The problem of stabiliy of Kerr under axilly symmetric perturbations

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The problem of stabiliy of Kerr under axilly symmetric perturbations
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I will talk about a recent result, in collaboration with A. Ionescu, concerning partial nonlinear axisymmetric perturbations of Kerr and will also give a progress report on my recent work with J. Szeftel on axi-symmetric polarized perturbations
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around In you can 2 I was thinking that much the introduction and thanks for 60 organizer for organizing this wonderful conference and then as the title suggests that I will be talking about the potentially equation so and it's a few words the following are going on was the slides so uh course by now I think people already familiar with this densely questions that would be talking only about the ancient vacuum questions in which case it was only about the form for money forward was a Lawrence imaging and they kind of breaching of key point is that this is the shortest equations and possible right now and yet you can even go there so I was the we we have to solve all of this aggression in the sense you have to find that takes which face questions now the 1st thing to do he said about the Canopus people in the know even those wonderful new job thinking that these hyperbolic questions that the Bowling and because of the matching England's 10 and therefore the list when any initial value problem for these initiative overgrown formulation in which you start was a streamlining the matrix of undermining the Jake given announcing must and tense as medical tents gave In the site like in the case of you always have to compare with say that the wave equation understandably equation here we risk I find the Pentagon will fight with his prescribing Jr the and cake which would be no 1 wants to construct the space that this would be the tide upon people that all the major points of its there so this is the initiative of course you have some constraint equations he'll have to be solved it's some minor detail know lots of people were going to the constrain equipment and a lot of interesting results but really interesting thing in the pollution problem mine so I assume that you have initial conditions in New York so now also another thing that I should say I just don't believe speaking to problems which are cosmology the problems and problems which isolated physical systems so I will talk about the letter In other words I assume the possibility of flatness that so what that means is that you are looking for matrix which shook call skin at infinity or intensively initially that means that His Michiko looks like the mitigating knee-deep in the 2nd column of the phone looks like the island of course you have some you have no precisely so formulating its but roughly that's what it anyway so it is a flatness indeed and don't bother then obviously we are interested in in solving this problem so there is a local existence results which will know president Yvonne Vogel can up 1952 which this is a list of you can construct given specific initially held some level of the French ability of course usual you can construct a solution after a certain time and then there is also boxing warm the panel existence which is called a multimodal hyperbole development strategies of Martin make possible extension that you can make by local existence arguments it's OK to keep in mind that that's so that's the maximum worldwide according to the island Of the space-time and obviously all all of general article of muscle degenerative is about about getting information providing information about these multiple we went by so now the next thing that you do whatever I have known that very difficult only a problem you look for stationary statesman of stationary states so they start states which in some senses are independent of time knowing of course Tiny generally there is a more complicated notion but stated you can make sense of it so it's actually relatively simple you say that the space-time are is stationary is a system that the which is space like which is sorry which is fine like that space like infinity in other words know you have a light connectivity .period then as you go a good to its Infiniti the vehicle feared is more and more like the standard during the tea in Eagle suspects but it's so that's so essentially sent means exist a observed that the dignity of the magic perspective these sequences in so that's kind of a special it's not the end Alco are examples so that there is the next thing that we know is that there is a distinct incredible were formerly of solutions of inside questions under this assumption an article declined was implicated essential which right so this was the funny so this is where the interested in parts of was stricken with an it is equal to 0 corresponds to shortages equal to 0 and of course Minkowski means that it would 0 equipped with adequate physiotherapist Bossi Spiderman right so so he sensationally states and then you know once you think that you need to know all the stationary states which of course is the problems of unmentioned here and there 3 obvious problems connected with the case solution the family then you can conjecture as people have done the so-called final state conjecture state conjecture the truth is that they should take any initial data but then not a single 1 is going to be very complicated legal actions of the very complicated ammonia is analysis of the incredible difficult but the hope is that eventually you are going to see In any final regional state of only going to seek a solution nothing so this is like the solid of resolution conjecture won but but I'm interpreting this fine as the conjecture in the sense that century things are stable because otherwise Bradley for his employer didn't make any sense of itself so that you go yes that goes
