 He's always on. Yeah, now he's working. Yeah, he's always on. OK, now Professor Zheboydev from Harvard University will start his lecture series on stringy aspects of gravitational scattering. It's a great pleasure to be here. It's my third time in Trieste. The first time I came as a student for this, exactly this school. But the only thing I remember is a name in the face of my roommates from the school. So everything else I forgot. So my lecturers have two parts, and they're quite separated. So I will be talking about gravitational scattering and several aspects of it. And so the first part will deal with IR. Yes, and please ask me, stop me. Because I planned a lot of material, but we better not go there if it's completely un-understandable. The first part is IR, and by IR we mean low energies. And say, late time, large times. And the second part deals with UV, so to say. So it's high energies. And they're quite disconnected, as you can imagine, even though maybe they are, in reality, less disconnected than they seem. So today I will obviously start with the first part. So let me just erase that. And the useful, before I start, maybe just may write a useful reference to this part because it's a course taught by Andy Streminger recently. And the videos are recorded, so you can watch if there are 10 or 22-hour lectures. If you are interested, just look. OK, in the low energies, we, of course, expect gravitational theories, even if we think of gravitational theory as being some effective theory with high derivative corrections. At large distances and low energies, it reduces to Einstein theory. And usually one way to introduce this topic is to draw, or for some reason it is drawn as a triangle. So let me erase it, which represents the plan of the lecture. So what is this triangle? So triangle, let me draw it like that. So each of the vertex corresponds to set of ideas. And actually, each of these ideas is very old. So I'll try to write a little bit larger. So one angle is a sovereigns. That can be memory effect, memories, and third in asymptotic symmetries. These are three recurring topics. So if you consider scattering in flat space, that's what I will be mostly interested in. These are three things which I usually discuss, though, in different subjects and different areas of physics. So soft theorems, they deal. And here, the subject is very old. So the reference would be 64th paper by Weinberg. The soft theorem deals with low energy limit of scattering amplitudes. So if you have some scattering process and you take one of the particles, momentum energy of, let me write it, energy of the particle goes to 0, this describes the soft limit of the amplitude. And so Weinberg found that this limit's universal, and that's known as a soft theorem. Then the second corner is, say, asymptotic symmetries. And in the case of flat space, again, it's a very old subject. It comes from 1962 in the works of Bondi, Wunderburg, Metzner, and Sacks. And as we will discuss also in details, asymptotic symmetries are the symmetries which will be relevant to consider for scattering problem. But so the more formal definition you can write is that asymptotic symmetry group will be set off in gravity allowed, and we will discuss what it means, allowed defiamorphisms over trivial defiamorphisms. And you should think about it as a large gauge transformation. That's how they are called, also, usually in the literature. And the third corner is, say, memory effect. Here the relevant paper will be from 1974 by, at least in the gravity context, by Zildović and Polnarev. And this describes a residual deformation of a gravitational detector, actually. As usual, I will also discuss in detail residual deformation of a gravitational detector. So now these three subjects, actually, they were very old and they were pretty separated. So theorems, mostly a particle physicist, are aware of this, but it's sort of less known in the other branches. And asymptotic symmetry is a subject and the questions which people who are doing mathematical relativity are doing. And memory effects, people who are doing astronomy and astrophysics and study gravitational waves, they would know about this one. But let me write it as true. But until very recently, these very old subjects, which are 50 years old, were developing sort of separately. And the recent development was that they actually all the same thing in disguise. And the driving force, I would say, of these recent developments were this rich interplay that if you have astrophysicists, you think about detecting gravity waves and it gives you one insight. If you're a particle physicist, you have a completely different picture. If you're a mathematical relativist, you have a third one. And the power of this recent development was this interplay which was constantly driving. You know something on one side, you translate what it means on the other side. And in this way, many things, new things in each angle were discovered. But why would you say, you can ask why? So why would we study this old material? It's already 50 years old. And how did people come about these things? And my picture is that there are two major steps. Well, of course the big question, if you want a big question in this field, is always, let's say, flat space holography. We would like to understand, we would like to have something like ADS-CFT, where we have a non-perturbative formulation of observables in asymptotically flat spacetimes, what is the S-metrics. But at the moment, it says that the project is very far from that, and it was the stage of, let's understand the basics of this problem. And the recent excitement come from first, there was a paper on 2,000 Nain by