 Thank you very much to the organizers and to ESUS for this lovely conference and the invitation. So I'm going to talk, what? For the weather. And the weather. It's going to improve. Exactly. But you know, we don't need to go to be out. Right, there's a motivation to come inside. So I'm going to talk about Einstein equations. I'm going to introduce some geometry into our PDEs. And of course, we all heard about the detection of gravitational waves. So it was confirmed in February this year that it was detected last year in September. And I'd like to make a connection to, let's say, geometric analysis on what we know from the PD point of view, analyzing the Einstein equations. What can we say about gravitational radiation from the mathematical point of view? So basically I'm going to talk, introduce first, well, what do we mean by a spacetime? What are the Einstein equations? And so again, this will be a point of view of PD, of this hyperbolic set of PDEs from a geometric viewpoint. I'm going to make a connection with, so what do we mean by energy in GR, general relativity? So there's maybe not a straightforward notion to think of from the beginning. So we have to think about it a little bit. I will talk about what I will call null infinity. So let me just very juristically motivate this for the moment. So if you think of gravitational radiation, so they are produced during the merger of compact binaries like black holes, for instance. So there's something happening here. We have two bodies, black holes spiraling in. They lose energy into the gravitational field, so they radiate waves. And these waves will travel at the speed of light along so-called null hyper surfaces. So you can think of this as a generalization of a light cone, basically. So this will go along null hyper surfaces. Let's say when we are far away, when T goes to infinity, so we can think of where we are sitting in an experiment. So this is null infinity. So we will look at null infinity. That's where we are sitting when we're doing experiments and we are looking backwards in time along these null hyper surfaces. So we would like to understand, from a point of view of analysis on geometry, how does our space term look like out here? And what kind of information can you gain from what's happening in the past here, for instance, in such a source? I will explain this a little bit more later. And well, of course, gravitational waves will also address something called the memory effect. That's a permanent change of the space time by the gravitational waves. And we also hope that this will be detected in the near future. And detection is one thing we hope, of course, for the future, that this will be a new tool to really learn about other parts of the universe where telescopes cannot see. Okay, I'll then address some results in the asymptotically flat case. So that's where we look at galaxies, for instance, or we think we are here. The galaxy creates curvature, but far away. The next galaxy is so far away, so we can assume the space time is flat far away. Another setting is the cosmological setting. When we look at the whole space time, the whole history of the universe, what can we say there? Well, first of all, what is a space time? So we look at the Einstein equation. So on the left hand side, and I'm going to talk about four dimensions. So three plus one, three spatial one time dimension. So we look on the left hand side, so we have basically two metric objects, if you like. So we have the Ritchie curvature, the metric we are solving for the metric, the scalar curvature. And we call this gene union on the left hand side, the Einstein tensor. And on the right hand side, so we plug in the so-called energy momentum tensor. So whenever you have electric fields present or a fluid or anything else that is not given by gravity on the left hand side, so you plug it into the energy momentum tensor and you have also to supply the corresponding equations. So for instance, if you have electromagnetic fields, you couple your Einstein with Maxwell equations and you get the coupled Einstein-Maxwell system. So we are solving the Einstein equations for the metric, like to create the space time, either locally or globally in time. And well, we'll see. So it's possible now to, I mean, if you have the Einstein equations, we usually write them as, well, if you look at it this way, it looks nice and compact, but we usually write them as a system of hyperbolic nonlinear PDE. And well, if you add the cosmological term, so how does cosmology come in? Well, Einstein in the first, in 1915, had written down his equation without this cosmological term, but then he wanted to study cosmology and he was actually looking for a static universe at the time. He didn't believe in anything that's moving. The universe had to be static. And to counteract gravity, he then plugged in this lambda term, which is giving you an expansion, basically. So he got the static universe by doing that, but for all purposes now, so we think since the 1998 observations that our universe is actually expanding at the accelerated rate. This is modeled then by this lambda term. Okay. And often I will just look at the Einstein vacuum equation. So if the right hand side, your team, you knew is zero, the Einstein equations reduced to the Richie scalar of the full space time being zero. And there's already a lot of interesting information or questions you can ask about just the Einstein vacuum equations themselves. Okay. Well, first of all, in order to study anything that has to do with real physics, we would like to do a mathematically rigorous problem. Of course, you have to study the Cauchy problem. So it all starts with the Cauchy problem for the Einstein equations. And as most of the audience is not working in GR with some experts as an exception. So let me review a little bit. What does it mean to set up the Cauchy problem for the Einstein equations? So it's a bit different in various ways. So an initial data set, what is that in GR? So we think, first of all, of a three-dimensional manifold. So you can think of t equals zero. You get a Riemannian complete manifold. If you look at asymptotically flat, for instance, that you're interested in. So you get a complete Riemannian metric, the induced metric on the initial hypersurface. And we specify a symmetric two tensor k. Maybe I'm going to write it down. And if you have