United as homage also so this this will be part of the fun that conjecture that are all sorts of things initial date complicate all sorts of things happened holes emerge and then they will call maybe some of them some other would go far away users who that but added that the liable any whatever it is that unlike was based essentially on on this picture and so what you know that there is an experiment but poverty based on the assumption that these speeches :colon now in order for that each other we going to have to do a lot of things so 1st of all the 1st thing that the simplest thing may be that you could ask is do these it is it is scant harmonies of rigidities rigidity is is this is a step sketch-comedy Beijing has was ordained as a station solutions besides also this solutions if you can find explicitly the site visits solutions which are finding in 62 63 by by Karen trotted even uniting 15 billion yuan right so I say to their duty project that those who the down was Satanist the 2nd thing is that very few of those discovery yeah of the camp comedy centers interconnected swamp admission of the consumption what makes the question is do we are going to get something up front because if you get something else then it's clearly not the case have no physical reality the existing mathematical universe that beautiful and you can study them if you want but they would have no no physical significant and of course that this 1 as the conjecture before the fact if any of these things that I'm writing now would be false and offensive to the of there is so beyond stability there is also call the issue of collapsing Cup again you actually formed Black also told I mean of course we have told them they form when you have a collapse of the stock for example you follow black code but is it true his speech troops that's another must questioned for hours and then if you want to go so these last 3 fundamental question which I call yard I like to call them reality so that the test Silvio at the office of the black if these are false then bye bye calls from few months now of course beyond that if you want to the finest conjecture that many many other things singled and many many steps like for example as you said you know taking all to block all it this would be just 1 block or this is the 1 the that the 1 body problem that has yet to study the development and then there is that is that there are other issues that have to do the cosmic censorship which is another fundamental conjecture in general which says roughly the only singularities you can have a better similarities besides specially set up a single light is of no new also fundamentally flawed understandable now it so happens that that that symbolizes Suez which are known in generating the sensibility and publishing once all the care funding it's like a family what is this what you from as a singularity In the Black also this is a remarkable thing about driving duties that the space above all Spitzer space that it's responsible the space that sees space between regions the buckle region and on the outside regions and outside the region it is but is a widely known similarities will occur in the buckle regiments essentially what we believe cosmic censorship is a will formulating the statement that the only single life you can have signed buckled Yukon have naked so-called naked singularity Of course not and and talking about here program for the next 100 years strengthening and this may be taking on more than 100 100 but but obviously there were lots of lots of steps and there are lots of simplification that you can make and of course there's a lot of them there is a lot of activity you can simplify many of these problems it was that the problems that have been most interested in the recent years on the street I'm so and I'm going to look at this stage of stability and stuff like that make me go to the site it's so cold in the can the only this was no vote renders this problem aside and said about his whereabouts or even really working late that's so here is that you can't from now they get upset with him the market will not have to be very careful with the media shouldn't even be used it's better than that act so I see you see isn't the case the students who are yet he dissented saying giving and so should never purchased and so users can finally expressed in so-called will relinquish called units which are like political Ordinance NDC Connecticut it's extremely explicit even tho it's a solution of a very complicated Parliament you have to remember that this thing just adjustment to blunt the fossil over this formula but the important thing is that all the coefficients that you see here of the are independent of the of that and of course was the fact that you have specialized in other words is like refuse accumulated indeed in the sense of this definition and you also see that the dimensions of politically flock was all of you see that this coefficient of explicit and are goes to infinity you get the Minkowski space In addition of special ID also have about systematic so the cat family is you see these coefficients also independent of defined which corresponds to the fact that the fact confusing which is you with the fee is indeed a king but other swatches solution is the party workers 1 equal to 0 emission becomes much easier you you have these dental but it's your causative have some kind of strange singularities of resisting American soccer he also but in this case is easy to see that when all is acquitted 20 moral R is equal to 0 these metrics of the become singular now the single singularity that the
