 Yeah, I'm grateful and I think it's a great privilege to have been invited to speak at this occasion Yvonne has been Holding the fort. I think that's something one can say for such a long time in the beginning all by itself now fortunately there are many young people Who are going to carry on? For example, we had his talk and I guess there's nobody besides Yvonne who has looked at the equations from so many different points of view so I'm going to speak about anti-desitter type space-time We heard about the Koshy problem. That is of course the most important thing We heard about characteristic problems. That is the title was about characteristic problems. The talk wasn't And now this is essentially a talk about Initial boundary value problems. So I start by saying something about anti-desitter 20 years ago anti-desitter space was a very quiet and peaceful Space and if you did some work on it, you could be sure not to raise too much discussions Then came Maldesina then came ADS CFT and Now there are thousands and thousands of references and you could think everything has been solved but the question which are asked by the Classical mathematical relativists are still unsolved and I want to talk about some of them I start by Recalling what anti-desitter is and Let's say n Equal to 4 or larger space-time dimensions. It's given by this manifold and This line element where R is a standard radio on it on this component It's a solution to Einstein's field equation with negative cosmological constant If you do Conformery scaling and an associated coordinate transformation. I've written down these things Then you get this really a conformity representation of this metric and one of the Important features is if you look at this thing you see this is a line element on the sphere and You can easily extend it smoothly to a semi-sphere and The whole thing then lives on R cross half of a sphere Where the boundaries included and the boundary itself is Different morphic to R cross the sphere two-dimension law so that's what anti-desitter space is I Tell you what I understand and I did the anti-desitter type space time I'm thinking I'm not doing this in the former way and principally these things have been around for 50 years, and I don't want to repeat everything So I shall refer to solutions to Einstein's field equation with negative cosmological constant Which emit in a similar way a smooth conformal extension Which adds a time-like hypersurface that represents time-like and null infinity as an ADS type space time Well, there are various ways to generalize ADS You one can often find the word asymptotically ADS space time This would be one of them, but one could think of the weaker notions by which I mean You have less regularity at at infinity This is not important for me for me It's just convenient to use this definition and I will stick to it But in principle one might think of generalizing or everything I'm going to talk about Now the main feature of ADS is That it's quite different from either the problems global problems on ADS is that they are quite different from Those are for desitter type or Minkowski type the solutions. I Show you the picture That is this is the Conformally extended under the zeta space time. We have this time-like boundary and it's clear and you find it almost Explained almost in every article that this thing is not globally hyperbolic if you feel a in any Accurinal hypersurface Yeah, there's always a possible for a time-like curve just to vanish at the boundary so that's the first statement and then there is a second statement and That statement you find almost nowhere except here And that is ADS does not admit is smooth finite conformal representation of past of future time-like infinity in Desitter space you have this in Minkowski space you have this but here you don't have this Nevertheless, you can find sometimes in the literature That people do a conformal Rescaling and then they get a point and they call this future time-like infinity and the point which they call past Time-like infinity, but it's more or less completely useless and misleading Because it's not smooth and the point of the conformal extension is that it's smooth This is clear that this poses problems This is not so clear But I guess those Who maybe in 20 years or 100 years are going to prove something global about ADS they will feel that this is important Okay, what can we about do about solutions? There's a certain history to this Pfeffermann and Greyham and later in a similar way in Greyham and Hirachi They studied formal expansions They assume a Gauss type coordinates system based on the conformal boundary and study formal expansion in terms of the Ingoing coordinate They get Taylor expansion at the boundary when the space-time dimension is even and they get a polyhomogeneous Expansion if the space-time dimension is odd polyhomogeneous meaning expansion in terms of a radio coordinate and its logarithm and if the if the data On on on on this boundary, which I call scry Analytic they get real analytically solutions near that boundary that Maybe useful for some purposes has been considered by quite a few people It's not what I'm looking at for some several reasons first I think analyticity is not a requirement which I I like It's not for our purpose important Secondly, what we are what they are doing is they study a Cauchy problem with data on a time like hypersurface and It's known that these Cauchy problems are not well posed And what it's even more seriously if they do this expansion With data on scry and then they extend into the interior There's no guarantee that this closes to a smooth interior or can be extended to another boundary at infinity There's no control on that So that's the reason why I'm not looking at this What what we have to do instead is to study the initial boundary value problem Where we look again at this picture