On Anti-de Sitter Type Space-Times

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On Anti-de Sitter Type Space-Times
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as tell them and I the I'm grateful and I think it's a great privilege to have been invited to speak at this location the Yvonne Haddad has been holding the fault I think that's something 1 can say for such a long time In the beginning or by hand said now fortunately there are many young people and who are going to carry on but for example was talking on the pitch and that it could have and I I guess says is that there's nobody besides the Yvonne who has looked at the patients from so many different points of view so the I'm going to speak about and it types based had about a kosher problem that is of course the most important thing we have developed a list of the characteristic parliament that is the title was about characteristic to top wasn't and now this is essentially a talk about the initial boundary value problems so I start by saying something about a 20 dissidents 20 years ago and is it a space were was a very quiet and peaceful space and if you did some work on it you could be sure not to raise too much discussion then came on the scene and then came the immediacy of the and now there are thousands and thousands of references and you're quoting everything has been solved but the question which us by the classical mathematical relative is arrested and salt and I want to talk about some of them was stopped by recalling what kind it is that In Mitsui and equal to fall larger space-time dimensions it's given them by this many followed and this line element where art is a standout radio call on with component it's solution .period sense the declaration was negative cosmological constant if you do on the income from rescaling and an associated quality transformation have written down these things then you get this really complementary presentation of this metric and 1 of the important features is if you look at this thing you see this as the lion element on the sphere and you can easily extended mostly the to the Sydney Steel and those things then lives on hot cross half of this year with the boundaries included in the boundaries of his differ Moffett to close this year two-dimensional so that's what I to do adjust basis so I had to tell you what I understand under on do and it is at that top space I'm thinking I'm not doing this in the form of a and principally these things have been around for 50 years and had don't want to repeat everything so Irish I referred to solutions transcends the education with negative cosmological constant which ended in a similar way moves come from extension which it time-lag type surface group presidents time like infinity as an 80 as the tide space I want them there are various bases to generalize CDs the 1 can often find the word asymptotic the ADS space this would be 1 of them but 1 could think of that can notions by which I mean you have less regularity I had to at infinity this is not important for me for me it's just convenient to use this definition and I would stick to it but in principle 1 might think of generalizing about everything going on good to talk about now the main feature of the is that it's quite different from either the problems global problems and since they are quite different from those for this at a time ,comma cost type solutions show you the picture that this this is the 2nd formally extended and it is a test This time we have this time by boundary and it's clear and you find it almost explained almost in every article that this thing is not really hyperbolic if you feel it in any action not that high percentage of the there's always a possibility for a time just 2 minutes and boundary so that's the 1st statement and then there the 2nd statement and that statement you find almost nowhere except here and there and that is 8 years does not didn't admit this finite conformal representational boss of future time infinity In this space you have this in of space you have this but here you don't have to nevertheless you can find sometimes is the recognition that people do from rescaling and then they get a point at the core of the future panic infinity and upon which occurred as tannic infinity but the smallest completely useless and misleading because it's not slows and the point of the conform extension is that its moves go this is clear that this poses problems this is not so clear but I guess those who maybe and 20 years 100 years going to prove something noble about 80 as feel that this is important OK look everybody do about solutions but there's a certain history to this fair feminine grace later in a similar way ingredient here actually
dates studied formally expansions they assume it does type :colon its system based on the confounded boundary and study former expansion in terms of the ongoing cornered they get Taylor expansion at the boundary when the space-time dimensions even and they get a putty homogeneous expansion if the space-time dimensions on the podium with genius meaning that expansion comes off a radio call on it and flow Greece and if the if the doctor on on on on on this boundary which I squad on another date they get really analytically solutions near the boundary that may be useful for some purposes has been could considered by by quite a few people that's not what I'm looking for the pot for summer reasons 1st I think city is not to a requirement which I like it's not fall apart this important secondly what really what they are doing is they study kosher problem missed out on the tarmac campuses this and it's known that the Scotia problems are not well policy and what is even more seriously if they do this expansion was .period on and then the extends into the interior there is no guarantee that this closest to interior or can be extended to