 I see many people are still connecting. Yeah. All right. Are you ready, Mallory? Yeah. Yeah. Good girl. OK. So we're going to start the recording. OK. All right. So we are very happy to have Bayer-Omke, who will start the set of lectures on the traditional waves from the early universe. Yeah, thanks a lot. It's really a pleasure to give these lectures. In a way, I've been looking forward to this since I was a master's student myself, visiting the ICTP summer school. So yeah, the topic of today is gravitational waves from the early universe. And I invite you, some of you have already figured out what's with the QR code here. So I invite you to either scan the QR code or just enter the address of this link here. This is to have a little bit of interaction during the lecture. So the first question that you'll find when you click this link is a question kind of about how much you already know about gravitational waves, which will help me to judge the correct level for the lectures. So the plan, the outline for the three lectures that we now have in the next three days is today we will cover gravitational wave basics. Many of you will have heard something about gravitational waves, but I do want to spend some time explaining what a gravitational wave actually is, because it's not quite as trivial as one might think. Then tomorrow, we'll talk about the stochastic gravitational wave background. That's essentially the gravitational analogon of the cosmic microwave background. And then in the third lecture on Friday, we'll see kind of how we can use this stochastic background to actually probe particle physics. Particle physics with very high energies. And so here's some literature relevant for the course and particularly is these first, I mean, essentially, this first item is more or less the first lecture, and then the second two items will become relevant for the other two lectures. And just to give you a feeling, so you can see how much maybe the other people here in this room currently know about gravitational waves, this is the current status of the pulling results. So I see we have a couple of experts, but it seems I'll be able to teach all of you something, I think. OK, so let's get started. So the first topic that we want to talk about is the wave equation, which is essentially the final equation of gravitational waves. And to do so, we will start from Einstein's equation. And I'll be doing, essentially, all of the lectures I'll be doing in handwriting. So I mean, you know best what's the best way for you to learn. But I do encourage you to take notes if you find that useful to follow, because I won't be writing down absolutely everything that I say. So Einstein's equation relates on the left-hand side objects that are related to the metric to, on the right-hand side, objects related to the energy content of the universe. So G here is Newton's constant, C is the speed of light. At some point, I'll switch to natural units, but let's keep them for a moment. And this entire left-hand side is also often called capital G mu nu. And so what essentially Einstein's equation tells us is that the curvature of spacetime here on the left, which is described by a metric tensor G mu nu, from which, as you can see, once we have G mu nu, you can essentially compute this entire left-hand side. So this contains such a new geometry of GR. This is influenced on the right-hand side by the energy momentum tensor of the matter content of the universe, which is T mu nu is the energy momentum tensor. And essentially, what GR tells us is that the way space is curved influence how particle movement space, but also the energy content of the universe itself can curve the metric. So this is really an arrow going in both directions. And now, what we want to study when we study gravitational waves, we want to consider a small departure from flat spacetime. So consider small departure spacetime. And then we want to compute, essentially, what we do next, there are five minutes or so, we want to compute what this left-hand side of this equation is. So concretely, we want to express this general metric as some flat metric, which I will call eta mu nu, where I'll be using the mostly plus signature, so this signature. And then I'm going to perturb this by something which actually depends on spacetime with the condition that this perturbation is supposed to be small. So this is essentially the validity of the analysis that we'll be doing throughout these entire lectures. And the convention will also be that whenever I raise my lower indices, I will always do this with this flat metric component here. So now, once you have an expression for the metric, you can plug essentially this into the left-hand side and you can compute what is this capital G mu nu object. And then the goal is essentially to find an equation which tells us how this perturbation of the metric, which at the end of the day will be the gravitational wave, how that kind of evolves depending also on the energy content of the universe. So here now comes the point where I can cheat because this is not a real blackboard lecture. So this is just showing you, if you've had a GR lecture, you'll have seen this. If you haven't had a GR lecture, you'll just have