singularity .period equal to To win is not only a single line of the consolidated gradually extend the solution any facts I put 2 and just the bond between a so-called black hole and they still region now they decide that the you'll have to take and he added features never seen this kind of tendrils backgrounds might be a bit confused by the important thing is to look at is maybe if you don't become a plum uses picture is to look at vise bodies so far Societe the bodies that I wanted to and which corresponds exactly to the bond between the buckled under the exterior of the space than I listen to and this is a buckle region and maybe the Whitehall delightful is not the physical so we would strongly felt that interested in it there is a cover to sing like a diet with zeal it here that the major country does becoming the the coverage of the magic becomes infinite that I would you again I would not be so this is about the broccoli that the black all by itself is of course the fascinating thing and that's where the similarities will occur but but I would be interested in what he thought about stability interested in the external part of the block all the way up to the horrors so this is a project to do it and do here but there are some other interesting numbers I was sweetened corresponds to the existence of so-called trapped 9 geodesic structure will eat lots of problems in Bosnia Somalia but especially at the end of finally these are to infinity which corresponds is expected to these boundaries so these bodies are really at infinity now the life here propagates a 45 degrees right so is it you see that inside you cannot escape your and eventually hit the singularity while of course if you're outside if you if you are lucky and you go in this direction you'll get into the black woman indicators in writing but can also escaped and became views of the city's was between the extent of it very roughly between the the black colleges I don't know 1 year
or so the test by study is very similar at least in the exterior region he said that the horizon now is that the article AA-plus which is the root of the spawning a means of getting they explicit but the broccoli slightly different in the sense that you don't have that seemed like a with 0 but I'm not going to get into this region at the interesting this and as you see the picture is a senior well this is a space like headless surface which will play the role of the No I want to give initially but later on I want to but because solution we want to give usually on something but so far there are the gained extended region alleges in class event horizon which is exactly the boundary the buckle region which is our lives and our books
here is a gain the same picture what this is also the 1 that you should look at but that's external region this is even more eyes and where I was equal to plus in Kent now here you also see the victor feared copy team which is the station taking the view that it's time like means it goes in this direction as you approach infinity for the season remember this is founded Infinity while now the market was thing that happens is a T becomes archery space like the horrors so even tho a study by the Panamanian Verizon space like and that these 2 lots of issues I mean libraries of interesting physical features but also validate lots of difficulties because of for example energy is always tight working but you think that with mechanics the so-called acidity is of course tied to the fact that the in places that you would be space-like elicited the wound up this appeal and so you have been doing this is something that I just want to draw your attention to the point but so this
is now the conjecture the office that this publicly conjecture so civility of statement extended Katz and others I'm interested only in this region I take this initial data but I think initially by the exactly of the Council motion rental consumption has of course and it associated initially Delaney space like others of France I think that they thought that the right and by the conjecture says it's more perturbations of a given can solution I have not seen the future development writing as illustrated by the Socialists Pfizer I can which called lost when as a cancelation saying but it would be compete in this direction and it to look in politically it look like like consumption but maybe a different 1 in other words I can start as a N N and the final states would be different from that and which meets again to allow me as a Michigan as we know learning the acceptance of this kind of situation you have to do come from another White so
what are the difficulties of the buckles stability problems but that the first one which is common to any genetic problems in particular general activity is a so-called digital provides having you right down the questions I what you directed like this it's useless to you want to write it in some kind of cold units but then as it the quality may who maybe we get a system of mourning him with equation shot carefully but it may be that guy that the instabilities in that system of coordinates would mean nothing goes into picked up the phone call their friends so you have to be very careful what do you mean you have to work as as much as you can in an environment fashion which is not easy given the nature of these equations of Americans you can uh this of course is the official visualizes claim was proper immunization to change marvelous you about the military's Robison only equation Bonino questions were very different brands at the end August of the new if you don't know you can try to arrives at the level of Natick efficiency and even coordinate system that's not a very good idea as it turns out you can use connection coefficients of can use curvature to canonize at the level the coverage and this seems to me this is actually the a little way to go and others claiming be more later on uh and then another problem of course is that distinguishes a cause union rights so in as Woods the characteristics depend heavily on the solution itself and the characteristics of course have to play a fundamental role in the story because this is where like problem amid the gravitational waves so vital we if met a few new propagate along not characteristics the so there is an on TV structure I mean we recovered about another condition these equations are can be viewed at some level as some system of Norlina wave equation so you expect the noncommissioned blending both in the wrong however if you try to use among condition in the coordinates his Bible chocolate Shaukat of with coordinates it's not that there is no condition basis with the so-called week now condition but there's a difference there it's