anything on the right-hand side present, so other matter fields you also specify, of course, these equations. Now the Einstein equations couple into two sets of equations. One is the constrained equations, which the initial data has to satisfy. And the other set of equations are the evolution equations, the ones you then solve into the future. And so the constrained equations, you cannot just pick any initial data, but it has to fulfill these constraints. And only then you can think about doing an evolution problem. And then a Cauchy development of such data is given as a globally hyperbolic space-time of this sort verifying the Einstein equations and the embedding given as these things. So maybe just to say at time equals zero. So we think of some space-like hypersurface h, zero, for instance, g bar and k. And this initial data has to satisfy the constraints. And so if I have a time vector field, I can introduce that. So this k is going to be the second fundamental form with respect to t then. Well, let me first talk about asymptotically flat systems, as I just motivated a few minutes ago. So if you look at what happens, for instance, within a galaxy or a cluster of galaxies, you think the next object is so far away, you think your space-time is going to be flat very far away. So what does that mean? So we think of asymptotically flat initial data in the sense that outside of a sufficiently large compact set, so h without k is demorphic to the complement of a closed ball in R3. So and then the g bar, the metric g bar and k have to decay g bar to delta ij and k to zero fast enough. And of course, I will make this more precise. So what type of decay, what type of asymptotically flat data we are looking at. So let me first say that, well, if you look at the history of general relativity and so the Einstein equations were put out there in 1915, and it took, well, there were many, many things happening in between in math and physics, but it took until the 1950s and 60s where Yvonne Joqui-Brouillard and her and Bob Garrosh actually really solved the well-posedness for the Einstein equations. If you look at, for instance, in 1919, we have Eddington's experiment confirming the bending of light, et cetera. You have on the physics side of GR many of these big things that happened or the expansion of the universe found by Le Maître in 1927. So all he thinks to have many of these successes and mathematically could ask, well, what's been going on mathematically? There were many people actually working, like Le Raie Schauder, many people looking at the initial value problem, but it's not that easy to figure out what should be exactly the right way to put up the initial value problem. So I'm skipping all this history, but it's actually interesting to read up on that. So there's many people who contributed in many ways. So basically in 1952 that's the first rigorous general mathematical theorem in terms of well-posedness. So Yvon Jacques-Cœur-Brillard then showed that if you take hg bar k as such an initial data set, which is, well, here this is formulated for the vacuum Einstein equations, but it's been generalized to hold for most other fields on the right hand side. So then there exists a spacetime satisfying the Einstein-Vacuum equations with h into m being a space-like surface with induced metric g bar and second fundamental form k. Then the general form of what, so whenever you do solve the Einstein equations, so basically you go back to one form of these well-posedness results for these local results. So then the other formulation is given here. So Yvon Jacques-Cœur-Brillard and Bob Geralsch in 1969 then came up with this formulation of the well-posedness. Okay, so that's the first rigorous step in general, I would say. Well, let me say something about energy in GR. Well, most of us, when we think about PDEs, so often we need to find, of course, energies that have to be controlled to get control on the solutions of our equations, etc. So if you think of doing something like that in GR and you take the most naive approach, so things might fail, so what's the problem here? So one problem or interesting feature is that if I'm sitting at the point in GR, I can actually transform away the gravitational field. So just by transformation I can make it zero. So how would you define energy at the point if you can actually transform away the field there? Well, that's one problem. So you can think, well, maybe I don't look at the point, maybe I go a little bit further, maybe I integrate over a sphere, something quasi-local. Well, it turns out that's one thing we can do, but also for asymptotically flat systems, for instance, what's been understood quite well for a lot of interesting space times is you can integrate, for instance, over a space-like slice, or you can also look at what's the, if you integrate locally over a sphere and you get some mass or energy, well, expression, and you can take the limit like the Hawking mass, I'm going to write that down later, and you take the limit to null infinity, there's something else called the Bondi mass, for instance, that will show up later. So there are mass and energy definitions for certain, actually, many interesting space times that we understand. But let me give you maybe a little bit of an idea of the confusion at the beginning by citing Einstein and one of his colleagues. So Einstein formulated some sort of an energy momentum theorem very early actually which is for a closed universe. And so most of his colleagues at the time did not agree with it, and here is what he says. So here this is a citation, while the general relativity theory was approved by most theoretical physicists and mathematicians, almost all colleagues object to my formulation of the energy momentum theorem. I'm not going to go into details, but he wanted some kind of an energy momentum conservation for some closed universe, but turns out that's maybe not the right way to do it. And so this is Einstein with Schoch-Schlimmetre, maybe a side note of actually Schoch-Schlimmetre, a Belgian mathematician and physicist who derived from the Einstein equations and Slifer's observations in 1927 that the universe is expanding. It was not Hubble. Hubble had a paper two years later, and he never talked about an expanding universe. Okay, let's still stay a little bit more with energy and conservation. So what do we know? So we can look at the