 Hello everybody, bonjour, bonjour, Thibault. So it's a great honor and pleasure to participate in this Happy Dammel Fest. And I would at the beginning like to thank all the organizers for putting together this great theory to celebrate Thibault and his many contributions to general relativity. And I would say that when you look at the program, the first thing that sticks out, stands out to me is the really broad spectrum of different topics over all of general relativity from very applied from gravitational wave detection to physics, numerical relativity and the more mathematical approaches. So I think, and also the speaker, so I think it reflects in a really nice way. Of course, the work by Thibault, which is really broad and has made so many contribution to different fields, but also on a personal level. So Thibault has collaborated with so many people over the years from different backgrounds in GLR. So I think it's a great model for us to say, well, we should go out and talk to people across the borders from applied to very theoretical on this journey to discover more secrets in GR. So let me start. I'm gonna talk about gravitational radiation. And so we have heard a lot already about radiation from our typical sources like binary black holes or neutron star mergers. What I would like to do today is to give you a little bit of different view of, let's say if you look at more general space times, which do not decay like mass over R, but just like mass, but just like R to the minus one half far away from the source. So this would include, for instance, if you have very extended neutrino halos around the source of a binary black hole merger, for instance, I will show space times and what we find in gravitational radiation that there's some new structures showing up if you generalize the space times that we're looking at. So of course we have seen a lot of progress recently since the 2015 detection of gravitational waves by LIGO and many detections by LIGO and Virgo since then. And I think it's interesting to live in this area or for many reasons, we have so many mathematical resources that allow us to attack problems in GR that have been unsolved for a long time. And on the detection side, we have this wonderful breakthrough by the LIGO Virgo collaborations and many detectors to come on board in the next few years. And also the event horizon telescopes that took this picture that zoomed into one of the black holes here on one of these pictures. So let me maybe mention also, I mean, Thibaut, as I said, made so many contributions to general relativity as is reflected in the talks of this week. I will focus here on gravitational radiation. And let me maybe mention a few things that Thibaut and collaborators have contributed over the years. We have heard a lot and I'm sure we're gonna hear much more about that, but it's really impressive if you just think of an overview. So I think to remember a lot of Thibaut's work has been really crucial to analyze data from the gravitational wave detectors. And this started already in the early days for the Halls-Taylor pulsar when Thibaut and Nathalie de Ruelle computed the decrease in the orbital period of the binary system. And in more recent years, so in the direct detection of gravitational waves. So there's of course this huge work by Thibaut and Luc Blanchet who described the motion of two black holes approaching each other. And also Thibaut with Alessandro Bonanov computed the final merger. And that's only a few things to name. I mean, there's a much larger literature by Thibaut and also with many co-authors who collaborated to really from the theoretical point how to interpret this data from the gravitational wave detectors. So of course here's our friends, the Einstein equations and let me maybe just briefly mention I'm gonna concentrate for most part of the talk on the Einstein vacuum equation. So when the stress energy is zero, but I will also, I mean, the results carry over I'll mention at the end an application also to the Einstein neutrino case in the neutrino case, what we do we model neutrino radiation on the right hand side here by a null fluid. So we think of neutrinos as well having a very tiny mass going almost at the speed of light. So therefore we can approach things by some null fluid in a good way. So maybe a little bit of a background. So most when I think of problems in GR I really coming from a mathematical background so I really like to think about it as an initial value problem. So what I will present here resulted of looking at some let's say physical initial data. So we have asymptotically flat initial data hopefully should represent some interesting physical situations. And then we evolve the Einstein equations to generate space times which should be interesting in terms of radiation and physical properties. So at the, and what we do here we generate whole classes of space times with different behavior and different radiation properties. And let me also mention at the beginning of every proof or every derivation you do here is of course the celebrated results by Yvonne Chocobriar on well postings and also her results by Chocobriar and Robert Girage. So when we think of gravitational waves of course we have the source that as we said we could have binary black holes, binary neutron stars that merge and so on. So they will send out gravitational waves which travel let me introduce here maybe a few geometric objects along null hyper surfaces of our space time light cones if you think of that to null infinity. And we would like to understand when we generate the space times what happens asymptotically here at null infinity that's where we read off the information about gravitational radiation. All right, so here's a little bit of notation that I will use. So if we think locally, so this surface S so if I'm sitting locally on it maybe surface which is different morphic