so so that have to find a correct not condition is also entitled to Indonesian to the taken as was in opposition also has to do is a problem of coordinated and gages now that even if you look but that being a problem in Canada to fix a background so this is much simpler you fix a breakdown this is sort of a baby the ization you assume that joint can renew that summing up ,comma for his image as a way extend the later questions we get this is already highly known for highly nontrivial problem which was only understood Due to mark the 30th maybe center 40 years of effort was all Anderson Austin is a century by mathematicians you by introducing introducing all sorts of new techniques which take into account the fact that the characters has limited synergies so the thing that hits the sound of it that heroes are not enough the Democrat who job by 2 symmetries of care the fact that there is a medical region in which you have seen is them issues even as the energy on the fact that you have presence of truck-mounted disease which is actually consulted be technically the most difficult of all so what that struck much of this at the center of don't escape eyes in the block or at infinity In the end stayed fact in spacetime at that time that low so far and as you know from geometric optics they will create lots of problems but so finally but not least you do expect no matter what kind of unionization is due to have some kind of leery stability connected as gages and this is due to the changing and right so the final as I said the final and would be different from the ones who put them up and that has to fill the B rating some kind of instability of the nearest equations and also the position of the final block all also would change have to take that into account also that the bill also considered but so far so now carries a little bit those news a little bit of technical stuff but this is not the only me here Societe typically when you have a space but the best way to underlies it is not using coordinates but using friends rent so wire frames importance so if Frankfurt simply effective feuds in space which not so that mythology is 3 . 3 please 44 he put 2 0 and therefore they correspond to somehow to propagation rental so that's why they're very political units is very hard to write things like that so that's where friends having bought and then once you have your also normalizes so that the industry for the sequel minus-2 right now and then once you have that you can also look at the space that that particularly to this still and you right down he won 2 2 was that the fees which are often mobility to themselves and of course but that he with his friends so that's that's that's what it's called enough rain which plays a fundamental role in just about anything he wanted to injure our knowledge base did so the 1st thing they could do is not friend once you have been
on Frank you can construct connection connection coefficients I was simply means that which I did virginity by out which simply means that you take you take that even the core of the relative to the train let's say of thought in the the gunman went so you take over and the unity of of the frame relative to the franc and then you projected a gain in the light of the threat to this year's if you that the so-called asked and then the next thing you do is you you look at the curvature so this is a coveted and you also the composer coverage of Kansas cancel the coverage of his is a Fulton said also legal was relative to the franc and you get the vise coefficients for example you know if you look at all before the 1 already to Seoul according to a copy delayed the firstly 142 before indeed this is called they all paving and it's something the boss of the coverage and what all there Our all various combinations of the components of artists the red a friend and I was very often in interactivity with see this kind of knowledge and so anyway so this so again what you have to keep in mind however may be you know I don't want to I don't want to scare you lose too many locations what you should keep in mind is that the gunman the Gunners the flies the equations which related to curvature so for example this this is a very important declaration before gunmen that's gonna is equal to curvature of course and I think it in a symbolic manner the case should think about bias components of gun buyers components of garment and Basel Basel coverages against it right you have a lot of you you confuse aboard blood boy was many equations sorry yesterday you you can feel the block vote was those lies equations and finally the designation so that there is another 1 medically street so something and then they something similar about the curvature so there is a history of heart muscle damage and on it is I believe a piece of curvature of the universe which would be no different from that of the street that is have to put it here and before 2 very roughly 20 times the plants are disposed of course that again as I said an you tread on all the components you can feel the blackboard the important thing 2 realize is that this on the most fundamental of all of these are in fact nothing about young gave ended its 98 identities and it's not too difficult to see you in a 1st approximation that there are actually but 1st of all they derive from adding that they can be derives from the the standard junket entities which are the government of the Sigma blasts tried to take a psychic presentations of these indices rental units just like consequently 0 and by the fact that you also have the honor instead you have this kind of kind of divergent of evidence of the gunmen about equal to 0 so this this actually did this equation falls from this 1 democratic government very cheap of genes it was 0 another with the flour for in advance of common questions anyway so this is what I want to say is that you can view this as glorified Mosul equations and somehow this is extremely important because it goes to bed that somehow the hyperbole ,comma tho the question is most explicit in terms of the coverage for the coverage of