another boundary at infinity there's no control of the so that's the reason why I'm not looking at this what about what we have to do instead is to studies in which a boundary value problem there we look again at this picture he said are the prescribed .period on this space like how but 1st surfaced which looks like the end on this time like boundaries yeah the basic question that this hall on the boundary conditions and that could be formulated to obtain would oppose the measure related problems for instance coupled to at future and of course we want these problems to produce and it is at time solution there exists quite some regional contests views on CDs on hasn't publicly idiots background or maybe some sense hasn't ordered ADS because I've listed here if you will maybe fuel which escaped me I apologize for this kind of note but complete it's all interesting and important role they discuss various didn't and pose and reverb on the value there you problems based on various choices of boundary conditions now the boundary as he is defined here by the terms of its components structure so it's clear that the component behavior of the test fields did play a role when you discuss the boundary a bigger problem if you have come from the covariant few declarations like next notification of young men equation then this boundary is as good as any other time boundary the Commission's just don't feel but you could imagine that you have nations which do not end Dec nicely was conform rescaling I call this conformity behave which is the notion that this is defined but there is still no 1 can imagine that 1 has such things and in that case a discussion really fairly difficult now if you want to say something about Einstein's occasions then you don't want to fiddle around receives complicated situations therefore I look at the VA :colon case for it and I wish I recall it pre scene results I mention this because if I had decided to adapt do do this after Montesinos and have been had been admitted in principle this idiocy 57 I might have passed different questions but that because this was before I just stand questions 1st how many of these issues solutions exist 2nd how can you correct them in terms of start-up and answered that wasn't his office's initial boundary value problems there was almost nothing known about general initial boundary value problem and this is a very nice example that comes in naturally in a geometric and that that I found interesting at the time OK he has said this was in existence resigned the truth is positive and negative land that's fairly easy then the consider three-dimensional computer there so there a history of measurement avoid there's a symmetric and there's a 2nd fundamental form for this occasion and that is supposed to be Oriental blow open and has had a chapter is supposed to be complete and the whole thing is supposed to have his moves conform the completion which is such that too as had the attention boundaries researchers zinc mine which is compacted compacts with many followed so that the whole thing is compact we assume that on ghostly defining functions of stigma and we assumed that these restating the result in a in saloons fields and this metric is to use the money generated by the bombers furthermore we require that the conformal abide by of which we can take a lead from the start that it can be we scared this component factory and what we get it has a smooth limited at the boundary so we assume such doctor on some space like lies then we need to introduce boundary .period time on all crusty we assume that there is given this three-dimensional Renzi conformance structure and then we have to say something all these things fit together and To this
picture again but we want to create this picture you have given something here and we have given something he and it's clear that these things somehow have to fit together and do you need to impose conditions see along this boundary that these things fit together nicely and the outcome of these conditions are referred to as ,comma conditions there
yeah ,comma conditions I am going to have to say more about
the current ,comma conditions later this ,comma conditions of course have to do with the few declarations and I should say I working computation the compliment picture I'm rewriting the the few dictation in terms of conformity and indications I have like all conforming to the vision the now assume that these things are given then there exists on the center of this film so it's as that should view it had nothing to do so as it's in a dialog in an open and other crops has stepped there exists a solution that should also be is moves conforming extension which looks the we wanted to do and they wanted to look at this solution induces on S & on this boundary Please forgive and gotta have to interview Mofaz again I referred to this picture what I'm saying is you prescribed .period here the prescribed got on the boundary and then begin distribution in some domain I don't know I don't say how far it extends in time it's just local the time OK the
death of 3 things we have to talk about we have to talk about the kosher Dr. HO do we get them we have to talk about the Lorenson Conte's former structure the boundary doctor wanted do we have to say about them and we have to talk about a common condition and I'm going to do that now