Bianchi identity that we know from geometry and apply it here to the Riemannian curvature tensor. And, well, we take, for instance, twice-contracted Bianchi identity and get this equation on the left-hand side of the Einstein equations. And then, of course, this implies by the Einstein equations themselves an identity of this form. So people at the beginning also thought, well, is this energy conservation yes, no, does it make sense? Well, it makes certainly sense of some sort, but to call this the energy and energy conservation is probably the wrong thing to do in GR. So maybe one more thing about that. So Neuthor's theorem, if you think of it a little bit more in a generalized way, well, what are we looking for? Within some setting of a Lagrangian theory, we always look for a continuous group of transformations which leave the Lagrangian invariant, right? So to such an object, we have then a quantity which is conserved. And in GR, the first problem, if you have a most general spacetime, you don't have any symmetries. So what are you looking for if you don't have any symmetries at all? However, nature is usually better than just the most general case. For instance, if you look at an asymptotically flat spacetime, your background Minkowski space has a lot of symmetries. And you can start looking at maybe quasi-certain isometries you can preserve basically in a way or another in the manifold or control. Well, here is maybe one more slide where Einstein, Hilbert, and Weil struggled over these energy components of the gravitational field. So they were thinking about this very hardly at the very beginning. And here's just one more quote by Hermann Weil. This is in his book Space, Time and Matter of 1921. And he looks at these components here. So he says, well, nevertheless, it seems to be physically meaningless to introduce this TIK. So that's the Einstein-Suter tensor based on some specific Lagrangian he had there. As energy components of the gravitational field for these quantities are neither a tensor nor are they symmetric. So in fact, by choosing an appropriate coordinate system, all this TIK can be made to vanish at any given point. Although the differential relations referring to the divergence of the Einstein-Suter tensor being zero are without the physical meaning, nevertheless, by integrating them over an isolated system, one gets invariant conserved quantities. So you see there was a lot of thinking reflection about what the right energy notion should be. And already one suggestion is, oh, well, maybe we should integrate something over an isolated system. So that was already a good idea then. Okay, but let's now jump to more modern days. Well, modern 61. And so if we think of one type of notion of energy and momentum, so this is the ADM energy linear and angle momentum that was introduced in 1961 by Arnawitz, Deisler and Misner. So basically, this is the definition, but the way you think of this, you integrate these geometric objects basically at the sphere, over a sphere at infinity. So you let R go to infinity. So you're in one space-like slice. And you integrate basically over a sphere at infinity. And you can define then energy, linear and angle momentum in that way. So you have to say, of course, this holds for many interesting asymptotically flat systems. And so energy and linear momentum hold actually very generally. And for angle momentum, you need a little bit more of decay to be defined. So what about null infinity? Because I told you at the beginning, heuristically so radiation, we would like to understand null infinity and explain something about radiation here, extract this from our space-time. So we would like to understand the Cauchy problem and then see how the space-time, the solution looks at null infinity. So I can also foliate my space-time not only by, let's say, space-like hyper-surfaces like this. I can also foliate my space-time by null hyper-surfaces. So I'm going to draw them like a cone. Of course, they have structure, but you can think of them as generalized light cones, for instance. So this is maybe, so you want, so you have this light, this null hyper-surfaces, et cetera, that I can generate like that. And again, so when gravitational waves travel at the speed of light, they travel on these null hyper-surfaces out to null infinity, and that's what we're interested in. So if you, the boundary definitions of this, so the adiom definition that you just saw on the previous slide have something like an equivalent, well, have a corresponding definition at null infinity, and we call this the Bondi definitions. It goes back to Troutman, Bondi, van der Borgen, Metzner and Sachs in the 1950s and 60s to define mass, energy and momentum also at null infinity. Well, there's some more constraints probably you have to put there to make things work. I'll say more about it. A famous theorem, of course, in GR is that mass is positive, and so this was shown by Shane and Yao in 1981, and then also by Ed Whitton separately. And then in the Bondi case, this was also studied by Shane and Yao later. So first of all, energy, what's now the real good way to talk about energy? So energy controls the curvature, first of all, but what kind of energy? So we use the Bell, Robbins and Tenzer quite often. So here's one solution to this problem. So we can think of the Bell, Robbins and Tenzer. So this is defined as follows. So this is basically, it's a four covariant tensor field, but it's basically, if you look at that, it's a quadratic in the wild tensor. So you look at the wild tensor of your space time and basically the Bell, Robinson is a quadratic in your wild tensor. It has a lot of nice properties. It is actually positive in this sense that if you plug in future directed time like vectors, so you get this positivity property and it also satisfies the Bianchi equations. So here's just a notation for what I use for the hardstools. Maybe let me point out, so this was used heavily in the work by Krzysztole and Kleineman on the Minkowski stability. So this is really the main object to control the curvature there. And nowadays, by many other people as well. So let me also introduce the current. So if K, this Q now, the Bell, Robbins and Tenzer are just introduced. So contract this now with three future directed causal vector fields and we obtain what we call the current in this case. So I'm going to call this J. That's