being the environment Of course it is obviously extremely important when so this is this is a this besides the hyperbole this is that the work out of the question and therefore that the the way to think about all this Is it somehow you 1st made the curvature using an adjustment in energy because is going 5 months questions in order for muscle admissions have an inducement and so you can do something seen that it's a bit more complicated but is there a senior once you have the curvature then you can analyze the garments by sinking of the systems for the questions whether there which has already given the that and then somehow all these other things here I these relations I mean the true however including here this of course is follows so that they could here follow from these ones but somehow this could be could be used in combinations of 1 in order to get all sorts of things that the solution is source of this thing over there so we I'm going to say something right not market so this was as preparation for an executive of but so so why wire this important agency immediately when you look
about them to the southern unstructured questions that I mentioned you can see near the when you apply all these 2 cats because in care he so 1 remarkable fact about care is that there exists a principle not afraid another East for together I don't explicitly them according exist end such that all curvature components fuddy-duddy above about all this coverage of 0 lead managed in their friend and only the so-called role in articles that survive this these the this sort of thing is this body 43 foreign there another 1 which is just like that but let me not getting the OK I just want to give you some some kind of a bit of a importance is Is isn't this principle of rain down organizes the coverage of cancer of the convention as always will be incredibly complicated when the care you know anybody was wanted to scare 0 few sure I think Ordinance you you have a never-ending series of calculations that leads nowhere so it's very important to have a conceptual understanding of of but anyway so this is the principal offering is essential swatches of doubt in addition of course you have connection coefficients some connection coefficient thousand 0 never mind exactly which ones but some are also 0 Tuesday important inch watches In addition you have that style right so you have that most that Ross Yechury 0 inch watches and also flares the commission provisions also would this so many things or if I'd only known vanishing components of gunmen would be well within their means I don't necessarily want to write on working with them that there are fewer and fewer of was a shiny costly space in addition you also have the do always 0 and then this 1 says 0 this store and your left just oozes through which are just Major jet these are just 2 overriding themselves softball calling and so this service into the country when that's that's kind of the story this is serious this friends having both now now it is and you want to think about the traditions of cancer 100 added to think about the division of Guelph contributed Friday to look at the level of domestic but that's a mess you don't get anywhere right so this is in the debate you imagine always space jobless face that Jesus was eventually which is where you have a frame which is Clausell principal enough friends other words you you have enough 341 at 2 such said all connection efficiency and the the curvature components but you have here now all all vaccine on the relative to a small part of that's on the 2 having the problem and so this would be offered similar and also that the part of the coverage which is not zeroing care you you want to say that the is a little from the position all the old kept for this would be the sole fits here but and then opportunity arise unionizing questions that we had before in exactly this kind of equations UT arise by saying well done lies begun on campus income at the event where all the all government these policy on alcohol use I can't exactly the coverage of comply plus something which is and of course you get the system of equations and now a billionaire eyes level which looks like this but for the Southern Union quantities it is very complicated large extent because of the presence of these things because this is a copying you knew this was a company which is very hard to understand In the costly space this would be a century 0 and that's why the subject costly space could be done but that's much simpler but in this case is it's far more complicated well so immediately read to you a little bit of what we see no not 1
important thing is that you see in this definition Is this
definition you can't say whether this trend is obvious manure which friend I'm talking about it's a very very frightened off which is good up to all the was 1 thing that you can
try to do is to see how To change to go if you want to go from 1 friend to another friend but a friend transformation and here in the story everything which is often seen on squared is good because all the square times are the times that we love right we like ammonia glitch but of course yesterday the complicated obviously but at least you unique scene seemed to be adjusting the problems of something that you would have some control readily Connecticut that I will neglect and then look at look at the only times like this these are all facing why that this is very simple and you just write down what the general transformation which preserve enough friends I want to go from 1 of Franklin was enough for me now that the matter is that he is a matter of fact which I think was a fundamental role in everything that is done was near its gravity he said the extreme now components of half a about half of our work for the resist this what this was all farmers in the face interchange E-4 was really the so-called of about so these components I invited after all 51 square dance with respect to this kind of transformation and so so in other words they somehow don't care about the gage I pick right the allocating level of a right so this is