1st talks about the doctor fortunately that's easy as pie because work had been done already if you know it has been now the the opposite observation is the following if you want to construct ADS type solution which at times reflection symmetry you assume that the 2nd fundamental form miniatures on the lies and the constraint then reduces just to desiccation and then you have to wonder what are the USM Toddie conditions and you find out the identity conditions are similar to those required on hyperbole lighter which IPO surfaces in a space was vanishing cosmological constant which extend up to squat on the other hand these gentlemen studied logjam the elections to from this land they could 2 0 this is the assumption that the 2nd fundamental problem this Pruitt trays and the trace it said does not ban if this is so dissatisfied and is constantly and then the momentum constraint is satisfied and Anatolia constraint just reduces the and they discussed solutions to the switch to the special conformance and if you look at this you see if you have these guys you get those guys that's amazing observation if you have hyperbole modeled after the give few quotient just reinterpretation of some Const 1 dozen cost share the studied marginal hyperbole about 10 and you can't generalize this correspondent Jim principally from the point of view of cozied up to a year in a good position there are still generalizations are possible but I'm not going to talk about an interesting client as the topology of this boundary is not restricted by the by these constructions the boundaries just as the boundary often only into the 394 is an old-fashioned conditions and so the same as the the is fairly fairly general but if you look at that this article and in this article in particular in this article that being constructed marginal Doctor Doctor which our efforts basement Infiniti in the sense that you have to put Maginnis expansion you could think of constructing PDS Tetsuya from his motion that hasn't topics but this is not not so easy and didn't makes the slightest the attempt to do this if you want to establish this picture and you have to institutions which are off here and you want to construct a solution which extends to the future of the domain of dependency so which is going beyond the ongoing not those of you have to be very careful and arrange things very carefully it's possible at all such said the smoothness does not travel into space so I I didn't make any attempt but to everybody is invited to attend so was a
bounty . nice positions it was received initial that there's a boundary .period said it appears that the nicer position then all constraints because we just say boundary a scared given by conformist structure so it looks as if this even part of the whole thing now there are few certainties behind that and this is what I don't want to discuss customer the proof of this result mission coded consist of 2 steps you arranged an initial value problems put for the PC which meant and means and particularly you truly introduced Sunday's condition and so on and that had tried to get DVDs and a farm to which it applies no knowledge about an issue about boundary value problem and then there's 2nd step there you have passed into the smallest such that you get it could their information now boasts things depend on very specific features of the 80 s based on the 1st specific features of do not buy KEB and cut baby the 1st and 2nd fundamental form on the boundary so complicated the prison Lorenson metric then its effect that in a suitable component gage the 2nd fundamental form on that boundary ventures if you know 1 gage you find just that the Tracy partner nations that is a component density but then when you look at the transformation a trace India sees you can choose geisha section this thing this has has squandered consequences consequences and that has to do is gage conditions you have to to make a choice and you have somehow to say whether boundary what I use this used come from a Jedi 6 conformity the 6 are conformally invariant and the associated with a conformist structure in a similar way more like the Jedi 6 or associated with a symmetric but it's a marginal class of self-regulation anyway you can prescribe dont have for them that they start off southern into the new definitions In what you'll find if you prescribe that have such that they 102 finishes wrestlers and started a program where the initial slice is supposed to intersect the the boundary i they stay on that boundary and that means you stop them and they generate the boundary for you OK various things you have to do you have to to confuse the come from the fact that it supplies also nicely if frame you have to make a choice that I see in minute how you do that what you get is something which I call come from the gods gage and a innovative very similar to the usual Gul but I mean there's certain differences and what's nice about this if you ride on the occasions in you ailment Penrose imitation you could also write them differently there are nowadays all kinds of limitations around the venue dedication which looks like this the tall is apparently on the come from 6 X as far our constant components on the duties 6 so here is just a diluted with respect to the parliament along the Jedi 6 and you is a set of fields which comprise the frame coefficients the connection coefficients this respect to the free and the Shelton and so I'm not going into any details that's remarkable that you have this kind of propagation and then there is this tends to hear the have seen this cancer already before we can require this to be smooth so this is our basic unknown in the conformity education in this bill and notation it's a symmetric spinner if you write down components it's that 5 complex functions and PCI is supposed to be this vector indignation it satisfies all of this fall this makes it clear if you have the freedom on the boundary the freedom can only be in these functions if you want to discuss the freedom you somehow have to adapt frame to true to the dormitory and what I do this I choose this double 9 frame which is in the human panels in the patient which is such that the only nonlinear Spain a product of the CIA show that denied that a class then I think they then generated time magnate Bill which is future future directed and attendant to the bunker the difference