two-sphere things are happening here. And so when I am in my space times I'd like to investigate if I go out so cease, you know, an outgoing null hyper surface so think of a light cone and I will look at outward null geodesics and inwards null hyper surfaces and inward null geodesics. So what I will call chi hat is when I look at the shear with respect to the outgoing null geodesics that's gonna be my chi hat. And if I look at the shear with respect to the incoming geodesic that's gonna be my chi hat bar. So the chi hat bar at null infinity. So this is what is usually used as the news called the news tensor in more physical terms. So this is my notation for the shears that will show up and play of course an important role in describing gravitational radiation. So let me cite a result by Christopher Kleinerman here about Minkowski's stability because that's these and another result is gonna be the source for many interesting space times I'm gonna look at. So at the beginning of the 19th, Dmitry and Sir True have this breakthrough result showing that he have asymptotically flat data which is small enough in a well-described version. You can create a space time or a set of space times which are complete null geodesically and culturally geodesically complete. So there's no singularity, et cetera. So you have global space times that tend again to Minkowski at infinity. And I generalize this result in 2007. So in my data, I will give you the formulas in a moment. I have a very slow decay of the data. And so we also were able to generalize space times which are globally asymptotically flat with the right behavior. So why am I stating this space times? So first of all, in these results and then in Kof's instability proofs we had to endure the pioneering results by Christopher Kleinerman. So and I also we had to endure a smallness of the data to endure existence of these global solutions which do not produce any singularities or black holes but which are asymptotically flat then culturally geodesically complete. However, something interesting comes out of these results which makes the space times even more interesting also from the perspective of studying radiation. So a lot of the behavior that has been derived in these results holds also large data. So you can put in large data and also if you have let's say black hole space times then you will have a portion of null infinity that will behave or where the behavior of the asymptotics is really very well described by the results of these space times. I will say a little bit more about that in a minute but here are the types of initial data that were created, looked at by first the Christopher Kleinerman case. So you have some master over ours of the Schwarzschild part in the metric. So this is an initial space like hypersurface and far away from the source or far away things look like decaying like one over R here plus some more decay and the second fundamental form has corresponded decay. Now what I would like to talk today about today is a different type of space times that are out of my work. So if you have just initially in your hypersurface in the initial data a decay which is only R to the minus one half. So of course I have the one over R decay included in the lower order terms but I only have this R to the minus one half and the second fundamental form which goes like R to the minus three halves. Still we could establish stability in this setting. And let me say a few words about now large data. So we have now generated a whole class of space times which is very general and very slowly decaying to infinity. So one thing to say here maybe a little bit more why we can use some of the small data well develop derivations also for large data. So for the large data you can show it's kind of an easy calculation that even if you plug in large data that there exists a complete domain of dependence of the complement of a sufficiently large compact subset of the initial hypersurface. So you can generate a sufficiently large portion of scry for which we have a very good behavior that comes out from the understanding of these space times because what we derived there at null infinity is independent of the smallest assumption. So you can have large data. So with other words, we have some solution space time with a portion of future null infinity corresponding to all values of the retarded time which I will call you in this talk which is not greater than a fixed constant. So with other words, so we have here a really nice setting of space times that we can understand them with some really good understanding of behavior at null infinity and gravitational radiation. So just a little bit of notation maybe I jump here better to say or maybe here. So these are my curvature components and you may be more familiar maybe with the new and Penrose notation. So in that setting, so the alpha bar is the one over r part. So the alpha will be the psi zero and alpha bar psi four is going in that direction. So whenever I have a three component that means that's a component and null component in the inward direction. Four is a null component in the outward direction and A and B are sitting on a sphere tangential to a sphere north of normal frame. So maybe this is more helpful the next slide. So alpha bar is then the curvature component that goes like one over r. So which is interesting for the radiation field for us. So in the crystal climb on space times when you go to null infinity, they showed that for the while components you have a decay as follows. So you get the peeling up to r to the minus three but then it kind of stops at r to minus seven halves. Here, tau minus by the way, you can think of tau minus as being retarded time u. So it's actually one plus u squared not the square root of that. And of course, if you ask, well, if you don't ask so much decay at the beginning you only get what you ask out at null infinity. So