very important because this is something that I have to start is otherwise I have no idea what to do it all as any encourages components of the colleges or otherwise of the Jan knocking by from it's also important and now based on the observations of the Sokolski was able to show in 1973 that this convertible bonds which I actually verified the copilot but among conservative media wave equations and sinners or there would be severe wave equations with all the times which Arnold conservative Donald Lagrangian other conservative 1 now I have however they were extremely excited physicist very exciting said debate centered uh so they immediately declared victory an article costing millions of people who were more these and that but you need a lot in the physical integer that somehow this solves the problem of stability completely I mean even tho this is just at the level being established I was actually waiting made prove something using these prove that that'll cost equations of this way ?question mark 170 and knowing spine-tingling knowing nods yes to that of Blair also recognizes is that this has afraid about the change of venue is reflected in this is his rounds this policy was I needed that the whole point is the flying violence there are certain components of the country this is going very much the fact that composed the collision which do not depend on all of these on on the stock otherwise you wouldn't be able to do anything at this level anyway so so so using this Whiting was able show that the Indian denies that it would mean everything which is exactly what they will that going malts so it and then in principle 1 can show you now using gages it all as a curvature inclination components can be controlled once you have these unwanted control far behind to this 1 they principle can control visible .period don't even this is not a sign but the question is does this solve the problem and of course we all know the answer if lot of expunging going molds for the unionized equations was enough to that is known in instability than the presence of shock waves extreme sensitivity of the data In turbulence influence would be ruled out because in all those cases we have more than that noise .period Ewing want to ensure that there are are no growing of in India's history and the 1 so is this doesn't solve anything but of course it was a very important it was a very important circle of observations and ideas anyway so what are the main results known stability of course we know that was Makovsky space the end which I should say that uh what was important was maybe not so much just the result but the fact that it uses a geometric methods which turned out to be important in many a promising but it's also important in this problem but the classical vector few methods non-trivial views of characteristics that the noting that is of privacy it is fundamentally important citizen was that you use of a useful uations this by the by now characteristics the General Eisenhower conditions which also played an important role on a fundamental reluctantly and some kind of very complicated bootstrap argument which of course by now we all know we can do it in many people from whites so now that for the months of the confinement and mention form stability so this is a stability that I mentioned earlier that if you fix it can't solution and look at the way equation just standard way which Monica solutions then you can show but you can show sufficient information about the solutions which are consistent with what you might need in the morning a problem of Kazunori virus which were difficult because you don't even know that equations but but but this is extremely important that these to study the understandably aquatic life and the founder wave equation newcomers than anything and so this was enormously important all these techniques which were developed by these
people aboard there was a new event that he had a message which developed as a consequence of this which is not just based on the military but is based on many of the geometric features of black calls for example that the horizon a achieved the horizon there is something that has to do with their goal region that the giants of the collision is something that has to lose a propping which I told you that trapping means that you have this knowledge of this basic seating and lead to all sorts of technical difficulties the FAA region which has to do is something that in a sense you could say was already understood in the summit of legal system airing more the most
and ahead I mean I mean it's not certain that provide this he stated that is very general very gentle very so um so what does the results we know that there is a scattering construction that dynamical block holds by government hold together with the unscathed but which means that you solve the problem for infinity rather than from initial data this is much stands out to be much easier but still quite remarkable that it was done then I these days as the result of Humiston myself which I call the week magic is on 2nd which is based on some posturing unionization you you don't solve the 4 problems but it's also something which is still in Armenia so in a sense it's it's a better approximation that just the wave equation on on and on the can of kids solution it's something new which is the then there are the reason for the model results of the muscles altogether the unskilled just appeared at the beginning of the year which is the newest of a fortune and finally work in progress of substandard myself I on the newly-established your Schwab but nonunion anywhere without trying all luck with full stability for our city polarized so this is this is actually In terms of stability of Cosby established also plays a role here but this many stability in Al-Qaeda's would be sort of a subset of that but we're trying our luck was footnote the From now the