to kind of believe me. But this is essentially how you start in from a metric, you compute the left-hand side of this object. And really, it's all just a big contraction of indices. So not particularly insightful, but let's just have a brief look at it anyway. So the so-called Christoffel symbols are essentially what this describes as sometimes the connections of GR. So the Christoffel symbols, this is the definition, is the function of the metric tensor. Now we insert the linearization of the metric tensor. So we replace with this equation here, with the constant part plus the perturbation. And now we expand to first order in H menu. And then at linear order, we get this term here, one power in H. And then we have higher order corrections, which will drop. Then once we have the Christoffel symbols, we can compute the Riemann curvature tensor, which we can derive, again, this first line is the definition. As soon as we have the Christoffel symbols, by taking, again, contractions and derivatives. And in particular, this second part of the curvature tensor always involves two Christoffel symbols. But since every term in the Christoffel symbol is already proportional to the perturbation H, it means that if we contract two of them, it will always be of order H squared. So since we will always only want to do the deleting order expansion, we can immediately drop these terms. So we can then plug this expression for the Christoffel symbol into here, this expression for the Riemann curvature tensor, do all these derivatives and contractions. We end up with this expression here. But in particular, we have some terms here, which are symmetric in some of the indices. So for example, this term here is a d alpha d beta. So of course, I can commute the derivatives. So this is the same as d beta d alpha. And then I have these three terms. And I have, then again, the same terms, we switch alpha and beta, which means because of the minus sign here, it means all the terms that are symmetric and alpha and beta will cancel. So this term will essentially disappear at the end of the day. Oops, I'll go back to the theme of transport here. So we will have this term and then the same two terms again where we switched alpha and beta. So four terms in total. And now we can, by a simple contraction, we can get from this Riemann curvature tensor, we can get the Ricci tensor, Rmenu, which is already one of the objects inside here. And then if we perform another contraction, we can get the Ricci scalar, which is the other object that was inside here. So this kind of blows up quite significantly. In Majora's textbook, you can find the full computation. But if you manage to do it correctly and not lose any minus sign, then after quite a bit of work, you arrive at the final expression for Gmenu, which is a minus one half box Hmenu plus eta menu. So that the box is just two partial derivatives, del rho del sigma Hbar rho sigma minus del rho del nu Hbar nu rho minus del rho del nu Hbar mu rho. So now you see why I took this shortcut beforehand because with all this indices writing, I am bound to make mistakes. And here I've introduced this object Hbar, which is, I define, this is a, I introduce this simply because it makes the expression more compact. And this is given by the original Hmenu, so a perturbation minus one half eta menu H, where H is a contracted version. So this is H alpha alpha. So I essentially subtract the trace and this makes my expression more compact. So this is the final result for the left-hand side of the Einstein equation to first order in perturbation theory, right? So then they have the correction terms, which are order H squared. And this is an important equation. So I'll give it a star so that we can refer to it later. Maybe any other questions on this very first part? So I repeat that kind of the goal here is to derive an equation of motion for this metric perturbation, which at the end of the day would be the gravitational waves. And then that will kind of tell us how gravitational waves are related here via Einstein equation to the energy density in the universe. And that then at the very end of the day, I'm meaning on Friday, we'll show us kind of how we can use gravitational waves to probe actually particle physics because all the particle physics is under in the right-hand side of this equation. Okay, so now we need to understand a bit better what is this object H menu, which is appearing here to first order in perturbation theory. And so for that, I have another poll for you. So you should now, if you still have it open, the poll thing, you should see a new question. Otherwise you can enter again this link here. So this is just, don't think about it for too long, right? So this is just to give me also a bit of a feeling kind of where you stand. So the question that I would like you to answer now is what are the symmetries of the linearized version of GR that we're just in the process of the rising, right? We linearized means that we kicked away all the order H squared and whatnot. And maybe I should do this blind, right? I should not let you see now everybody's just clicking what everybody else is checking. Yeah, so there's a question in the chat where does one see the poll? So I don't