Barnish and Thoasa who discovered sort of Virasora symmetry in the sky. Virasora, let me call it in the sky. I don't know if it means much, but basically they made this analysis of asymptotic symmetries. Olden, they found that similar in the stories from the 70s, which we just mentioned in the previous lecture, that you can extend the group of asymptotic symmetries, extend the group of global conformal transformation in the sphere to the local ones. And then the question was, they call it super rotations. I'll talk about it in details. And so it was very suggestive. What does it mean? Do we have a two-dimensional CFT? And people, my impression, got a little bit excited. And we're trying to understand what it means. And then there really is a flood. It opened in 2013. In the paper by Straminger, who sort of was able to understand at least two points of the corner. And then if I told you that all the things are the same, so you should draw the connection between them. So he was able to understand, to know enough about this mathematical relativity and particle physics at the same time, to understand what is a relation. And he explained that these two things are the same. And if you wish, this is the word. The soft theorem is the word identity of this, of this asymptotic symmetry. Then it was understood that if you think of this kind of residual deformations, which people discuss in astrophysics, they also naturally appear in this asymptotic symmetries as well. Sometimes it's called vacuum transitions, even though it probably causes more confusion than it helps. But let me just follow the literature so people understand the connection between asymptotic symmetries and memories. And memory, as I told you, it's a residual deformation of the detector to the response of detector at large times. And soft theorems, it's low energy. It's large time. And as you can imagine, that low energy and large times are related by Fourier transform. So these two things were kind of Fourier for each other. Not exactly, but. Sorry, yes. So yes, I have to explain residual. Let me explain it here. So if you have a gravitational detector, it's just say two particles moving on the geodesics. And then when the gravitational wave comes, they start oscillate. So the distance between them changes. And then the gravitational wave is gone. And there can be a residual effect that the distance between them is different from the original one. So residual is, in a sense, when we have some gravitational wave. And then the radiation comes, and the radiation is gone. And we again sort of end the vacuum. And this is exactly. So in this wording, this vacuum doesn't refer to a vacuum of quantum field theory, because it's statement that the radiation modes are not excited. And so the transition between one state without radiation to another state, well, people who study gravity waves are knew that there are sources of gravity waves. Well, actually, generic sources, for example, the one by LIGO, would lead to such an effect that there is a deformation, residual deformation, from the original state of an antenna to the different one. In this case, LIGO is sensitive at around 100 hertz. But this is a very low energy effect. So that's why it's with Earth-based interferometry. You cannot quite measure that. So now, basically, after this paper came out, since the end of 2013, there are around 200 papers on the subject. And if you have all these papers, you can sort of, I wanted to do this review, you can sort them by first choosing, so in which, say, which corner are we in? And write a paper on either some theorem or asymptotic symmetry on the memory. Then you can choose which theory you would like to consider, because even though I'm talking about gravity, just for no fundamental reason, you can talk about gravity. You can talk about QD, which is an even older subject. You can talk about supersymmetry. So you can choose your theory. So after you choose your corner and your theory, you can choose, so if we have a small parameter or a large parameter, we can expand in this parameter. And so there is what is called leading and sub-leading and sub-sub-sub-leading, et cetera. So theorems and the same translates to all other corners. You found the soft theorem. You question what is the asymptotic symmetry that corresponds to it. You found an asymptotic symmetry. What is the soft theorem? What are the memories? So you can travel around. Well, and of course, say, the number of dimensions also say, you can choose the number of dimensions. In some corners, you feel that number of dimensions is completely irrelevant. For example, if you're a particle physicist, you know that there are soft theorems in any number of dimensions. But if you're a mathematical relativist, you will find it very confusing, because say, people thought that there is no BMS symmetry in D larger than 4, et cetera, et cetera. So it's basically consort all the papers written on the subject in this way. And so this is a picture that's supposed to cover this, broadly this field. And I will focus on gravity in 4D in my lectures and try to explain the story in more details there. But please ask me questions now. So let's say we can take D equals 7, we take a gravity, and then you can ask, OK, so what can we say about gravitational scattering in seven dimensions? You can ask, you can study scattering amplitude. So you can study the asymptotic structure of field at null infinity. And then, depending on D, it happens that these questions, they depend on D and they depend on the theory. But in principle, and then you proceed to which order, because the leading order, everything is known, the subleading order. And of course, also, there