thermos once again erroneously so good as in the central point of their approach which is of adjustment should is a remarkable discovery of new which is in a standard two-tenths of a daredevil stood a medieval courageously member of file about adjust components of the curvature and becoming the Kokoszka questions so yeah you take something which involves stood in the facility which turns out to watch very fine conservative equation in other words it said it's a quiz which is just the way the question each watches applied to the speed plus a potential Tennessee in this this potential has good properties something isn't it is is called with a potential is so this has actually good energy estimates and more evidence was also you can start doing applying all the technology developed in the last 15 years which only before In order to control the decay properties bombers indicate properties of the speed once you have that then you have to deconstruct all the other quantities and PE of course is just available to the people will 1 component of a college said to go back and go down there and get all the other quantities it by of those squandered out which are gage dependent and therefore you have to make a choice and of course that's a highly nontrivial but in fact remarkably they the this other thing they defer to a competition of termed uh elaborate argument proves its which to me that it's like the most important thing they say just the calculations and by then the list of the deliveries about 170 pages just do this listen this stuff so it's a very it's a very complicated things way
I know that I said I want to talk about that images I want to see the Pfizer the problem symmetry because of a a lot simpler I missed it explain why so Osasuna Jimenes means that you assume that then it should still on looking at each other divisions of of Kent Bush watchers have an additional symmetries which corresponds to a cannon but the fusing of course get itself has its unity and its political so it intense out dead using is easy you can construct the so-called immense potential which is a complex scale so it's exports Iwai wakes his just due these intelligence to something as simple to complex and as the canceled 580 wave equation Soviet-type it's it's actually the nonlinear wave equation in fact it before people will recognize immunity that it's a wave not whose values hyperbolic space until it is
part of the same introduction they consolidated their work a little harder and good decomposition but also the matrix of Dimitri decomposes itself into a 2 plus 1 entry and the scale of this complex and no navigate the wave mark which is this 1 but you know if you know of course you to the wave of possible the curvature so you you have the reaching part of these 2 plus 1 metric trees here is Stites to wave marks receives relations before people I should say for people who love way months of receipt usually the way mountain goats space and there are lots of beautiful results which were approved in recent years but this would be a great challenge to apply their message the Waveland the 2 waves of which is coupled actually was admitted this is a in a system is much more complicated there but only an interested in something that's here when it so this is the way of now she had 2 problems that could have been working on in recent years so when I suppose Jews with equal 1 of our number 1 what you do but again I cannot do this stability injured it's still
immensely complicated in this case but uh but it you you can try to simplify things a little bit by keeping the fight to satisfy this equation and simplifying the metric to be the cabinetry that they deem to be Kennedy that's so then EU
you get now the system the duties actually feet and you have the best potential carried which exist other care itself as an assault sugar was made to the perfumes that sits in a chair and you look now for solutions of this equation so this is the equation dining components for 5 this is how it looks Enrile and then the solution that you already know that you had which is exactly the 1 although them as potential that can connected so in other words you have this system of ammonia with ammonia wait months but you have a solution a specific solution when it takes to be a N and Y to you get a specific solution all you want but that so this mother analog mother from a number 2 is if you assuming additional you assume polarization so if you go back to if you go back to what
we have here I I'm going to assume that what is equal to 0 and it's easy to see that the
system is consistently another was like What is your isn't true solution so must not have enjoyed was your items they way equal 0 so this is this is actually when you do lot Cincinnati polarization you really it's honest this not because you are to be sold to with the 2 1 cyclist right other so the problem now becomes when equal to 0 so this the simplifies a lot so that the waiver quit serving a public system is now just 1 wave equation which is the ambition of flies the I find like the statute console explicitly but exponent shaving and adjust their social standing wave equation and of course you you tied to the Ricci curvature of the space-time matrix of analysis at 2 plus 1 metric but you tied by about it so this is a system that you have to solve understood this liquidation and it's easy to see that in this context the final status to be swatches so it is really a problem of the stability of Fortune solution in boxes Najibullah it's but so