here Where we were prescribed data on a space like hyper surface which looks like this and on this time like boundary The basic question that is how are the boundary conditions and data to be formulated to obtain well-posed in national boundary value problems for Einstein's field equation coupled to suitable meta fields and of course we want these Problems to produce anti-dissicter type solutions There exists quite some literature now on test fields on ADS or on asymptotically ADS background or maybe some sense asymptotically ADS backgrounds. I've listed here a few Maybe there are a few which escaped me. I apologize for this. I don't know whether it's complete It's all interesting and important work They discuss various ill and well-posed and little boundary value problems based on various choices of boundary conditions Now The boundary is defined here by in terms of its conformal structure So it's clear that the conformal behavior of the test fields will play a role when you discuss the boundary value problem If you have conformally covariant field equations like Maxwell's equation or young mill's equation Then this boundary is as good as any other time like boundary. The equations just don't feel it But you could imagine that you have equations Which do not interact nicely with conformary scaling? I call this conformally ill-behaved, which is a notion which is ill-defined But still one can imagine that one has such things and in that case the discussion will be fairly difficult Now if you want to say something about Einstein's field equations Then you don't want to fiddle around with these complicated situations therefore, I look at the vacuum case first and I Shall recall a pre-Maldesina result. I mentioned this because if I Had decided to do this after Maldesina and have been had been Influenced by all these ADS CFT stuff. I might have asked different questions But because this was before I just asked the standard questions first how many of these Solutions exist second How can you characterize them in terms of data and then third there was this initial boundary value problem There was almost nothing known about general initial boundary value problem And this is a very nice example that comes in naturally and in a geometric way And that that I found interesting at the time Okay Here is a Inexistence result we choose a positive and negative lambda. That's fairly easy then We consider three-dimensional Cauchy data So there's a three-dimensional manifold There's a metric and there's a second fundamental form for this equation as Head is supposed to be orientable open and I said H head is supposed to be complete and The whole thing is supposed to have a smooth conform a completion Which is such that to as had we attach a boundary surface Sigma which is compact a compact smooth manifold So that the whole thing is compact We assume that omega is a defining function of Sigma and we assume that these Rescalings result in in smooth fields and this metric is Non-degenerate on the boundary Furthermore, we require That the conformal by white and so which we can calculate from these data It can be Rescaled this is conformal factor and what we get it has a smooth limit at the boundary so we assume such data on Some space like slice Then we need to introduce boundary data on r cross ds We assume that there is given a smooth three-dimensional Lorentzian conformal structure and Then we have to say something how these things fit together and I refer To this picture again We want to create this picture. We have given something here and we have given something here and It's clear that these things somehow have to fit together and to you need to impose Conditions here on this boundary that these things fit together nicely and they are these conditions are referred to as corner conditions Yeah corner conditions. I am going to to say more about the con corner conditions later on These corner conditions of course have to do with the field equations and I should say I Working completely in the conformal picture. I'm rewriting the the field equation in terms of conformal field and the equations I have I call conformal field equations Now assume that these things are given Then they exist on a set of this form So it's s s now that should be a hat not a tilde So it's an it's an interval an open interval codes as said There exists a solution that should also be hats With a smooth conformal extension Which looks the way we want to to do Wanted to look and this solution induces on s and on this boundary piece the given data up to a difium office Again, I refer to this picture What I'm saying is We prescribe data here. We prescribe data on the boundary and then we get a solution in some domain I don't know. I don't say how far it extends in time. It's just local in time Okay There are three things we have to talk about We have to talk about the Koshy data. How do we get them? We have to talk about the Lorentzian Conformal structure the boundary data What do we have to say about them and we have to talk about the corner conditions and I'm going to do that now First talk about the data Fortunately, that's the easiest part because the work had been done already If you know it had been done now the observation is the following If we want to construct ADS type solution, which are time reflections in metric You assume that the second fundamental form vanishes on the slice and the constraint then reduces just to this equation And then you have to wonder what are the asymptotic conditions and you find out the asymptotic condition similar to those Required on hyperboloidal hyper surfaces, which are hyper surfaces in a space with vanishing cosmological constant which extend up to square on the other hand these gentlemen studied vacuum the vacuum solution with lambda equal to zero with the assumption that the second fundamental form is poor trace and and The