is an E, which is sort of important E, which I forgot, which is classical versus quantum. You can start studying loop corrections and what happens, how to think about that. One thing also is I want to say from the beginning that I don't want to present the subject as something which is already written in stone, that everything is clear. I think that many aspects are pretty confusing. And probably in a few years, it will be a better way to think about it. So I chose my way through and tried to explain the ideas. But probably, maybe there is a better way, or it's developing quite rapidly. So any questions with this picture in these corners? OK, so now if you have this picture, you are already pretty far. Because we will not just discuss what exactly is the asymptotic symmetry, what is the memory, et cetera. And I would start with gravity in four dimensions. I would follow several steps. And these steps will be of increasing complexity, because so that we can slowly understand what's going on. Can I read here, or is it fine? So let me discuss gravity waves in flat space. OK, so let me remind you how the Lagrangian finds the theory look. We have curvature plus potentially corrections, matter and some high derivative corrections, which I will not be care about, because they're all irrelevant. In particular, as I will discuss a little bit, you can consider theories which are gravity. When I mean gravity, you can also add high derivative corrections, et cetera. But the nice thing about this IR part that is pretty universal and things which I will be talking about, they're pretty insensitive. You can add any high derivative corrections you like. It doesn't change the story. The questions of motions, of course, take the well-known form. T minus stands for the stress tensor. And now to discuss this effect, let's consider a very simple example. Because we consider a flat space plus some small perturbation. And well, there is some reference frame where it's small, so it's enough. And now we can, yes, also the I will be working in this lecture in a signature minus plus plus. OK, so if we consider this kind of perturbation, because as you know, due to different variants, we have a gauge symmetry. And the field equations would be there is a redundancy in the description of the spacetime of the type. So as a standard step, we can fix the gauge, which has many names. Lorentz, the Donder, Harmonic, I believe they're all the same. So let me fix the gauge. And then there is a residual, so that's a gauge condition. There is a residual symmetry, which, again, you can shift. You can make some differ morphisms with vector field. Laplacian of it is 0 and eliminate formal components. And in this way, we can end up describing what is called transfer stressless gauge, where h0 mu 7b0, trace will be 0. So it's very simple, and dj, dj will be 0. And the equations for, if we plug to the Einstein equations in the vacuum, we will find simply that box, htt, hij is equal to 0. So if we start in four dimensions, we have 10 components. This eliminates four. This eliminates four more. So this indices, i and j, they're transfers. So if you choose the direction of propagation of the wave, we have two components. And this corresponds to the fact that graviton has helicity 2. We have two components of the gravity waves. Now, that's you all know, probably. Now, how do we measure what is a gravitational wave detector? Gravity wave detector would be simply, as I mentioned there. We have a pair of particles. Say, each of them follows a geodesic. So if this is a massive particle, we characterize it by four velocity. Formical minus 1. And the geodesic equation is 4d tau, 0. So I hope this is also familiar. Now, when we study this memory thing, or just in general, we're talking about response. We're talking about LIGO. That's what we have. We have just particles that follow geodesics, which are given by these equations, where this capital gamma Christoffel symbols for the metric. Now, questions? That's right. Exactly. So in flat space, this gamma is a 0. Particles, I just say, stay at rest. And then when the gravitational wave comes, we will now discuss the distant changes. That's how we know that there is a gravity wave. Also, if we have some spacetime and this notion of distance, this is the notion of a distance between detector or synchronization of the clocks is coordinate independent because you can sit here and they can exchange signals and say each other, I observe this time, and I'm here. And then by measuring time, it takes light. They can measure the distance. They communicate via exchanges, but it's not very important now. OK, so now let's consider the solution to the correction to the metric due to the presence of the gravitational wave. That's very simple. So I chose this coordinates as the following. So u and v are just u is t minus x3, v is t plus x3, and z are coordinates in the transverse space. And because the gravity wave is just a function of retarded time, this is a solution or finds any questions to the first order. And then the wave travels in x3 directions. And all as an exercise, you can check that if a particle stays at the fixed position, say, z bar 0, this is, if you take this metric, you compute the Christoffel symbols, you find the geodesic motion in the spacetime, you can check z and z bar as a geodesic equation. Now imagine that we start with two particles, which separated by some distance in the z plane by some distance delta z. So there is a distance delta z and delta z and delta z bar. Now as I just told you, that this separation in coordinates z, it's solved geodesic equation for any time. But as time progresses, the metric changes. And in