least some of them some consolation so corollary of this would be that the Christodoulou formulation strong cosmic censorship is true in the neighborhood of the curve my warning to show that a version strong closing session this fall's cost it suffices to show that its forces in the neighborhood some solution because conjecture about generic initial to show that it's true you really have to show you something about the whole Montanans based solutions so complete proof of this version of struggles censorship appears to be something far away each other soaring to go over a little bit and makes it the fastest and if you do that for 15 and so was a little young see the hopefully hopefully you very soon I tell my collaborators not to have babies that debate on this yes the beginning of the criteria on the horizon in that area is scheduled to be held in Wednesday you have some sense of of was desirable area here you have a strong Europe singularities prior to the of symbols and the singularity of this and we are about variety of singularities authorize just because you feel well and in what was so as not to get into the philosophy which I heard a lot about the Indiana last weekend but actually your question is in fact an so am I I didn't mention but there is another interesting thing about the case we have pose a cosmological constant which is that it is now you have many parameters of the solution you have the court was constantly the rotation parameter in the mouse and it turns out that opportunistically as you sort of move towards externalities In this in these parameters and you sort to some heuristics that this suggests that actually the singularity that you'd expect on the cushion Verizon gets weaker and weaker so there in particular in you you expect that the closer you are to extra melody than the higher sort of LP on you you will be regularity you have for the rest of us some particular that if you're sufficiently Klaus then they will be announced soon and and that's really troubling because that tells you that it's the air Christodoulou formulations struggled session it's probably not true that it if you have commercial cost so there could even be an argument that there should not be comfortable people want a philosophical argument the for sort of White most Wilkinson is not dead that could even be used but in any case that's that's to say that there not be Unfortunately the answer the final answer is not as clean as we would like it to be and you know as a result of guards opened the door to all sorts of speculation what what is interpretation of this singularity should here is at the end of space-time Rossignol and the this sort of it's not as definitive as would have been had very strong should interest this will allow the continuation of the as those of the question is sort of what you what types of continuation of physically admitted and this is something you stated that there is a lot of alone Of course is in good form of this morning who wants the metallurgical coal used to be although only just as he was led the that's very may start we start with this assumption resolved is a euphemism for the former facilities it doesn't nothing outlining its certify of whether this came from an flood regions it is not relevant anymore and therefore you're here but there's 1 you expected just the response system with dozens of liquidation is just 1 I in the morning for writing there's also an and suppressing the fact that you also them since this should be a curse fear but and this is a limiting sphere has decided your your your solution is approaching :colon but you can't think that this is sort of what franchises the free dot on the horizon and what you really need is a freedom of the is decaying sufficiently fast and it's not you don't need to of the shoulder that although this is the 1st time since start of this is this is what you have to add more doubt than you can try to add the here of course as we all know sort of solving discovering problem it has its own difficulties so we went to some of the leading lady poses a real and what the the matter time of the disease in the region sessions with the same thing Kelly things