my current. Then I can apply some diversion theorem on a bounded domain in the space time and I can also then find what is my curvature flux. So if the omega here is just the boundary of this, so if this contains a portion of a null hyper surface, so we remember that, with some affine tangent null vector field L, then the corresponding boundary term is indeed the curvature flux through this null hyper surface C. And we can write it down as like this integral. So this is the curvature flux we can define like that associated to some null vector field L. So these are some important objects to think about and often when you would like to prove, I'd say, control, Robbins and Tenzer actually controls the curvature and from there we can go on. So maybe back to the Cauchy problem for one tiny moment. So if you look at initial data of this type, so I have not yet told you how much decay you need, et cetera, for certain questions. So if you look at Christodoulou Kleinermann's Minkowski stability proof and Sipser's generalization to Einstein-Maxwell case, so the initial data has this kind of form. So you look at the initial data where for large R, the G bar has a form like 1 over 1 plus 2 M over R here, the first term, and it decays like, let's say, R to the minus 3 halves and the corresponding decay is also for K with some regularity. And in my generalization of this result, so I have less decay, so I only have a decay of R to the minus 1 half here and some less regularity. And now you can ask what do you need in order to see some interesting things in terms of radiation or memory effect. So it turns out, I mean, here you get all the information and in fact Christodoulou used exactly the last chapter of his book with Sergio to derive this nonlinear memory effect of gravitational waves. So not only you find the existence and uniqueness of solutions, you have a perfect description of how the asymptotic behavior looks like. So in my case, well, I do see some things, you can compute a few things, but in order to get the full picture, this is kind of hidden in the decay which is very low, very slow decay here. Just a comment, question on notation. Subscript O subscript 3 means that's true for three derivatives of the R piece. Right, so I mean, with each derivative you go down by one power of R. R or U, I mean, well. So here are, let me also write down just the evolution equations, the constraints and the laps just to give you an idea. So I mean, we can then write down, we said the Einstein equations couple into constraints and evolution equations, so phi is just my laps function. So I have an equation to evolve the metric g bar given by this right hand side, I have an equation to evolve the second fundamental form Kij and so here this is actually the R on each space like slice on the right hand side. And if I choose a maximal foliation, meaning I have the trace of K0, so the constrained equations reduced to this set of equations. And in addition I have an equation for the laps of a maximal foliation here to look like that. So again, so you can put this into hyperbolic form, etc. and work with that. Now let me also mention just briefly the stability result by Krzysodulow and Kleinerman. So this was a big breakthrough of course. Question is, can you find, can you perturb Minkowski's space by just a little bit to get a global solution which is chodesically complete for all time? So, and the, well this is one very simple version of the theorem to state it fully, you need a few more pages, but you could say that every asymptotically flat initial data which is globally close to Minkowski globally close in the sense with weighted subolive spaces, etc. certain norms have to be small enough. So this gives you a solution which is a complete space time just tending to Minkowski at infinity along any chodesic. So, and interesting enough, so again, so I was just a few weeks ago I spoke to a few physicists like Stroming or at Harvard and they now talk all about the Krzysodulow-Kleinerman space time. So they are at the moment very much looking at, they are looking for solutions which are exact, there's no approximation anything. And it turns out as I said before, the last chapter of this book gives you the most precise information about some interesting physical space times. And of course you can always try to, I mean, do approximations, etc., but it's interesting also to know if you maybe relax, go do something else a little bit, how does your solution really look like? So, I mentioned the generalization by Sipser, well in my generalizations there is, you don't really see as much of radiation, but you really read these things off that well. There's many, many people that I'm not citing everybody who have worked on related problems. So, let me just say this is actually interesting for us because we get a precise description of null infinity and that's what I would like to stress also. Okay, maybe, well, just here on the blackboard I already explained this, but we usually work with two type of foliations in this setting. So, given by a space like hypersurface, now it's called sigma here, which I call h on the blackboard, and also null hypersurfaces. So, we will be interested in outgoing null hypersurfaces and for simplicity they look just like cones on my slide. Well, one thing again to remember, I use a maximal time function, the trace of this case is going to be zero, each space like slide is a complete Riemannian hypersurface corresponding decay. And then I'm interested in the intersection of these. So, when I intersect a space like an null hypersurface, I get, of course, a second, well, an object which is different morphic to a sphere in this case. So, STU is just the intersection of these. And it will be important in what comes. Also, let me just introduce a little bit more. So, I mean, I can look at the null vector field and I'm writing null vector field L in the outgoing direction and I call L bar the one in the ingoing direction and maybe this will be B4 later and this will be B3. Then I can complement this with an orthonormal frame just on the surface S here. And I will actually decompose now my components, my curvature and my geometric components with respect to such affiliation where this is given g of L and L bar is negative 2. Okay. And well, now let's look at the Bondi mass and something I mentioned before. This is going to be interesting for