think you can see the results. I think you can just see the questions. And the questions you can see when you go to this link here, then you should be able to see the questions and participate, right? So there's a big majority for infinitesimal coordinate transformation and global Poincaré transformation. And that is correct, but just kind of interesting because general GR has this first symmetry, right? So this is the symmetry of all coordinate transformations. Somehow when we linearize it, we lose part of the symmetry. And the reason we lose it is the following. So let's talk about symmetries. So we start in full GR, we start with the general coordinate transformations. But then we, this actually breaks down because we do this linearization. So we divide this, we said we divided this G menu, right? We say we divide it in some eta menu plus H menu, right? Which means we somehow define a constant background and then fluctuations on top of that background. But to define a constant background, you need to do an averaging process. And to do an averaging process, you kind of need to define over which time slice you are doing the averaging process, right? So in some sense you are doing a time slicing and then you're averaging on each time slice. And hence you no longer have the full symmetry, but you only due to this first step, which is associated the background, you only have remaining the SO3 symmetry of global spatial rotations. And this is precisely where we lose the general, the biggest symmetry of GR is precisely because we do this linearization process. So now that we have global spatial rotations, we can use the fact that we only are interested in the linearized theory, which means we can do a Fourier transform and all the k-modes of the Fourier transform will behave independently. And hence this Fourier transformation, we can kind of single out an individual k-mode and we can study only that k-mode. So the second step is essentially to do a Fourier transformation, which singles out one particular direction k that one can then study. And then with respect to this particular direction k, one then has an SO2 symmetry, which describes the Helicity eigenstates with respect to this vector. So essentially you have some object, like X for example, right? And then you perform some rotation by some angle theta around this vector k that we've selected and this object here will pick up a phase due to the IM theta X. And now depending on what this M is, this essentially classifies these objects X. So if M equals zero, so essentially if the thing, you do this rotation and the thing does not rotate, then the thing that you were looking at is a scalar. If M equal one or plus minus one, it is a vector. And if M equal plus minus two, it is a tensor. So to say is that we have, we started with a full symmetry, then we can realize that actually in our linearized theory, this actually breaks down to a reduced symmetry, but in within this reduced symmetry, we can classify our perturbations by essentially a quantum number, which is essentially the helicity under rotations around the vector which is specified by the afferent transformation. So let's do this concretely because what we're actually interested in is say this G mu nu object, right? Or the H mu nu object, one of both. So for G mu nu. So G mu nu, we can write as G zero zero, G zero i, G i zero, G ij, right? Where ij run over from all spatial indices from one to three. And now if we do this step one here, and we do this in a very physicist's way, where more formal people will be horrified, the way of doing this is essentially drawing lines here and saying I distinguish between temporal indices and spatial indices. And then to do the step two, so to actually define these perturbations, what I need to do is essentially write down all possible perturbations that give the correct index contraction. So let me let me do the example and then I think it will become, ah, before I do that, before I do that, I have a question for you. So the question is in the poll again and the question is how many independent degrees of freedom do you think are inside this metric tensor? Okay, I'll continue and then we'll see the result at the end. So the, we look at the G zero zero component, then in the, if we just set that metric then this is just minus one. And now I want to add a fluctuation to that but I have to have to add a linear fluctuation such that it has no spatial indices because as soon as spatial indices at a spatial indices on the left-hand side and the right-hand side have to match and the left-hand side does not have any spatial indices. So all I can do is essentially add one scalar perturbation. Scalar means no indices, which means that it rotates with this or it changes with this M equals zero with the scalar under these illicit rotations. Now, we can do the same for a spatial component. So here we can get, from here we've got essentially one scalar. So we do this with this mixed component here. So this is the same because this is symmetric. The same as this, right? So there's a, in, if we just had the flat space time this would be zero, right? The metric tensor is diagonal. So there's no leading water term, there's only perturbations. And now we