particular, we can consider this function h, something which maybe starts from 0, nothing comes. Then there is some gravity wave. And then there is a ring down, and then there is some residual transformation. And consider such. And this is time. Now as h changes, the distance between two particle changes. And this can be seen that by just evaluating, if you take two points at the same time and the same position x3, and evaluate a user metric to compute the distance, we find that the L of u is L0 z square plus this effect of the wave delta z square plus h z bar z bar delta z bar square. Now this is called gravitational strain. And so for the, again, just as a warm up for the LIGO event, this is 10 to the minus 21, the maximal peak. So it's pretty small usually. That's why you have to have a very sensitive detector. But now notice that if we say, this function of h of u can be arbitrary here. In particular, we can choose the original and the final value to be different. In this way, the gravitational wave comes. The distance oscillates. And then after the gravitational wave passed, we had some residual deformation. This residual deformation of a gravitational detector is called memory. So that's a simple. Yeah, it's a distance. It's a distance between two detectors. So we have a space time with a fixed metric. And we have two positions of the detector. So for example, I send moreover, as I will discuss, it's a flat space. So we're just in Minkowski space. We use perturbed metric. We use perturbed metric, yes. But let me now ask your questions in a slightly. Actually, how long can I go here? Is it better stop or is it fine? OK, you don't see. OK. So what is the final space time? Well, the final space time is, let me go here. The final space time is simply that. It's ds squared minus du. Yes, and I refer to Wikipedia for Christoffel symbols and Riemann, so I don't write them. Please check Wikipedia or any other book you like. So the final space time is this, where h final is this final value of the metric. And it's a constant value. It's just a number. Notice that we are working in a linear theory. You can check as an exercise against that this is a flat space. This is just flat space. And you can make it if you have morphisms, which looks like that, which will bring the metric into the usual flat space form. So we can either use, and basically in this simple example, we have the original space and we have a final space, which are both flat space times. But they differ by this defer morphism. And you might think, and in many usual texts and treatments of GRs, that two spaces which differ by defer morphisms, they should be physically equivalent. And somehow the whole driving subject or recurring topic in this story is that sometimes it's meaningful to talk about this large defer morphism, such as a physical. And you should be careful when you talk, even though there is just a flat space, and the two flat spaces which differ by defer morphisms, they can be meaningful. For example, here when we do this, if we follow the detector and we do this transformation on the metric, it will become just the original metric of Minkowski space. But as soon as we are acting on this with this transformation, the detectors, on the positions of the detectors, they will be not invariant. And in this way, we can either detect this change in the length by looking at the metric or equivalently, we can make it defer morphism. The endometric is Minkowski, but this residual memory is encoded in the positions of the detectors. So this large defer morphism in the context of the whole Minkowski space will, well, let me just mention them yet. They will be analogs of, say, something which is called super translations. But here now we are not discussing. We are doing a very simple, very mundane local analysis. Nevertheless, we have sort of main ingredients with slowly emerging. Now, let's ask what is the relation? Yeah, why also one thing which, so you might think that electromagnetism is simpler than gravity. And this is one of the instances when that's actually not. Because, as you know, in quantum electrodynamics, gauge transformation, when it act as a detector, in which in this case would be a particle, say, it adds to the phase. And the classical phases of particles are not observable. And to observe them, we should do some kind of Aron of Bohm effect. So in QED, the analog of memory is Aron of Bohm effect. That's a dual description QED. So H is small. H is small. I worked for first order and H is a small parameter. And this is Minkowski space. Well, here, yes. But if H is. Do you expand? Yeah, yeah, I am. Yeah, I'm sorry. Here I was working to first order in H. Everything so far. We can do, at the end, I will do all the statements precise and full in full nonlinearity. But here, for simplicity, I just work the first order in H. Let me not call it largely thermomorphism here. Sorry, I'm sorry. Let me. It's confusing. I don't want to call it largely thermomorphism. Yeah. Right, at the moment, this is an assumption. But we will see that that's what happens. I will show that for the sources, which corresponds to the problem of scattering, this is what happens. So, and moreover, we will see that, well, that's more or less in gravity that is what happens. That's the only thing that could happen in the difference. For example, there are sources which go to zero at the end. So at the moment, it was just an assumption. And now the simple comment, of course, that if you want to do a Fourier transformant, if you define the Fourier transform of this wave, then this residual deformation, this delta H, is encoded in the pole at low frequencies. It's, again, trivial statement. But