radiation. So, if you look at one such null hypersurface, call it Cu in our space time and we'll let the time go to infinity. So, we would like to see what happens to local quantities which define locally what's the limit up there. So, I can define what is called the Hawking mass. So, quasi locally. So, I integrate over such a sphere basically, STU. What is the trace of chi and trace of chi bar? So, what's that? So, maybe I should introduce that as well. So, two important objects when I draw this picture again that, well, you see me draw often. So, if I have L and L bar, the generating vector fields in the null direction. So, I can say, well, okay, I have the chi of X and Y. So, if X and Y are in the tangent space here of this S, so this is going to be given by chi, nabla, XL and Y. So, it's a second fundamental form in the null outgoing null direction. And I can look at the bar version, of course, doing the same thing, but with the ingoing null direction. So, these are the corresponding second fundamental forms in the null direction. And interesting for us will be the shears. This will be the traceless parts of these objects. So, by a hat, I just use this to call this the traceless part. And then, of course, I have the trace of these objects which comes up in the definition here. So, one way to write down the Hawking mass as an integral over, of course, a local over such a surface is given like trace chi times trace chi bar. So, the trace of these objects. And, well, it's given like that. Now, questions. So, for interesting space times, let me not be too specific at the moment. So, this Hawking mass actually tends to the Bondi mass when we go out to null infinity. So, if you look at the blackboard over there or here, we go out to null infinity. So, then, this is going to what we call the Bondi mass. So, it has a finite limit for the space times we are interested in. Now, what is future null infinity? We heuristically already introduced that. So, I will call this I plus. So, this is defined to be the endpoints of all the future directed null geodesics along which r goes to infinity. And it has just the topology of r cross s2 with the function u taking values in r. So, you can think of the function u being parametrizing your null infinity. So, and our Bondi mass sits up here and is depending on u, of course. So, each null hypersurface has a finite Bondi mass. Of course, if your space time is not, let's say, nice enough, this could blow up or not be defined. But in the cases we study here that I mentioned, so this is all finite and well defined. Okay, now the Bondi mass measures... So, what does it do? So, it measures in this sense the amount of mass which is remaining in an isolated system as measured at null infinity, at the given retarded time out here. So, basically, if I say, well, out here, this is maybe cu which is hitting, well, this is null infinity for one such null hypersurface. So, I have a certain amount which we will also see the Bondi mass loss formula by saying taking the rebate with respect to this function u. So, this will give us how much radiation actually has happened. So, here it is. So, if you just look at Einstein's vacuum, we don't care about other fields, just pure gravity is acting. So, then we have the so-called Bondi mass loss formula. And again, this is something... this terminology is also used... I'm just copying the terminology of Serge's book with Dimitri. So, this is also in the setting there, derived and well understood. So, you have a Bondi mass loss formula. So, you take the derivative, and on the right-hand side you have this Xi object is actually now the limit of one of these shears. So, this shear with the depart version. So, you sit on one cu, let t go to infinity, and you take the limit. So, this is what is called Xi of u. And correspondingly, this other guy has a limit as well. So, let me just write it down, because it will show up. So, this is called sigma, then. So, these are objects at null infinity. So, I explained this already, so we have the second fundamental form. So, this is the traceless part which gives us the shears, then we have also the torsion we can identify. So, I told you that already. So, let's skip further. Now, what is gravitational radiation? Now, I gave you a setup of the spacetimes we are looking at. But gravitational radiation is now a fluctuation of the curvature of your spacetime. So, your spacetime has a lot of structure in terms of remodeling and curvature, etc. And so, when a gravitational wave travels from the source, so it's changing the curvature of the spacetime. So, it's like a wave packet that may come for, let's say, one second, traveling through. And now, we would like to see what we can learn about it. Okay, also, let me also mention the so-called memory effect for the moment. So, okay. We have heard about LIGO. And LIGO looks like an L-shape. It's an L-shape detector. So, this is the same distance like here and of a 90-degree angle. And if you, for simplicity, assume now that the gravitational wave source is coming from the third perpendicular direction. So, then it acts like a planar wave out here. So, the wave will... of course, it hits from different angles, but in order to simplify the discussion here, so let's assume it comes from this direction, perpendicular. So, what will happen is that these masses will move in the plane because it's like a planar wave when it hits our detectors on Earth. So, these masses will move and what they were able to measure in this experiment is really by laser interferometry, the distance of, let's say, mass 1 from 0, mass 2 from 0. This is where, actually, you have a beam splitter here by laser interferometry. You measure the distance. And, of course, when the curvature is changing, the space time is changing and it reflects in the displacement of these test masses. So, the test masses, you can think of, they're floating on their geodesics. So, you see what the geodesics are doing by looking at the test masses. During the time of the pass by of this wave, now you can ask what happens afterwards and this has not yet been detected but let's hope in the near future, maybe. So, what you think is, maybe in most cases, you would think, oh, well, things go back to their geodesics or as before and everything looks the same. Now, there is a prediction called the memory effect, saying, no, this will not happen. It will be