trace itself does not vanish if this is satisfied and is constant Then the momentum constraint is satisfied and the Hamiltonian constraint just reduces to this and They discuss solutions to this which are to this which are conformally smooth at infinity and If you look at this you see if you have these guys you get those guys That's a million observation if you have hyperboloidal data. They give you Koshy data. It's just a reinterpretation of some constant now Anderson and Korsche they studied more general hyperboloidal data and cana generalize this correspondence so in principle from the point of view of of Koshy data, we are in a good position. There are still generalizations possible, but I'm not going to talk about this an Interesting point is the topology of this boundary is not restricted by these by these constructions The boundary is just Boundary of an orientable three manifolds. There are no further conditions and So the seam is a three manifold is fairly fairly general If you look at this article and in this article in particular at this article They have been constructed more general Data data, which are rough at space like infinity in the sense that you have polyhomogeneous expansion you could think of Constructing ADS type solution with more general Asymptotics, but this is not not so easy and I didn't make the slightest Attempt to do this if you Want to establish this picture and you have a solution which are rough here And you want to construct a solution which extends to the future of the domain of dependence So which is going beyond the in-going null hyper surface You have to be very careful and to range things very careful if it's possible at all Such that the non-smoothness does not travel into the spacetime So I didn't make any attempt, but everybody is invited to analyze that So with the boundary data we are in a nice position There was with the initial data There's a boundary data. It appears that we are even in a nicer position. There are no constraints required We just say the boundary data is a datum is given by a conformance structure So it looks as if this is the easy part of the whole thing Now there are a few subtleties behind that and this is what I don't want to discuss now The proof of this result which I quoted Consists of two steps you arrange an Initial value problem for for the PDs Which meant means in particular you to introduce some gauge condition and so on and then you try to get PDs in a form to which you apply No knowledge about initial about boundary value problems and then there's second step You are forced into this more or less Such that you get a covariant formulation Now both things depend on very specific features of the ADS spacetime The first specific feature is the following Denoted by KAB and Kappa AB the first and second fundamental form on the boundary So Kappa A KAB is a Lorentzian metric Then it's a fact that in a suitable conformal gauge the second fundamental form on that boundary vanishes If you're in the wrong gauge you find just that the trace three part vanishes that Is a conformal density? But then you look at the transformation law for the trace and you'll see you can choose the gauge set such that this thing vanishes altogether This has a consequence Consequences and that has to do with the gauge condition You have to to make a choice and you have somehow to say where the boundary is What I use is I use conformal Gd6 Conformal Gd6 are conformally invariant and they are associated with a conformal structure in a similar way like Gd6 are associated with the metric, but it's a more general class of equations Anyway, you can prescribe data for them that they start orthogonal to the initial slice and what you find if you prescribe data such that they are orthogonal to the initial slice and start Where the initial slice is supposed to intersect the boundary I they stay on that boundary And that means you start them and they generate the boundary for you Okay, there are various things you have to do you have to choose the conformal factor It supplies also nicely a frame you have to make a choice there I say in a minute how you do that what you get is something which I call conformal Gauss gauge and they say in a way It's very similar to to a usual Gauss gauge, but I mean there's certain differences and what's nice about this if you write down the Equations in you a human Penrose notation You could also write them differently There are nowadays all kinds of notations around Then you get equation which looks like this Torr is a parameter on the conformal Gd6 X alpha are constant coordinates on the Gd6 So here is just the derivative with respect to the parameter on the Gd6 And you is a set of fields which comprise the frame coefficients the connection coefficients with respect to the frame and The shouten tensor. I'm not going into any details But it's remarkable that you have this kind of propagation equations and then there is this tensor here We have seen this tensor already before we can require this to be smooth So this is our basic unknown in the conformal field equation in the spin notation It's a symmetric spinner if you write down components. It's Five complex functions and psi is supposed to be this vector and the equation it satisfies are of this form This makes it clear If you have a freedom on the boundary the freedom can only be in these functions if you want to discuss the freedom you somehow have to adapt your frame to to to the geometry and What I do is I choose this double null frame, which is in human Penrose notation Which is such that the only Non-managing scalar products are easier so that the null vector L plus the null vector L and N Generate a time-like vector, which is future directed and tangent to the boundary The difference of these vectors is supposed to be normal to the boundary in inward pointing And if you have this these two guys are