00:00

Resultante

Stabilitätstheorie <Logik>

Punkt

Flächeninhalt

Allgemeine Relativitätstheorie

Vorlesung/Konferenz

Strömungsrichtung

Innerer Punkt

Numerisches Modell

01:53

Resultante

Krümmung

Punkt

Hyperbolischer Differentialoperator

Flächentheorie

Natürliche Zahl

Familie <Mathematik>

Fortsetzung <Mathematik>

Gleichungssystem

Kerr-Lösung

Erweiterung

Extrempunkt

BAYES

Inzidenzalgebra

Raum-Zeit

Computeranimation

Deskriptive Statistik

Temperaturstrahlung

Umkehrung <Mathematik>

Existenzsatz

Kommutativgesetz

Asymptote

Kontraktion <Mathematik>

Analogieschluss

Horizontale

Vervollständigung <Mathematik>

Krümmung

Geodätische Linie

Schwarzschild-Metrik

Singularität <Mathematik>

Ähnlichkeitsgeometrie

p-Block

Frequenz

Zeitzone

Arithmetisches Mittel

Randwert

Exzentrizität

Einheit <Mathematik>

Verschlingung

Rechter Winkel

Sortierte Logik

Hochvakuum

Körper <Physik>

Skalarfeld

Eindeutigkeit

Charakteristisches Polynom

Gravitationswelle

Aggregatzustand

Darstellung <Mathematik>

Subtraktion

Allgemeine Relativitätstheorie

Stab

Anfangswertproblem

Schar <Mathematik>

Auflösung <Mathematik>

Term

Physikalische Theorie

Symmetrische Matrix

Topologie

Unendlichkeit

Gegenbeispiel

Spannweite <Stochastik>

Kugel

Symmetrie

Endogene Variable

Inverser Limes

Affiner Raum

Gleichungssystem

Minkowski-Metrik

Einfach zusammenhängender Raum

Erweiterung

Kurve

Konvexe Hülle

Raum-Zeit

Linienelement

Hyperebene

Validität

Physikalisches System

Unendlichkeit

Singularität <Mathematik>

Diagramm

Symmetrische Matrix

Flächeninhalt

Minkowski-Metrik

Normalvektor

Innerer Punkt

Numerisches Modell

15:35

Punkt

Krümmung

Flächentheorie

Regulärer Graph

Gleichungssystem

Kerr-Lösung

Extrempunkt

Erweiterung

Skalarfeld

Computeranimation

Numerisches Modell

Statistische Analyse

Wellengleichung

Auswahlaxiom

Verschiebungsoperator

Deterministischer Prozess

Parametersystem

Statistik

Horizontale

Geodätische Linie

Schwarzschild-Metrik

Singularität <Mathematik>

Heuristik

p-Block

Frequenz

Mechanismus-Design-Theorie

Linearisierung

Menge

Einheit <Mathematik>

Verschlingung

Sortierte Logik

Hochvakuum

Einstein-Feldgleichungen

Eindeutigkeit

Geometrie

Theorem

Sterbeziffer

Betragsfläche

Allgemeine Relativitätstheorie

Validität

Zahlenbereich

Funktionalgleichung

Topologische Mannigfaltigkeit

Physikalische Theorie

Gegenbeispiel

Physikalisches System

Unterring

Perspektive

Spektrum <Mathematik>

Gleichungssystem

Verschiebungsoperator

Erweiterung

Finitismus

Raum-Zeit

Eindeutigkeit

Zwei

Hyperebene

Penrose, Roger

Unendlichkeit

Uniforme Struktur

Symmetrische Matrix

Minkowski-Metrik

Analogieschluss

Horizontale

Numerisches Modell

20:54

Stereometrie

Heuristik

Stellenring

Extrempunkt

Randwert

Kontinuumshypothese

Familie <Mathematik>

Statistische Hypothese

Gleichungssystem

Extrempunkt

Inzidenzalgebra

Raum-Zeit

Gradient

Superstringtheorie

Theorem

Wellengleichung

Vorlesung/Konferenz

Einflussgröße

Verschiebungsoperator

Parametersystem

Horizontale

Singularität <Mathematik>

Schwarzschild-Metrik

Heuristik

Störungstheorie

p-Block

Frequenz

Ereignishorizont

Linearisierung

Transversalschwingung

Randwert

Forcing

Sortierte Logik

Physikalische Theorie

Beweistheorie

Hochvakuum

Einstein-Feldgleichungen

Parametrische Erregung

Lineare Abbildung

Theorem

Stabilitätstheorie <Logik>

Physikalismus

Gruppenoperation

Klasse <Mathematik>

Abgeschlossene Menge

Physikalische Theorie

Unendlichkeit

Spannweite <Stochastik>

Flächentheorie

Hyperebene

Störungstheorie

Gleichungssystem

Minkowski-Metrik

Gammafunktion

Physikalischer Effekt

Lipschitz-Bedingung

Thermodynamisches System

Pseudo-Riemannscher Raum

Raum-Zeit

Linienelement

Eindeutigkeit

Gasströmung

Unendlichkeit

Singularität <Mathematik>

Energiedichte

Mereologie

Energiedichte