in the space times that I looked at we have again peeling for the first two components so that one over r and r squared term but then it levels off at r to the minus five halves. We don't even get the r to the minus three for the other components up here. Nevertheless, something can be said. So interesting enough. So we have this non-peeling curvature components. And if, so the next few minutes will be diving deeper into the structure. So it turns out that this non-peeling curvature components the first order parts are actually non-dynamical and we can actually dig deeper in mathematically to understand what's lying beneath here. And there are some dynamical structures which will show up. So here are these are the Bianchi equations in my notation and maybe to explain what needs to be recalled here. So row, so again, this would be like the psi two in the Newman-Penn-Rowe's notation. And row is an electric component of the wild tensor and sigma is a magnetic component of the wild tensor. So I have the Bianchi equation with the derivative in the incoming null direction. And on the right hand side basically the beta part is another curvature component as we have seen pi hat is my shear. And you can forget about the lower order terms is another is our other structure components or structure components seat down epsilon. So I will come back to that but we don't need to remember all the structures in my notation, but it's interesting that I will from the Bianchi equations we can just extract a lot of this extra information which is hidden in them under the non dynamical parts of this courage components. So let me say something here about the shear. So again, the chi hat bar in my notation when we go to null infinity, this will give us the news tensor. The chi hat, the chi bar are the shears that I introduced in these more general space times with slow decay. We find that the leading order terms of the chi hat is like goes like R to the minus three halves. And let me explain my notation a little bit if I have this square brackets. So this means these are terms at which are decaying at the order but which are non dynamical. Non dynamical means they are not depending on the retarded time you. To our minus, you should think of retarded time you here this curly brackets. So these are dynamical terms which give you terms that's decay at the order given in these brackets. But so they are dynamical they are changing with retarded time you. So now if we go to null infinity, this if I take the corresponding limit of the chi hat bar at null infinity, this will have a well-defined limit but chi hat will not. So in all the space times that you look at where you have a decay of mass over R, this will be an R to the minus two decay and you have nicely defined limits at null infinity. So no problems there. So here we cannot do that but we can do something else. Maybe just one slide, which gives you a little bit of a difference if you have small data or large data, what would happen if you find this small assumption in case of small data on the derivative on the incoming null derivative of the curvature component. So row is an electric part of the curvature component that would go like one over R to the three in let's say mass massive space times where we have compact binaries. And in my case, just like R to the minus five halves. So if you assume a smallest assumption, then your row and sigma terms will have components as given in equations 16 and 17. So fine, but if you now plug in large data, just to give you an idea what happens between small and large data at the curvature level of decay, there's other questions of course. So if you plug in large data, then what will happen, you will have a lot of different terms also that will decay like R to the minus five halves and U to the minus something. So depending also on retarded time. So that's one of the main differences that will be important. So maybe this is one of the most important slides of the talk because I would like to explain what happens now when you go to null infinity. So again, if we are in a space time that is that are well studied like mass over R, then all these curvature components and geometric components have well defined limits of null infinity. We don't have to talk about any problems. However, because I have a very slow decay, some of my components don't get a well defined limit at null infinity, but there's something we can do. And I'm gonna show you that at the example of this shear that we call chi hat. So CU here is a null hyper surface. So think of this in the little picture from being four. We have CU is a null hyper surface going out to future null infinity, outgoing direction. And I can kind of cut this null hyper surface with sphere. So at each time T, I can cut out a surface SU which is the few more factor around sphere. And so what we do here now is the following. So we will sit on such a sphere and go out to null infinity. And then we take a neighboring null hyper surface which is not far away and also sit on a corresponding sphere, go out to null infinity and compare what's happening here in between. So if you do that with chi hat, so if I would like to take the limit R squared of chi hat, this would blow up in the setting that I'm interested in. So I cannot do that, but this will not take a limit at null infinity. However, if I'm taking a point at one of these spheres on a null hyper surface and I take another point on another neighboring null hyper surface on a corresponding sphere and I get out to null infinity with these both points then actually the difference of this value for chi hat at neighboring points, this will attain a limit. So it's not too bad. So, and the limit will be given by this formula. So what is that? So chi hat, this again is that shear, D3 is the derivative with respect to the incoming null geodesic. And if I take this derivative and I integrate