of these vectors is supposed to be nominee to the bond rating in or pointing and if you have this these 2 guys attention to the boundary and if you do this then in India and the associated human Penrose invitation you you find that the equations MIT boundary conditions which all of this fall to fall minus some function acreage by 0 -minus The Times players your body is equal to the where the functions and agency are subject to some restrictions and the use of boundary .period and that can be prescribed completely free the there's no constraints on the I want to explain roughly how this thing comes out we take the the antique patients in study patients satisfied by the by the stance of W these 8 occasions the Sept splits into 2 parts 1 party has covariant derivative into the direction of fans and to maybe I should draw that they should by the way we fixed everything and is also what pointing he is
element forward and the maybe have chosen and I 1st things look like this and invite pointing an assault pointing this means there those fields which occurred here cannot be prescribed
the already fixed by what's in the interior you know you're not allowed to touch them this operator is In what and you see upside does not occur here so far as the 1 which you may prescribe and you can try to feed and information on so 0 India has the impose conditions and agencies and you can do it in such a way to get Energy estimates that the Basic of the existing through so we have finally they have problems that's why the problems and what small it preserves the constraints engaged conversation that's a constraints are preserved this initial boundary value problems much more complicated than in the initial value problems so we're lucky that rots in that case well not contiguous producers there's just 1 1 problem that I have to choose an interim and I've chosen them such that plaster his attention to the boundary but there are many times like make those tension to the boundary and there's no natural way to fix it unless you have something like circus cemetery where there's a unique time neglected orthogonal drew 2 2 the orbits of sociological so that that's that's a problem and that makes the formulation given their not not covariant is that 2 guys they indeed they tried to produce a space-time space science by following these recipes and if they are asking how they do we get the same solution or Moffett solutions are not accept feeling uneasy to get an answer but in this case healthy ATS type solution it is possible there and I think that's very specific to EDS space time and realize on the specific features and that's the following month by BA BE I know that you alliance could talk and of symmetric KB on the boundary so that's the object which is it Greenwich estate the boundary is conformity by W. start maybe I do know that the magnetic part of city life attends W lists respect to the boundary that is I take the one-sided view of this ,comma make a contracted tries in upon being nominated unit ,comma and then I get that object that's as the spatial stenciled on the boundary and a special feature of India's touch space is there you have such a relationship so the Cup contents or on the boundary is related directly to the magnetic apart from the bombers OK now what you can do that and supporter and you look at the boundary conditions and you pick a particular choice of these functions agency which were fairly general you find if you take this particular tries then you can rewrite this in this very if you use communications so of a real and imaginary component of the and you see this is sitting in a combination of the electric power and then you find something notes that's a lot of this is really amazing how the patient know how to do that if these components unknown to you and you look at this differential identity which in any case then you see that this thing reduces to about hyperbolic system if the background is given not of Hugh need to integrate the background as well and you know Karadzic structure locations for the normal conform a couple connection on the boundary and you get a hyperbolic system which allows you was that had given on the size to integrate into conformance structure of could be on the boundary this just effect and secretly this user's again this condition was 2nd Ltd conversely if they come ,comma structures given to you you know now how to arrange the dig the gage which I discussed and there attended is the whole thing tends to that these kind of boundary conditions are together was initialed either equivalent to giving the conference structure so isolated that motel complicated than 1 might think I shared use this relationship again because it is the making things a little bit clearer but juridical variants of the the come from the structures no I want to talk about protecting boundary conditions In that sense if you said include 2 0 any of these boundary conditions could be understood as reflecting boundary conditions the sides which you may treat prescribed stress obtained as a genial conditions of the things which are public transport but you want to have gage independence that's the reason I look at this moment conditions and I require this so Quaker equality on the intersection of the finishes lies with the boundary and then I have this these things together imply that so I will refer to this condition has come from the area said reflected boundary conditions to this combination good corner conditions we can make it easier if you have kosher .period you can Cagle it's a solution at any order you want and the former expansion is determined uniquely in terms of the gage on the other hand if you boundary .period also a across a former expansion in terms of optical elements and what you have to make sure that these expansions coincide and Bledsoe steering says this can always be done and