it will be exactly what's happening again and again. So again, if you take a function of this type, you do a Fourier transform. You will see that this change is encoded in the pole of the function. And you see that in this way, we see that there is a soft theorem and the soft radiation is related to this large time effects. And when I will be talking about Minkowski space, of course, it's the full space. It's much complicated. But if we are observers who are sitting locally here on Earth and we have an antenna, that's what we see. Now let me move, start making things more confusing. And to make them more confusing, let me switch to a global picture of a spacetime and draw. So when talking about scattering problem in Minkowski space, it's useful too because Minkowski space is infinite. It's useful to put it in a finite region. And this is done by means of the so-called conformal compactifications. If we want to discuss a causal structure of spacetime, so we rescale the metric by conformal factor. And the known fact is that it doesn't change the causal structure. And by judiciously choosing conformal factor, we can make infinity a finite distance. So let me discuss Penrose's diagram of Minkowski space. The way you see it is the following. This is, let me maybe write some useful coordinates so that it is a metric of Minkowski space. So we have time, radial direction, which goes from 0 to infinity, an angle on the sphere. And on this diagram, the time goes horizontally. This is the slices of a constant time. This is a t constant. This is a constant R. And the interesting part is the boundary of this region, which is infinitely far away on this diagram. And there are several things which people usually discuss. So they are called I plus. I plus is a future time like infinity. That's where massive particle end. Then there is something which is capital calligraphic I plus, which is called Scri plus. Scri plus. It's an null infinity. That's where light ends. So the great thing about Penrose diagrams is that light propagates at 45 degrees. Then there is I naught, which is called spatial infinity. And then there is I minus, which is called Scri minus. It's a past null infinity. And then there is I minus. One thing which you might maybe remember is that, and will be important for us, that even though on this diagram, all these three regions, yes, sorry, and at every point here, this is a two-dimensional diagram. We have four-dimensional space. So there is a two-sphere hovering everywhere, which denotes the angle on the sphere. So one important fact about this diagram is that even though this null infinity is and spatial infinity merge at the point, they actually, this configuration is singular. And when we will be approaching, if you imagine a big barrel or big cylinder in spacetime and you start making it large, and then trying to discuss how we approach I plus and I minus, I don't know, for example, if you have some observer that send light here or here, we will find that this null infinities will be actually infinitely far, even though when we will be, even though they merge here at the point, in a sense, in certain situations, it's important to remember maybe to think about approaching this point as a limit from an observer that fix time who are going at large distances and send lights to the future and to the past. And then we will see that this data, even though on this diagram it can be at the point, it can be actually pretty far. Sorry, let me maybe not, the moment it's not saying more about that. And now the scattering problem corresponds to the something of this type. We have a, what is the scattering problem? Well, we specify the data. Something comes in, some particles. And then comes out, given an initial data, what are the waves coming from minus infinity? The problem is to find the result, square plus. And now to discuss this global structure, yes, and for example, let me write what, yeah, I will be scattering just some massless particles. I will be thinking, even though we can scatter massive particles coming from here, anything, we can scatter whatever we like, but simplicity, we can scatter gravitons, photons or any massless particle. But now, so I would like to repeat the previous exercise for this kind. Oh, this is not, here it's not 45 because here it's not massless. It could be a rocket or it can move on the sphere. So, so it's, maybe it's a, we can draw a better picture. Imagine, let me consider, that's what I want to consider now. So, we have a point, we have a light cone. So, for this kind of picture, we will just have one line. So, I wanted to resolve them sort of on the sphere. And now, we can imagine particles coming along null rays and going in some direction. So, they collide at the point and then proceed to null infinity. This to the leading order, let me describe this process as just a given stress tensor in Einstein equations. And the stress tensor is, say we have some of our particles integral over the proper times, their momenta, is there for velocity. And then we have a delta function that's the position of the particle, which is. So, this is a stress tensor for a set of incoming particles for t equal less than zero. So, it is localized at the position of the particle, which is given by unit times tau. It moves in time. And it's stress tensor is related to its momentum. For velocity. And then as they collide, we have the same kind of sum, but we have a set of final momenta with some other choice of positive times. Now, we're asking the following question. What happens if we, before we observe the local gravity waves, but what happens if we have this gravity wave coming from the event of this type, where