tangent to the boundary and if you do this Then in the in the associated human Penrose notation You you'll find that the equations at mid boundary conditions, which are of this form So I fall minus some function a times by zero minus C times by zero bar is equal to D Where the functions a and C are subject to some? restriction and D is a boundary data and that can be prescribed completely freely. There's no constraints on this I Want to explain roughly how this thing comes out We take the Bianchi equations. These are the equations satisfied by the by this tensor field W these are eight Equations The set splits into two parts one part has covariant derivative into the reaction of N and and Maybe I should draw a picture By the way, we fixed everything and is outward pointing Here's our manifold and the way we have chosen N and L Things look like this and is it in what pointing and is outward pointing This means That those fields which occur here Cannot be prescribed. They are already fixed by what's in the interior. You you're not allowed to attach them this operator is inward pointing and You see psi 4 does not occur here psi 4 is the one which you may prescribe and You can try to feed in information on psi psi 0 and impose conditions in A and C and You can do it in such a way to get energy estimates. That's the basic of the existence truth So we have an boundary value problem That's well posed PDE problem and What's more it preserves the constraints engage condition That the constraints are preserved is Initial boundary value problems much more complicated than in the initial value problems So we are lucky that it works in that case. I don't know. I'm going backwards What is this? There's just one one problem I have to choose L and N and I've chosen them such that L plus N is tangent to the boundary But there are many time-like vectors tension to the boundary and There's no natural way to fix it unless you have something like spherical symmetry Where there is a unique time-like vector orthogonal to to to the orbits of the symmetry group So that's that's that's a problem and that makes the formulation. I've given there Not not covariant if there are two guys They they they try to produce a space time space times by following this recipe and if they ask each other Do we get the same solution? I mean odd if your morphic solutions are not It's a priory not easy to get an answer to this But in this case of 80 ADS type solution, it is possible And I think that's very specific to ADS type space time and he relies on a specific feature And that's the following one By BAB, I denote the dualized coton tensor of the metric KAB on the boundary So that's the object which if it vanishes tells you that the boundary is conformally flat By W star AB, I denote the Magnetic part of the tensor W with respect to the boundary That is I take a one-sided dual of this connect a contracted twice With an in inward pointing normal unit normal, and then I get that object That's a spatial tensor on the boundary and the special feature of ADS type space time is That you have such a relationship So the coton tensor on the boundary is related directly to the magnetic part on the boundary Okay, now what you can do then is the following you look at your boundary condition and you pick a particular choice of these functions a and c which were fairly general and you find if you take this particular choice Then you can rewrite this in this way if you use real notation So dA are a real and imaginary component of D and You see this is a linear combination of the electric part and Then you find something else That's I say it's really amazing how these patients know how to do that if these components are known to you and You look at this differential identity which holds in any case Then you see that this thing reduces to a hyperbolic system if the background is given Now you need to integrate the background as well and you look at the structural equations for the normal conformal Carton connection on the boundary and you get a hyperbolic system Which allows you with the data given on the initial slice to integrate the conformal structure of KB on the boundary? That's just the fact and Secretly this uses again this condition on the second form the metal form Conversely if a conformal structure is given to you you know now how to arrange the the gauge which I discussed and It is the whole thing tells you that these kind of Boundary conditions are together with the initial data equivalent to giving the conformal structure So it's a little bit more complicated than what might think I Shall Use this relationship again Because it's making things a little bit clearer, but the covariance data are the conformal structures now I Want to talk about reflecting boundary condition in the sense if you said D equal to zero any of these boundary conditions could be Understood as reflecting boundary conditions The psi which you may prescribe is just obtained as a linear combination of the things which are outward transported But we want to have gauge independence. That's the reason I look at this boundary condition and I require this Equality on the intersection of the initial slice with the boundary and when I have this these things together imply this So I will refer to this condition as conformal as Reflectic boundary condition or to this combination Good corner conditions We can make it easy If you have Cauchy data, you can calculate the solution up to any order you want And the formula expansion is determined uniquely in terms of the gauge On the other hand if you have boundary data You also have a formula expansion in terms of of the coordinates and what you have to make sure that these expansions coincide and Borrell's theory says this can always be done And there are many a boundary data which you can prescribe so that this is satisfied But I shall say a little bit more on