then many diabolically doctor which you can prescribe so that this is satisfied but I should say that the this little the now we have no other candidate in existence uh problem now reside locally in time and no you may get more ambitious and do you you of course would like to have nonlinear stability everybody wants to have city-state's so if you have of this you also want to now the 1st thing is to doing our stability that has been discussed but she that she had I don't want to go to and into the recently that the different roads and other related to nonlinear stability aspects which had to do was immediate and both assumed asparagus symmetry and that's a good idea because it simplifies the analysis which so simple anyway and it may also simplify the numerics if you do from a new mode calculation so this implies come from the flatness of the boundary so that smallest reflecting boundary conditions and they have a scalar Fiat to retain some dynamics otherwise there is nothing in there is a result policy good and some will be easy it shows the stability of fluctuating EDS for spherical symmetric and gun system Misamis entrance on the climb government it's an interesting results I'm sorry I'm not going to talk about it before the following reasons it has an outer boundary which corresponds but it also has a new name Valerie and a boundary that will rise and so .period tational radiation can escape and this makes us problems are different from the pool has problems and a phenomenon I want to discuss to simply not occur so I forget about it and to consider some work in which only the outer boundaries consider and in fact the analysis stopped as soon as some in inner boundary is going to be that that's work by these long and possible loss of the they studies very symmetric and 7 at scalar field system is run by the crowd of negative homogeneous satirically hasn't topics for FY the news doesn't happen this year .period time category it's a solution American and what they find they found find bed 5 attendees small initial they can fall on trips surfaces this is I think an amazing reside was a little bit carefully here attorneys small in America this is something delicate but they married they represented there .period has very convincing that should be should be true he was out the door the solid these guys they perform a pocket of a different audiences that smallest pointing into the same direction but it is exhibits also some on an issue about which seemed to develop into globally small solutions and then they give evidence that the development of trapped surfaces as they observed result from an energy transform a transfer from low To high-frequency Moats a that's a lot for them for those who haven't seen that before but that's how it is and this immediately at the started the made on other people .period American at the soldiers and sometimes they also do their part of the analysis was reflecting boundary condition for the food Eisenach imitations and they get similar the conclusions in a sense is this what they do mimics what their views on the last ball skidded and high I would say there's lots of of space to to do more complete work on locally 190 print they also detonated taking nations they reduced produce results was a complex in you and they observed and this is land that modern shown that that if you have got that close to the specific about which had been exhibited clear you get global existence again in the American cents but to nobody seems to have any doubts about it so data reside reflect and also offered to confirm conjecture ADS is unstable gains the formation of bricks holds for a large class of arbitrarily small perturbation just to make it clear I think that didn't a generator later I had pointed out that there is a class of of of of of downtown which seemed to be coming into develop into something no I don't think so I don't think so there is a will there is a kind of an island of stability and apparently good so I that I should say not to live room and generate a wrong impression that I find these results extremely interesting because of the bad they have done the questions they raise and I have not cited doubts that this is something pointing to something really concrete however the problem with his statement here if that is meant to apply to general perturbations it may be too strong I think what the dead rather suggests that ADS was reflecting boundary condition is unstable against the formation of the codes for large class of her arbitrarily small perturbation no I have been wondering all the time and when I looked at papers about it figures that most people immediately look at reflecting boundary conditions they take that natural some people the even to this is conditions as India's boundary condition it's clear they are very convenient you get a very defined closed system was no information coming in or going out the question is is enough reason to concentrate on this or should 1 look at these things and more generally as this is the end of the year and the facilities of the original holes for small they hearings itself is yes there was a for a large class is somewhere wake here and that this may change if unit for marginal boundary conditions I I'm going to discuss this so I can understand that people consider these boundary condition and in it's safe it's an interesting problem the only thing which I find it disturbing is that may generate generate the impression that's all wants to say about so aptly Lulea these reflecting boundary conditions are not part of EDS you put them in by hand opossum from there on the Mall but the intrude on explored difficulties and this I think it's something for John he's going to have to list on the could looming difficulties that this is 1 of them the difficulty is that this is the point we are not used to from the coaching problems that you Yukon