need something which gives me one spatial indices, right? So the one way I can do it is by taking one spatial derivative of a scalar or by taking a vector directly. So B is now a scalar, SI is a vector. Vector meaning it has kind of no, the divergent should be zero so that there's no scalar component inside. So this gives one scalar, this is this P and one vector, this is this SI. And then we can do the same game with the pure spatial components. So now the flat space part would just be one. Now we need to add something which is two indices, right? So one way of doing it is by adding a delta function. Another way of doing it is by adding two derivatives and another way by doing it is by adding one derivative and one vector. So for example, like this Tj again to make it symmetric or we can add an object which is already by itself attends a perturbation. So here we have now two scalars. This is the psi here and the F, we have one vector. This is the capital TI here and we have one tensor, this is this Tij. And now we can count all of this together. So we have four scalars in total. Then we have two vectors but each vector has two degrees of freedom. And then we have one tensor, the tensor holds us two degrees of freedom. So in total, we have 10 degrees of freedom as one might have anticipated from a symmetric four by four tensor, right? Because if I have a symmetric four by four tensor, you can kind of count how many entries can you have just by doing this trick here. So one, two, three, four, right? Now you count the dots and it should be 10 because the downstairs components here are already specified from the fact that it is symmetric. So the answer that essentially the majority vote here is correct, right? So we have 10 degrees of freedom in the metric tensor to start with. Okay, but now, okay, and then why am I going on about this? I'm going on about this because in the end we want to study at gravitational waves which will be one of the perturbations inside this metric tensor. But preferably I would not like to deal with 10 degrees of freedom simultaneously but I would like to reduce this to a system which is few degrees of freedom and actually understand better what I'm doing. Any questions so far? Yep, there's a question by Mehmet. Please go ahead. Hello, so when you classified the types of representations that we just counted, you used the SO2 group and probably it will be tied to the fact that we have a master's particle but you never told it explicitly, right? Correct, yeah, yeah, yeah, yeah. We have, right, I'm already anticipating that there's no master here, correct? Another very concrete question but just out of curiosity, if you had a massive gravity or I don't know if we didn't have this masslessness then this classification would break down or modify it? No, so the classification works as long as you work in linear order in the perturbation theory because this is really a very, very general kind of way of classifying cosmological perturbation. So this does also work when you have a mass. What happens when you have a mass is that however, the degrees of freedom, as not the total degrees of freedom, my total degrees of freedom are fixed but for example, if the graviton had a mass then it would have more, well, we will come to that later, right? But it had potentially more degrees of freedom. I see, okay, thank you. Yeah, to briefly answer to the chat today, okay, this is Tjti, so del i Tj del j Ti and yes, all 10 degrees of freedom are independent. Yeah, this is what creates the mass, right? Because if we have to actually treat 10 degrees of freedom this is a quite complicated system but the goal would be to reduce this. Okay, so we can now use gauge invariance or yeah, we can use it in a particular we have to be aware of it and with gauge invariance what I mean is the invariance under coordinate transformations. The reason being that at the end of the day if I perform a coordinate transformation I kind of, the physics may look different in a new system but actually the physics that I'm describing has to be the same. So consider an intesimal coordinate transformation and I have the feeling this will answer some of the questions in the chat. So we take some vector, some spatial vector x mu, right? We shift this by some constant epsilon mu and the physics has to be invariant under the shift the total physics, right? In the full theory and this epsilon mu we can now do the same trick as we did above and we can write this as a spatial component which would be a scalar and a spatial component which can contain a scalar and can contain a vector. This is the most, but again we have this condition at del i fi equals zero which tells us that in this coordinate transformation we have contained inside two scalars and one vector which means that essentially we have two scalars and one vector in the system which are not physical, right? But we can essentially can go to different bases and then they will appear to be gone so they will be gauged away and hence we can impose a gauge condition which will make our life a lot easier. We can essentially force these degrees of freedom to disappear and we can impose precisely Lorentz gauge and as an anecdote note that the spelling here is