this later on now we have no Initial in existence problem result locally in time and now you may get more ambitious and you You of course would like to have Nonlinear stability Everybody wants to have this these days. So if you're working on this you also want to have it Now first thing is to do linear stability that has been discussed by she bushy at wall I don't want to go to into this Recently there were two different works Related to non-linear stability aspects which had to do with ADS and Both works assume spherical symmetry That's a good idea because it simplifies the analysis, which is not simple Anyway, and it may also simplify the numerics if you do conform a Numeric calculations, so this implies conform a flatness of the boundary So that's more or less reflecting boundary conditions And they have a scalar field to retain some dynamics. Otherwise, there's nothing in it There's a result beholzegel eagle and Smully visi It shows the stability of schwarzschild ADS for spherical symmetric Einstein Klein Gordon system with some assumptions on the Klein Gordon mass It's an interesting result But I'm sorry. I'm not going to talk about it before the following reason It has an outer boundary Which corresponds to our sky, but it also has an in a boundary in a boundary That's a horizon through which gravitational radiation can escape This makes this problem different from the poor ADS problem and the phenomena I want to discuss to simply not occur so I forget about it and Consider some work in which only the outer boundary is considered and in fact the Analysis stops as soon as some inner boundary is going to develop That's a work by Bison and Rostrow Rofsky They study the spherical symmetric Einstein muscle at scalar field system with lambda Negative homogeneous theoretically asymptotics for Phi they Use Gaussian type initial data and calculate the solution numerically and what they find they found find that For arbitrarily small initial data they can form trapped surfaces This is I think an amazing result I Was a little bit careful here. I will turn you small in numerics This is something delicate, but they vary they represent the data It's very convincing that it should be should be true even without without these guys here They perform a perturbative analysis That's more less pointing into the same direction, but it exhibits also some Initial data which seem to develop into globally smooth solutions and then they give evidence That the development of trapped surfaces As they observe this result from an energy transform a transfer from low to high frequency modes And that's a lot for for those who haven't seen that before But that's how it is and this immediately Started I made other people start to work on it Diaz Horowitz and Santos. They also do a perturbative analysis with reflecting boundary condition for the full Einstein vacuum equations and they get similar conclusions in a sense this what they do mimics what Bison and Ross Borowski did and I would say there's lots of of space to To do more complete work on it Buche Lena and Liebland. They also did numerical calculations They refuse reproduced the results with a complex scalar field and they observed and this is a little bit more than shown there That if you have data close to the specific data, which have a exhibited here You get global existence again in the numerical sense, but Nobody seems to have any doubts about it. So The result that Bison and Ross Borowski to conjecture AdS is unstable against the formation of black holes for a large class of arbitrarily small perturbations Just to make it clear. I think Huh Generic Nick I pointed out that there is a class of Data which seemed to be I mean to develop into something No, I don't think so. I don't think so There's a there's a kind in the island of stability apparently. Yeah, yeah So I I I I should say not to Generate a wrong impression that I find these results extremely interesting Because of the work they have done and the questions they raise and I have not slighted doubt that this is something pointing to something really concrete However, I have a problem with a statement here if That is meant to apply to general perturbations. It may be too strong. I Think what they did rather suggest that AdS with reflecting boundary condition is unstable against the formation of black holes for a large class of arbitrarily small perturbations Now I have been wondering all the time when I looked at paid bus about a ad s That most people Immediately look at reflecting boundary conditions. They take that's Natural Some people refer even to this these conditions as adS boundary conditions It's clear. They are very convenient. You get a well-defined close system with no information coming in or going out The question is this enough reason to concentrate on this or should one look at these things in a more general way I don't quite understand this statement. So you say adS is reflecting boundary condition is unstable against the formation of black holes So the formation of black holes for small data will be an instability result. Yes. Yes You see For a large class It's somewhere through a wake here And then this may change if you emit for more general boundary conditions. I am going to discuss this So I can understand that people consider these boundary condition and in itself It's an interesting problem. The only thing which I find disturbing is that may generate the impression That's all what's to say about it So a priori these reflecting boundary conditions are not part of adS. You put them in by hand They are not forced to know from us on us. Moreover, they introduce unexplored difficulties and this I think is something for Jim is going to have to list on Looming difficulties. Yeah, this is one of them and the difficulty is a following We are now used from the Koshy problem that you can separate the problem in two parts