that the space time will be permanently changed and the test masses will be permanently displaced for that matter. So, this is what is called the memory effect of gravitational waves and while there are various studies, many people have, in the meantime, worked on that. So, let me maybe just say a little bit about it. So, in the linearized version or linearized theory of the Einstein equations, the first people to find such an effect were Sildowic and Polnareff in the 70s and then in a fully non-linear problem setting that was Dimitri, who really used the last chapter of the Minkowski stability book of his uncertainties to, I mean, you plug in, the asymptotics also are true for large data one can show. So, he derived from all that he derived the so-called non-linear memory effect. So, people called it linear and non-linear and it was, so the first effect was supposed to be so small never to be detected, but then this effect is actually large enough it's also small but large enough to be detected, hopefully. And so, people always thought this is really a linear and non-linear thing of the same effect, this memory but it turns out, so with Garfinkel we looked at that and we found that these are two different effects which have to do with the linear one we call regular, so this has to do with one portion of the wild curvature, so the one portion of the electric part of the wild curvature changing over time and the null or formally non-linear effect has to do with fields that really go out to null infinity so things that change null infinity so these are two different things and we found this also in the linear Maxwell equations two equivalent, well not displacements but kicks. So anyway, so you can ask well what do electromagnetic fields do or neutrinos, so they will actually add to the second effect which we call null effect now and many people have worked on that, there is work by Blanchier D'Amour, Brackensky, Grishkouk, Thorne and many people I'm actually probably missing but it's interesting that recently froming around collaborators kind of also took up this work on memory and so they have the idea that the memory effect is actually part of a triangle where they look at water identities and BMS super translations and now a lot of people are looking for memory effect in other field theories in ADS CFT and anything you can think of so it seems that there is something interesting but maybe not, it's just the beginning of understanding of what this means in other theories okay, well there are two types of memory and let me now come back to the Riemannian curvature if you think of, let me decompose this a little bit and kind of lay open the structure of radiation I would like to show you again, three denotes just E3 is just the ingoing null vector field right down here and E4 is the outgoing null vector field so if I'm just in the Einstein vacuum equation so the Riemann curvature is my wild curvature and I decompose here the curvature components with respect to this foliation now the most interesting part to remember is this alpha bar so there's a part which goes like 1 over r the tau minus you can think of like u it's 1 plus u square root so there's a part that goes like 1 over r and this will be the interesting part for us for radiation so and again you can look at different space times and see what happens with this so if you stay within the so-called Christodulo-Kleinemann space times you will see well the alpha bar part has a limit at null infinity I call this capital A of u and this is a symmetric trace free to covariant tensor field at null infinity other components also have certain behavior I'm not going to look at that but if you change the type of space time and you do another Cauchy problem to really understand fully what's happening at null infinity so then you can do more general settings so with Garfin-Klevy also came up with a different method so the most rigorous way to study that is of course to do the Cauchy problem for each set of initial data you have well this can be cumbersome if you're just interested let's say in the radiation so we have another method which is an approximation by perturbing the wild curvature which is Gaetschen variant to actually for more general space times also look at radiation at null infinity okay maybe here is just one short note on one of these methods so Christodulo-Kleinemann actually introduced an interesting set an interesting theory looking at let's say elliptic equations on such a surface but then propagating along null or space-like directions so if you look at the trace of chi with the parameter s in the null direction so this is given by on the right-hand side we have some the shear squared we have a trace of chi and then this is now just anything if you plug in a null fluid so this comes from the energy momentum tensor on the right-hand side and this comes if you have Einstein-Maxwell equations this will be a component of the electromagnetic field so in general when we look at the Gauss equation how do these things look like so I can write down the curvature of this intersection I call stu in terms of as follows I have the trace chi chi bar on the right-hand side I have here the product of the shears plus w I call just a component of the wild curvature and contribution from t so I have either quadratics in shear or trace of chi one pure component of wild other than the alpha bar alpha bar is the one with least decay or contribution from the t on the right-hand side so and if I look at the null-kudatz equations in this notation so we can define what we call a mass-aspect function or its conjugate so c does just torsion so this is given again of this structure and we can write this with help of the Gauss equation we can actually write this mass-aspect function like this and now if you look at the null-kudatz and the corresponding conjugate kudatz equations so they have a form like that so you have either quadratics of Ricci coefficients on the right-hand side or the derivative of one of those here's another quadratic or a pure component of curvature but not alpha bar not the worst decay and contributions from t if any so I already told you about the limits so the limits again so chi hat and chi hat bar the shears have limits like on the left blackboard here at null infinity now the structure let me introduce a little bit more about the structure so if psi is a component