Horizontale

Hyperbolischer Raum

27:40

Resultante

Randwert

Extrempunkt

Flächentheorie

Ausbreitungsfunktion

Gleichungssystem

Baumechanik

Erweiterung

Analysis

Raum-Zeit

Vektorfeld

Numerisches Modell

Charakteristisches Polynom

Theorem

Minimum

Wellengleichung

Vorlesung/Konferenz

Tangente <Mathematik>

Korrelationsfunktion

Lineares Funktional

Horizontale

Singularität <Mathematik>

Störungstheorie

Kommutator <Quantentheorie>

Ereignishorizont

Linearisierung

Randwert

Forcing

Sortierte Logik

Rechter Winkel

Beweistheorie

Hochvakuum

Verzweigung <Mathematik>

Körper <Physik>

Einstein-Feldgleichungen

Identifizierbarkeit

Nebenbedingung

Theorem

Sterbeziffer

Extrapolation

Gruppenoperation

Derivation <Algebra>

Bilinearform

Symmetrische Matrix

Physikalische Theorie

Kugel

Symmetrie

Fächer <Mathematik>

Freie Gruppe

Delisches Problem

Gleichungssystem

Minkowski-Metrik

Analysis

Leistung <Physik>

Schätzwert

Fundamentalsatz der Algebra

Erweiterung

Thermodynamisches System

Hyperebene

Physikalisches System

Fokalpunkt

Menge

Unendlichkeit

Energiedichte

Singularität <Mathematik>

Symmetrische Matrix

Last

Minkowski-Metrik

Parametersystem

Mereologie

Innerer Punkt

Numerisches Modell

38:01

Quadratischer Raum

Zahlensystem

Kugel

Ortsoperator

Sortierte Logik

Krümmung

Vorlesung/Konferenz

Zusammenhängender Graph

Gleichungssystem

Physikalisches System

Sigma-Algebra

40:19

Nachbarschaft <Mathematik>

Gleichungssystem

Kerr-Lösung

Raum-Zeit

Vektorfeld

Arithmetischer Ausdruck

Regulärer Graph

Existenzsatz

Gruppe <Mathematik>

Wellengleichung

Vorlesung/Konferenz

Analogieschluss

Horizontale

Grothendieck-Topologie

Kategorie <Mathematik>

Krümmung

Stellenring

Ereignishorizont

Linearisierung

Randwert

Polynom

Sortierte Logik

Konditionszahl

Verzweigung <Mathematik>

Projektive Ebene

Aggregatzustand

Theorem

Stabilitätstheorie <Logik>

Gewicht <Mathematik>

Renormierung

Sterbeziffer

Verzweigung <Mathematik>

Abgeschlossene Menge

Diagramm

Transformation <Mathematik>

Physikalische Theorie

Stabilitätstheorie <Logik>

Kugel

Zusammenhängender Graph

Schätzwert

Chinesischer Restsatz

Mathematik

Physikalisches System

Singularität <Mathematik>

Chirurgie <Mathematik>

Diagramm

Mereologie

Kantenfärbung

Horizontale

Innerer Punkt

Numerisches Modell

50:20

Nachbarschaft <Mathematik>

Resultante

Stabilitätstheorie <Logik>

Physikalismus

Abgeschlossene Menge

Gleichungssystem

Derivation <Algebra>

Kerr-Lösung

Bilinearform

Drehung

Raum-Zeit

Stabilitätstheorie <Logik>

Kugel

Regulärer Graph

Endogene Variable

Ereignishorizont

Analytische Fortsetzung

Minkowski-Metrik

Analogieschluss

Parametersystem

Horizontale

Erweiterung

Kurve

Linienelement

sinc-Funktion

Heuristik

Physikalisches System

Flüssiger Zustand

Ereignishorizont

Konstante

Singularität <Mathematik>

Flächeninhalt

Forcing

Sortierte Logik

Beweistheorie

Horizontale

Varietät <Mathematik>

### Metadaten

#### Formale Metadaten

Titel | The stability of the Kerr Cauchy horizon... |

Serientitel | Trimestre Ondes Non linéaires - June Conference |

Teil | 07 |

Anzahl der Teile | 23 |

Autor | Dafermos, Mihalis |

Lizenz |
CC-Namensnennung 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. |

DOI | 10.5446/20821 |

Herausgeber | Institut des Hautes Études Scientifiques (IHÉS) |

Erscheinungsjahr | 2016 |

Sprache | Englisch |

#### Inhaltliche Metadaten

Fachgebiet | Mathematik |

Abstract | The stability of the Kerr Cauchy horizon and the strong cosmic censorship conjecture in general relativity I will discuss recent work on black hole interiors for dynamical vacuum spacetimes (without any symmetry) and what this means for the question of the nature of generic singularities in general relativity and the celebrated strong cosmic censorship of Penrose. This is joint work with Jonathan Luk |