now from U0 to U, so you think of this as a light cone or null hyper surface CU going out to null infinity and one CU0 going out to null infinity, then this has a limit, which is finite. So we can also show that actually the leading order term, this R to the minus three half term is actually non-dynamical. So there is a lot of dynamical terms showing up beneath the non-appealing parts. And what I showed you here for the chi hat part is also true for the non-pealing curvature components in this setting. So the non-pealing components are actually all non-dynamical. And we can sort of think taking them away and go a step deep and see what actually gets out to null infinity and impacts gravitation radiation. Then we find the dynamical parts and that's what we can now use. So there's a lot of structure in the dynamical parts. So maybe let me now switch here a little bit and actually go out to null infinity and try to tell you what is now new about these structures when we take this more general space time. So the claim is that there's new things showing up that would not be seen otherwise if you have stronger decay. So we have heard some interesting talks about LIGO, Virgo and future gravitational wave detectors. Here's just a schematic picture. So this mirror is suspended by pendulums by LIGO, for instance, or our test masses like in the other picture here which follow basically the geodesic motion of our space time, right? And so one aspect of gravitational waves that I would like to talk about is the gravitational wave memory effect, for instance. So this is a permanent change of the space time. So you think gravitational waves travel through the detector. So there will be a permanent change in the space time that will be left behind. That's what you call the memory and LIGO and Virgo have it on the to-do list. So we hope that in the next few years we will see a detection of memory. So maybe let me just mention the pioneering names in memory. So gravitational wave memory, memory has been derived in the 1970s and 90s on the theoretical side. So there's the work by Yakov Soldovich and Aleksandr Polnirev by Dimitris Christodoulou, Thibaut D'Amour and Luc Blanchet. So there are more names that followed up this early works on gravitational wave memory in the recent years. This has been a real explosion of work on memory-related research. And so we have learned about contributions to memory from various matter and energy field and what happens in some Lambda CDM cosmology, et cetera. Let me maybe just mention one paper here by Lasky, Thrain, Levine, Blackman and Chen. So these are just how to detect gravitational wave memory with LIGO by stacking events. And we have also found together with Barfingkel analogs of this memory in pure electromagnetic theory. So I cannot mention all the names. This has been really a growing area here, but I wanted to just share the most, let's say the pioneering contributors here. So what do I mean by memory from the mathematical point of view and what is now new? So I mean, we have seen memory for these cases, but let me explain here what the difference will be from what I will present and what we have seen so far. So in my notation, so first of all, the delta X here. So when we think of neighboring geodesics, what LIGO is doing, so it's measuring distances of neighboring geodesics. And we know we can do that by integrating twice the geodesic or the Chakubi equation. And when we do that, we get what I call here as a delta X, so that's just a delta. And in my notation, the permanent displacement in this gravitational wave memory effect will be related or equal to minus some D zero. That's the initial distance between the test masses divided by R, that's our distance from the source, times and here, so this is what I call here chi minus minus chi plus. If you are in a space time, which decays fast enough like one over R, then this will just be the shear at null infinity. In my space times, it's related to the shear, not quite that, but there's some geometric object that we can investigate. So if we are in a fast, in a slow, in a fall of like one over R, then we find some, we have all this intricate and complicated structure locally that will actually fade off at null infinity to something more simple and more clear at null infinity. And we can distinguish between what we call ordinary memory, which is sourced by the change of the radial component of the electric part of the wild tensor and what we call null memory, which is really going out to null infinity, which is sourced by the energy per unit solid angle radiated away to infinity. So this includes the shear and components of energy momentum tensor. So what is new now, if you have much slower decays that on all these level, on this level, you have more terms that will contribute to these memories which are usually zero in stronger fall off of the data. So let me maybe explain just very briefly or remind us that we can decompose the wild tensor into its electric and magnetic part. And so the electric part, so this is just the wild tensor here, epsilon on my volume element. So we can decompose into E, which is the electric part and H, which is the magnetic part. Again, so then the Jacobi equations can be written like the delta X and the derivative with the electric part and right hand side. So in my notation, the row that has shown up already is actually the normal, normal component of the electric part of the wild tensor. And sigma is the NN component of the magnetic part of the wild tensor. So row and sigma are given here for electric magnetic part of wild tensor. Lydia, please for me. Okay, thank you. So when we talk about memories, we call electric memory, memory that will be sourced from the electric part and magnetic memory will be sourced from the magnetic part. So let me maybe introduce here, go back to the Bianchi equations. And here the Bianchi