of second fundamental form or torsion phi is a component of the wild curvature and t just energy momentum so then you can write down and n is just now a normal to s into the space like slice so you have a structure of the equations when you look at the propagation here for chi hat so you have a quadratic again in psi or a derivative of such a psi term and we have just curvature or t itself and eta hat is actually the chi bar over there, chi bar hat so similar for the other equation if you look at this type of equations and take limits now I would like to understand what happens at null infinity so let's just multiply by the r that is needed and then we see on the right hand side so alpha bar has a limit which is called a so we get for these equations we get limits actually for many different space times this is actually true that the behavior at null infinity so we have this behavior between the shears so we take a derivative with respect to u and one shear here is related to the curvature at null infinity which is the one part of the curvature now energy radiated so we can say well in the pure Einstein vacuum case so the energy radiated is integrated from minus to plus infinity let me show this here so if you are in pure Einstein vacuum you only have this shear part but if you add electromagnetic fields you have a contribution from the electromagnetic field or one portion and if you have let's say neutrinals that you can model by some null fluid you would also get a positive contribution to this energy which is radiated away and this is so we integrate this from minus to plus infinity at null infinity okay here is maybe a theorem just generally I mean in the Einstein vacuum case you can just forget about the s but so what happens if you have different fields in the Einstein equations so we can say that this sigma so the shears of this chi hat over here so this is the limit of this shear so it has limits at plus and minus infinity for u so this has limits and this will be related to this permanent change of your space time after gravitational wave has passed and well s is now be any tensor of function which depends on the fields with the right decay and t will denote any lower order components of the stress energy tensor well what the what the theorem says well we also have some function phi which is the solution of the following so we have f minus its mean value f bar over a sphere at infinity and this strange notation means this is just at the sphere at infinity and then this difference sigma sigma minus is given by this equation so in other words the radiation here or what is radiated away this energy comes into the difference of this shears at null infinity and this is directly related to some displacement of test masses so maybe let me skip the ideas of the proof and do something else so you can prove this you need some to investigate some hot systems locally and then go to null infinity and take the limits of these and see what happens out there well now it can be shown that for now we are still in asymptotically flat space times so the permanent displacement if I write this down again you remember from before so maybe here so if you have test masses of this type and the gravitational wave comes and travels through so now the claim is and this is not a Laplacian this is just a delta the claim is that the permanent displacement is given by this right hand side and you see this right hand side is this difference of the shears at null infinity that I just developed in the theorem and in the theorem is linked to the energy irradiated away which is then sourced in the pure gravitational case it's sourced by the limit of this Xi so of this null second fundamental form here and if you have extra fields like electromagnetic or neutrinos they will actually add to that okay and so there is also something called the ordinary memory that's what people thought was the linear memory before so and this is now a completely different thing so the null memory is what the big portion of this permanent displacement actually is and well maybe here very briefly how does it actually now relate to experiment so we have well by really starting the Cauchy problem and looking at the null infinity so we can really derive a lot of geometric or analytic information but how does this now reflect anything in the experiment well if I set up an experiment so let's think of three geodesics in spacetime and denote them by gamma zero one and two so just geodesics on which my particles are floating and well T is my future unit time vector field sitting at gamma zero and then I'm looking at what's happening so I look at these geodesics I introduce an orthonormal frame field so I can construct the frame field along the gamma zero geodesic and well I can then set things up so that I can measure nicely what's happening with the distance between one and zero and two and zero now I can well under certain circumstances I can replace a geodesic by the Jacobi equation and I have two derivatives on the left hand side which give me a Riemann curvature component on the right here interacting, contracting with XL and interesting enough so depending everything is really sitting in this Riemann curvature component so you really need to understand the Cauchy problem how your spacetime looks like to really understand what's happening here and the interesting thing is maybe let me also add now a null geodesic or a null fluid so basically your team you knew is some positive function here K i, K j or K is a null vector and looking at the twice-contructed Bianchi identities if I plug in a null fluid so we have shown that well the null fluid will also contribute the Einstein equations for a null fluid reduced just to this equation here so the spacetime which is given by some constant times the energy momentum tensor where the null fluid is sitting now if you look at the portion of this null fluid which has let's say the right decay behavior so it decays like 1 over R squared and something in U and we like to understand how is now the Riemann curvature related to this portion of the null fluid which comes in through the right side of the Einstein equations well we know that the Riemann curvature can be decomposed into the traceless part which is the wild curvature and we have Ritchie curvature and scalar curvature here and so well if you plug everything in and we look at the let's