particles coming from infinity collide and then they go to infinity again? Well, to do that, to find what h is, in this case, we simply need to solve, we simply need to solve the, well, equations of motion for the h. And of this type, indices. And then, so we get the, we need to, if we have a question of the sort, the h is equal to the integral over space time. And here, we arrive to retarded, retarded propagator and the source. That's what we need to do. So that's how we find the field from a given source. Now, let me, what's a little bit in more details? What's going on? This integral is, of course, very, it's very easy to do. Something subtle that might happen unexpected. So let me focus my attention on the detector, which sits. First of all, so this is the usual coordinates. Let me introduce so-called retarded coordinates where we can use the variable u t equal minus r and the metric becomes minus du square, minus two du dr plus r square. Now, if we take the limit u fixed and r to infinity, that's a limit where we are approaching this scri plus. So that's where the light goes. Well, I would like to compute now the metric in this equation. h is a function of, h is a function of u and r, some angle on the sphere. So to do it, recall that the retarded, the retarded propagator is this. It's a delta function localized along the light cone. So if you have some source, if you have a source, you can draw a light cone and from the source, and that's where the green function is localized. So it's very simple to do this integral. It just gives us a delta function. So what we need is actually to compute something like this. We need to compute the integral d tau and then we have a delta function of x minus u mu tau square. This is just this delta function. And the four integral here is gone because of this delta function. Okay, it's very simple. This integral is completely localized. But what's interesting is that, let me just write some in details. So we have x square minus tau to x u for momentum minus tau square velocity. And now in this coordinates, if you take this limit, to take the limit of large r and fixed u, we find that this say x u becomes a product of r times n u. So it's linear in r. And here n is related to the, it's a four vector related to the direction of observation. If we have a detector at the position given by some three vector, unit three vector, then x u goes to r, this n four vector n dot u. And then an x square goes in the large r limit. It's also linear in r due to this piece. This term r times u. In this way, you can find the tau. You can let me leave it as exercise. You find the tau, you plug, you solve that this equation should be equal to zero. And then you compute this integral for the sum. And of course I almost did the computation for you, but the result is that, the result is very simple that, so h, let me actually write immediately as, so h will be equal to say theta. Each of you theta of u sum over final things, e final h a b, final u, final b over u final times this four vector n, plus delta theta function of minus u over the initial. And this thing is, so there is a jump. It starts with a constant value and it ends in the other value. So there is a change as we discussed before. And this change is just what is known as a soft factor. Take the difference between the two. Now several comments in order. First, you see, yeah, that's an important thing. Thank you. That's r. So the leading, we take an r to be large and the metric decays with r. So decays linearly with r, with this change. Moreover, it's, well, now let me do the following remarks. So first, this thing is known as a soft factor and it appears also, for example, in the, if you take maybe, let me just postulate it. Now, I haven't derived it, but you will get the same expression if you compute. You take this pre-factor. You take a limit, soft limit of the amplitude, energy of one of the graviton goes to zero and it's related again to this exercise we did where we saw that it's, this residue is related to the pole in the Fourier transform and here now we are saying that for this kind of scattering with the source, the source of the pole is controlled by a soft limit of the amplitude. Now, another important point in this, I will use my, I will end on it. It's a very, it's very crucial actually. Maybe I would even say that in the original paper by Strominger, that was a crucial, crucial insight, which is not completely, well, actually, yeah. Yeah, so here I, what is A, right? Oh, great. So let me explain what is meant here. So this is, this model that we consider with the sources, of course it's a model of a scattering amplitude. We consider scattering amplitude with this momenta, pi one, p one, p i, this is final one. Now, this is what is meant by a n. This is a scattering amplitude with the same momenta as we used for the source. Now, omega is a statement that we consider amplitude where we attach one more particle, which is a graviton in this direction of sight with this momenta. So it's a graviton attached in the direction where our detector is inserted. And now this is an identity, this is a statement that this factor that we found here, which come from the sources. It's actually completely universal and this change, it is totally insensitive. So here's the sources we're interacting at the point. But the result for the change, which was in this, we obtained in the simple model, it's completely universal and is insensitive to the, the details of the process. And actually this would be completely general. That's the meaning of the formula. Yeah, I wanted to explain this crucial insight, but maybe I'm out of time. I better start next time. Thank you.