equations for the electric part, that's the electric part of the memory. And if I just, some of the remainder terms will just go to zero at null infinity. So let me concentrate of the highest order terms. You see that alpha bar is the one of our curvature component, beta bar is the one of our two curvature component and I have a derivative of row on the left hand side. So you see, if I wanna take limits at null infinity, r to the minus three, okay, it's good, but r to the minus five halves doesn't actually have a limit, that's a problem. So let's see what we can do. A short computation shows I can get out of this term a trace chi times chi bar hat. The chi bar hat again will be the news tensor later on. So then I have a better decay for that but I still have this problematic r to the minus five half decays. Well, I throw that on the left hand side and then because in the space times, I know that the right hand side behaves well. So this is of order r to the minus three, u to the minus one half. But then on the left hand side, I have these terms which I mean, you see there's all kinds of terms which will not have a limit at null infinity but they cancel out. So just by this equation not by the properties of the right hand side, we know that these cancel out. But then what we do next, let us take the limit of this equation at null infinity. So we multiply with bar to the three, the whole equation, the Bianchi equation and take the limit on one null at the surface out to null infinity. So when I do that, I call the limit of the right, let's write p three and I will integrate that with respect to retarded time u, that's gonna be p. And so when I do that with this equation, then I can maybe jump to the next slide. So on the left hand side, I will have a lot of terms which actually will not be finite. So maybe just in the last two minutes, let me explain what we will get. So if we are, this is the limit of the left hand side, if we are in a space time that behaves like a one over r decay, then this first two terms, they would not be there. So there's nothing like p row one. What we would have is just one single component of a contribution from the electric part of the curvature tensor, which would be in F. But now we have new components and actually they will be growing with retarded time u. So we have components of the curvature that grow like retarded time u or even retarded time u to the beta between zero and one half. And we have a new contribution at the finite level, not only from row, but also from a shear term, which is rooted in the shear terms like chi-hat, chi-hat bar. All of, so this term especially would be zero if you have a stronger decay. So maybe in the last minute, let me show you also that we get magnetic memory. So I can do the same procedure with the magnetic component of the curvature of the Bianchi equation. So the same procedure gives me a limiting equation for the magnetic component. And what I have here, so just to give you an idea, so I have the limiting equation for the magnetic component, the magnetic memory. This will be completely zero if I'm in a spacetime, which is like one over r. So in a one over r spacetime, this would be all zero. This equation would not be useful at all. But if you have slower decay structures show up, this is not zero. And more than that happens, so explaining before when we go to null infinity, if I look at the quantities called q up here in this formula, and if I'm sitting on a sphere at null infinity and take a point u zero, I can evaluate q at u zero. But now if I go to a neighboring, let's say sphere on a neighboring null hypersurface qu, this is still finite. But if I take u to plus or minus infinity, the difference of this qu zero and qu will be growing like square root of u. So maybe I can solve these equations at null infinity. What is new really is first of all, we have magnetic memory, which is zero in a stronger decay, like one over r. But we have magnetic memory, which is growing like square root of u. And these new terms are sourced from the wall component of the magnetic component of the wall curvature. I have more contributions from sheer terms at finite level, which usually are zero if we have stronger decay. And also the electric memory grows at the same level. So maybe I don't have much time to talk about neutrinos, but the same results carry over and we find some new structure for neutrino radiation. If you have a huge extended neutrino clouds around a black hole, for instance. And let me just finish here by saying happy birthday, Shwayeus anniversaire, Thibault. It's really a great honor to have known you over these years and I've always enjoyed our discussions and I certainly learned those a lot from you. Thank you. Thank you very much. We have time for a quick question. Is there still a relation between memory and symmetries, like BMS symmetries in this generalized case? Yeah, that's a good question, right? So when people talk about BMS symmetries, right? So there was also a lot of question about, for instance, super translation ambiguities, et cetera. So one way to think, maybe let me give you one way that a lot of people ask me if they could think about memory like that. So a lot of physicists like to think about memory like you're in a Minkowski space, gravitational waves come through and it transforms you into another Minkowski space. You have some transformation. What would happen? So with my findings, for instance, that shows, well, you don't, I mean, my memory is growing with retarded time view, right? So what would that be? So that's a good question. Maybe let me say one more thing. I mean, in terms of what is finite in terms of the BMS definitions of null infinity. So we still have finite energy and finite linear momentum for the space times, but angular momentum is not known. Okay, thank you. Let's move on to the next talk then and thank you again. Thank you.