say worst components we find zero is just a T element so we find that well if you look at that well we have some component in the pure wild curvature but also through the Ritchie coming from the null fluid if you look at the Ritchie component we go and look at the corresponding Einstein null fluid on the right hand side so which component we plug in the worst components and we use null foliation again and when we plug everything in and go back to the null L and L bar notation we can see well there's two things that happen first of all I have this alpha bar curvature which is going like 1 over R I can define the limit and write it down like this and I have a component of the null fluid which goes like 1 over R squared so we would say well wait a minute so we have 1 over R in the curvature and 1 over R squared here so it does not contribute so this is lower order so it's true for the instantaneous displacement so when the gravitational wave is traveling through so this is indeed lower order for that but it turns out that for the cumulative effect afterwards the memory this is actually of the same order how does that work so let me write down okay I have maybe two more minutes so let me write down the second derivative here in the Czechubi equation again on the right hand side and now this is the notation for the curvature component at null infinity that 1 over R component so now we have this nice relation between the shear and the curvature and we know also that the lemus of this psi goes to 0 when u is very large so this means that if you have test so here we have just if you plug this in this means that if you integrate here once or you substitute this means that the velocity will be 0 after the gravitational wave train has passed so something happens but for large u the velocity will be 0 so they go back to rest I substitute twice here the shears and plug that into this equation and when we take the full limit I get exactly what I told you before namely this displacement here which is this permanent displacement is given by the difference of these shears which again behind the right hand side is this energy which is radiated away so what exactly contributes from the spacetime what exactly contributes to this to this displacement okay maybe I should say one more word just at the end so with Garfinkel we studied also cosmological spacetimes the question is if you are in a cosmological setting are these things still there how do they interfere how would for instance the cosmological constant come in so does this play any role so we look at Friedman-Lehmann-Robertson-Walker plus dositor so with positive cosmological constant and we found for the dositor spacetime so we can write down the metric like that so the dositor spacetime models the basically inflation period of the universe so that's one way to think about it so in that case we found in dositor that well there is a factor 1 plus r h 0 the Hubble radius which are the Hubble constant so which is giving you an enhancement of this memory effect so the memory effect is multiplied by this factor so it becomes bigger something similar maybe I jump through that to the very end but something similar you can say for the FLRW case so this is working progress we're writing it up but for FLRW which is modeling basically our present universe so you can also see that this memory will be enhanced by such a factor so maybe I stop here thank you does the error is there a lasting effect on the area of the triangle pardon do a triangle so first of all let me maybe make this a little bit more precise so if oh yeah exactly so if I have just a circle and the wave is hitting from here so what it does basically it's kind of stretching and squeezing it right so it's like a planner wave so this is what it does and you can think of this like well at pick points on that and now the claim is that with the memory so you would have this has a certain well displacement in this direction and this is by 90 degrees in the other direction so this will be permanently displaced and then you can actually compute what the angle would be but what the area would be but it's not clearly bigger or smaller is it? Not really I think maybe you could compute that so what means after LIGO they saw that it saw their one signal and now they have to recalibrate their machine well that's actually the problem so this memory is at low frequency and also what they saw in let's say if you look at the strength of the signal they saw let's say they dealt the lambda over lambda is like 10 to the minus 21 of what they saw now there's a paper proposing to to measure memory with LIGO that just came out a few weeks ago so they propose something like at the order 10 to the minus 22 and also all the noise that you can think of is at low frequency for LIGO so it's probably not the best way to look for it but nevertheless so some people are proposing to do that and they have some kind of a filter they put there so I don't know they are improving the instrument Yeah right absolutely I mean the place to look for that would be the space project that NASA and ESA has been working on and I think the Pathfinder was launched by the European group so in space this would be easy well easy technical details now modulo those out so this would be really easier to see there also with the frequency and no noise Could you say a little bit more about what you meant but that there is a memory effect even for Maxwell's theory? So actually we thought about we looked at the pure Maxwell equations which are linear of course and there's some if you take let's say Jackson and take one of the formulas play around so you get easily what is called the linear what we call the ordinary memory and this would not be a permanent displacement but you could think instead here these masses are not charges just test masses following the geodesics but what you would do for the Maxwell cases you have charge test masses and you would get an overall velocity after the passage of an electromagnetic wave so a kick so we found the ordinary thing and we also found a null kick which is then again bigger and corresponds to the null memory here actually and that's for the pure classical Maxwell equations of course you can now ask what about QED well one thing seems to be there the other we don't understand yet okay some more questions well if not let's find the speaker again thank you