separated the province into parts disillusioned problem and as a constraint problems and there is a clear separation between of course you have to show that the that the constraints propagated by that usually is not too difficult and after you told us how to do it no but in this case we have a problem if you have an initial boundary where the program and impose restrictions on the boundary .period tell this separation cannot be maintained and long and an extreme situation the following if you have reflecting boundary condition is not that they only prevent the flow of gravitation in all out of the system across the boundary what they do all they require the Koshy diet that to satisfied beyond being hyperbole loaded beside infinity rather strong additional form of conditions at base that finish and this comes about the music theater and an immediate that this comes about as it follows everyone to have reflecting boundary conditions I told you will be a great requirements on the section and section of the space like lies in the boundary and we want to have sci-fi sequel to the and perhaps most solutions you have to require for all now if you now look at the wood messianic nations you can take late expression for this in terms of the the initiative and that means you get an infinite a sequence of all off differential conditions on the initial .period so trimmed tomorrow you that this this is a complicated thing I've not the slightest idea whether 1 could do something with a component and up until fairly recently doing was also not so clear and maybe something has been done I don't know so this is a real problem and that I did not catch no we have had this in other situations where we prescribed Koshy .period and doing it would usually may be assumed some conditions besides fall off and demand was no problem but the situation is different here then we just describe the starter and let things go and the Commission said OK I'll do something this year we insist on this at all times does so we are constructing the occurs in this set of reflecting the Italian on this set of kosher .period which satisfies these conditions so the question is these additional fall of conditions do they possibly contained the seat of formation of services I have no idea but the possibilities in that problem because studies by bees under DOS Lonski U.S. situation solution is going out is reflected to going out is reflective and assuming that the observation is correct that the dissolution is reprocessed in a way that the that that the energy is going always into the year I am Oates so that's but what's done by these a doctor I have no idea and in which way it comes about but I guess it may be a real problem Anderson very interesting but also difficult 1 no good the other problems which they're not more definite less definite than that 1 I think if we have fixed celebrated on reflecting boundary conditions that precludes all kinds of investigations often more journalism durations situations which are possibly of interest in applications having said that you could ask Are there any applications of this and that is 1 of the main problems what is the meaning of the stadium touch solutions some people say saying all they do all the crew represent the isolated gravitating systems sound people talk about bones instead and then some of them still refuse reflecting boundary conditions if I have an object which I call a star whatever it is and it it's confined by the reflecting boundary conditions I think that's a physicist can forget about it there's no way out of this thing to interact with the rest of the world that so if you if you'd suppressed this injection is not so clear just to say what it is again from the mathematical point of view of all these things are interesting and maybe it's a 1st step to understand the whole thing but there is a question and 1 of the questions you have no idea what this thing means physically which gives no hint of what you should require what would be a reasonable assumption there is another problem if you look at this at that time only ties based on their that they conform the boundary it splits into 2 components there's 1 component where accreditation can enter and there's another component red leases space he is 1 component and principal radiation can end suspects and complicated is to make a distinction between entering and leaving radiation OK and then the question is what do we do Of course I don't know I don't even know what stability should mean in this context that might advice would tried to characterize those boundary conditions and out which solutions which stuck close to India's stay close to 80 years for our times L this is a task is not to easy but if 1 could correct devices 6 . 2 1 could also get an impression of what's in there it is so usually any food and how we should understand this and what I do think if you look again at the might begin a remark in the beginning if you look at the global cause obstruction of ideas I think it very general only be reasonable to require global rebound the time may be relative to EDS I mean I don't think it makes sense to require that things are the following falling off and go to 0 as he goes to infinity to homeowners OK that's what I wanted to say the LAT and on and on and world peace it probably will be on the and you I don't know you mentioned that this move will go to the heart bond of this policy and now I mean if it's in he wanted to have some sort of smoothness and then he is forced into it there may be other documents I think what you have to have some energy defined on these things and that energy which they defined only finite if you impose such boundary condition is that correct so there may be different reasons why you do this but to me that's how it is all right thank you at hand


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