uh so there conjecture number 1 which is probably 1 of men along the stationery wave is stable and out generosity is missing but divisions so gained given small initially thought close to them as potential care policies I have a surface at Macy's the global solutions such as Pfizer could rise in on my colleges with rivers and unfortunately I don't have time to talk about this I just want to say that went on on this we haven't made a lot of progress but we proved that years all the important thing is that answers a school so this is the conjecture is a very consistent at the senior level and I have my student John Solomon is now doing the full only these aren't so this is something that was a grandest until got to but I won't have time to talk about these dozen interesting things about that but I would have done so if we let me go back some of from number 2 which is a work in progress returns of 10 so this is a game this is a system that you want to answer your before matrix of this form I still fly this is the David with respect to fight exactly the Killing vector field so X's independent defined and matching G's independent of right and you have this nice reduction and the questions me look like this and be that 2 plus 1 dimension of aggressions now in principle you could think that this aggression so simple and there should be no problem to solve this which is what we sought but it turned out to be totally wrong and there's nothing you can do Was this equation if you try to lean arising diplomats said you never get anywhere so so we really have to go back to the stand-up picture using curvature and connections also which is what did and so again we
introduce framed by a similar to before it wasn't enough Rainey's 340 keyed up but now just isn't it because of the reduced picture you can do this just isn't to brass 1 picture which Singh flies all of this components of connection and curvature I seen very badly with our skill now so this is a skill this is thin end anyway don't want to bother to bore you with all the things that makes the most of what we have this fraying but you can write down this Sunday questions for details but I don't I mean I don't confiscated them a bit complicated by 360 takes just a little bit too adjusted
but make is I want to go back to this business Salfordville Bill Kowalski question so now are the still Sokolski equations which now asking so in my case of half of our skins and these are the questions that actually satisfied I can't have our game just like in the Jones case they they would be invited up to offset all of its announced was about to be violence all the problems up to the terms of 4 all all factions clan and they this this kind of equations so this is a founder the wave equation actually you can't ignore the metric to be the fortunately confined to this you conceive being swatches and these are some of the unit itself off about the people of Ireland similarly here it is very easy to see that this aggression is more conservative so there's not much you can do at this level so
I said the idea is to find the quantity to find a good quantity which arises is the only component of the curvature witches said that the 1 which is close to care is non-trivial and I cannot meet saying that these costs to all carriers kind of misleading because raw care only in a system of coordinates I can talk about the true value of but I don't have a system of coordinates the so I cannot even talk about it right so the issue is how to lean arises all doing this have to find a quantity which is invited and which is to launch which is small which is the on charges and so this this was sort of a major thing for us so we found that there is such a quantity which is at the level of it is the level of this X which was was a 214 In the previous slide and seizures the releases Eliza clearly call violent expressions of this way expression miners role which is the 1 it's not unionized miners is this what it intends to be 0 wanted has to be any viable for the glad right and it satisfies the way a good way that another good way the cushion unfortunately but 1 of which is still the way immigration to and it's unfortunately it's Steve tied to that quantities of about so we're hoping to get something which doesn't have anything other than suddenly just over square by then to be much more complicated however there were some very simple relation that possibly between this from this quantity which have been introduced our final about just candles no simple all these like this and inspired by the fact that we knew that the firm was halted Williams the already had a good quantity which was a at the level of the secondary to fusion which was level secondary we were we realize that but we had to do it's just to differentiate base with respect to screen then the French was strictly for commutes commuters get rid of the stance because of these relations and you get finally which which looks very simple and very feisty it obviously it's a way that which was a potential the potential is a nice it has good images tonight and if we can do more of violence in the things and once you have done then you have to work a little harder
and get I get know the full story about that approach would be given the time I had to stop so subject and if you want the the can on the question of what the final say that yes this is the reason you didn't know I didn't I didn't say I said that I want to prove that its unique London convinced of the adventurous and losing to the local news stations the Kuwait that did this is different than it whether about 601 is whether vows it any association states besides care that said he didn't want it and in so far we don't know and we have only partially his of of the usually village which took the form of food across the whole of the the use of force in the north of the version I have to ask people about the timing and really we are we have unfortunately where we are still very far from that that kind of result unfortunately so far so we are really had 1 body problem and promised when it must undergo G R is just at the level of the 1 part this that two-body problem something that maybe if we go to the Tuesday called the world we live and as it would be very very nice if we start getting into into the two-body problem also but but so far no no I haven't seen any serious attempt into the that in the mathematics more opponents Of course that needless to say that there this experiment this recent D'Amato explained and like is betting is that it 3 assumes there is not of course on medical experiments impressions that the two-body problem is stable and therefore In particular the 1 the problem will also step so that stability is not on the question from from a physical point of view the state but we don't have brought about because we don't have any CIA spokesman who results at this point In or of