There's a revolution problem and there's a constraint problems and there's a clear separation between them Of course, you have to show that the that the Constraints propagate but that usually is not so difficult any longer after you want told us how to do it But in this case we have a problem if you have initial boundary where your problem and Impose restriction on the boundary data This separation cannot be maintained any longer and in extreme situation is the following If you have reflecting boundary condition It's not that they only prevent a flow of gravitation in or out of the system across the boundary What they do they require the Koshy data to satisfy beyond being hyperboleutal at space like infinity rather strong additional fall-off conditions at space like infinity and this comes about We still see it and immediately This comes about as follows If we want to have reflecting boundary condition I told you we require this on the section intersection of the space like slice in the boundary and we want to have psi 4 is equal to psi 0 and To have smooth solutions you have to require this for all K Now if you now look at the evolution equation You can calculate expression for this in terms of the initial data and that means you get an infinite sequence of differential conditions on the initial data So Jim tomorrow This this is a complicated thing I've not the slightest idea whether one couldn't do something with a conformal method Up until fairly recently Gluing was also not so clear. Maybe something has been done I don't know so this is a real problem and I Did not touch this at all Now We have had this in other situations where we prescribe Cauchy data and do an evolution where where we assumed some extra conditions besides the fall-off and There was no problem, but the situation is different here There we just prescribed these data and let things go and the equation said okay I do something with this here. We insist on this at all times Yeah, so we are constructing a curve in this set of reflecting I mean on this set of Cauchy data which satisfy these condition So and the question is these additional fall-off conditions Do they possibly contain the seed of the formation of trap surfaces? I have no idea, but the possibility in that problem Which was studied by Bison and Ross Wolofsky you have the situation Solution is going out. It's reflected. It's going out. It's reflected and Assuming that the Observation is correct. The the solution is reprocessed in a way that the that the energy is going always into the Higher modes. Yeah, so that's what's what's done by these Data, I have no idea in which way it comes about But I guess it may be a real problem and this isn't a very interesting but also difficult one now the other problems which are Not more definite less definite than that one I think if we fix our work on the reflecting boundary conditions That precludes all kinds of investigations of more general situations situations which are possibly of interest in applications having said this You could ask are there any applications of this and this is one of the main problems What is the meaning of these ADM type solutions? Some people say oh, they do they represent Isolated gravitating system Some people talk about bones on stars and then some of them still use reflecting boundary condition If I have an object which I call a star whatever it is and if it's confined by Reflecting boundary conditions. I think the astrophysicist can forget about it Yeah, there's no way of this thing to interact with the rest of the world Yeah, so if you if you suppress this interaction, it's not so clear just to say what it is Again from the mathematical point of view all these things are interesting and maybe it's a first step to understand the whole thing but There is a question and the one of the question is you have no idea what this thing means physically Which gives you no hint of what you should require what would be a reasonable assumption There's another problem if you look at the zeta type or Minkowski type space time There the conformal boundary Splits into two component There's one component where gravitation can enter and there's another component where it leaves the space time Here it is one component and in principle radiation can enter and leaves the space time And it's very complicated to make a distinction between entering and leaving radiation Okay, then the question is what do we do? Of course, I don't know. I don't even know what stability should mean here in this context my best advice would Be try to characterize those boundary conditions and data for which solutions which start close to ads stay close to ads for all times This is a task it's not too easy But if one could characterize these these data one could also get an impression what's in the Solution manifold and how we should understand this Yeah, and what I do think if you look again at the my Remark in the beginning if you look at the global causal structure of ads I think it will in general only be reasonable to require globally bounded in time Maybe relative to ads. I mean, I don't think it makes a sense to require that things are Falling falling off and go to zero as he goes to infinity or minus infinity Okay, that's what I wanted to say I Would like to ask a question coming back to the work of your to be done Yes, you said that the boundary condition depend on the system study Yes As far as remember he mentioned that there's no freedom in choosing boundaries. Can you comment on this? No, I mean if if he wanted to have some Sort of smoothness and then he is forced into this there may be other arguments I think what you had you had some energy defined on these things and that energy which they defined It's only finite if you impose such boundary condition. Is it correct? Yeah, so there may be different reasons Why you do this, but I mean that's how it is