actually this is named after the wrong Lorentz actually with the other Lorentz that first did the Lorentz gauge but this is kind of the way it is now established in the literature. So we can impose Lorentz gauge are very similar to electromagnetism and this is a gauge fixing on essentially the scalar and the vector sector but we're not touching the tensor sector. In fact, we can see from here that the tensor sector so the Tij up here is gauge invariant because this coordinate transformation did not include any tensor degrees of freedom but imposing this condition we can now go back to the expression that we computed for g-menu, right? And you can see that kind of we have lots of terms with the root of our acting on the h-menu and the only term that is left once we do this is that box h-menu is the only thing that is left on the left-hand side and then on the right-hand side we have as before 16 pi g C4 T-menu. So this is now our result for the linearized Einstein equation. Okay, we just have a very simple equation where the left-hand side we're acting with the box operator, right? So this is nothing as this is del alpha, del alpha we're acting on our perturbation h-menu and then on the right-hand side we have the energy momentum tensor. Question for you now that we have imposed Lorentz gauge how many degrees of freedom do we have left? So you can again check in the polling system. So what this equation do with this equation is very powerful equation it describes both the sourcing of gravitational waves so the right-hand side sources the gravitational waves which will be on the left-hand side and it also describes if we set the right-hand side to zero it describes the propagation of gravitational waves. Yeah, and again the majority vote wins so we had 10 degrees of freedom beforehand. This condition here, this is four equations Lorentz gauge with new indexes is free so it's four equations. So at this point we are left with 10 minus six it is four degrees of freedom. We can now do one more step to simplify the system and that is that in fact, but this step we cannot well we can always do the step but the step can be more or less complicated depending on the environment. So now let us just consider the situation in vacuum. So in vacuum we have T-menu equals zero and this means- There's a question from Ivan. Probably from before. Hello, can you hear me? Yeah. Could you repeat how many degrees of freedom do we have left after fixing the gauge? Here, four. I don't understand why. So we had 10 beforehand, right? Yes. And this here of four, oh, wait. I thought 10 minus four equals six. Yeah, yeah, yeah, sorry. This is what I wanted to write. Okay, yeah, no, yeah, exactly. This is also why everybody was correct, right? Can I ask another question? Yeah, and thanks for finding this out, yeah. What are GUs? Gravitational waves, this is a W. Okay, thank you. Yeah, thanks. Yeah, multitasking is complicated for lecturing. Okay, so in vacuum the energy momentum tends to zero and in that case we have a residual freedom because essentially if we have, we can in any epsilon, which satisfies that box epsilon is zero, right? In any epsilon that satisfies this, we can always add, maybe we should call this epsilon prime, right? We could always add this to our transformation of both and it would not change anything here about the Lorentz gauge, but kind of it still gives us residual freedom to manipulate our H-minu tensor. This is exactly the same as the way it works in electromagnetism. The residual gauge freedom, where I recall that with gauging here, I'm always referring to these coordinate transformations of general relativity, right? There's no, whatever, a billion, non-a billion groups, any of that, right? No, you won, it's really just a GDR symmetry that we're talking about. And to fix this residual gauge freedom, we can, in vacuum, we can do the very simple prescription and that is this prescription is called a transverse traceless gauge and the prescription is that we set the trace to zero, so H alpha alpha, if you want, is zero and we also set the mixed components or the zero i components to zero. So you can see this is again now four equations, right? This is one equation, this is three equations. And at the end of the day, if you add this condition and the Lorentz condition, then you can always impose results in the following conditions. So in this transverse traceless gauge, as soon as you have one time-like index, the thing is zero and also the trace is zero, in particular the trace over the remaining spatial degrees is zero and of course we still have Lorentz gauge. So this TT is for this transverse traceless gauge and these are the final conditions for the, what is called the TT gauging. And so now one more time, right? The question is, how many degrees of freedom do we have left? You can again, one last time, I will ask you for your favorite number here. Yeah, so the majority vote, yeah, very good. The majority vote is looking very good. So exactly, so we have two degrees of freedom left because we had six here at the end of the previous step. And then this here are four equations or also here, this thing essentially are four equations. And so we have two equations left at the end of the day. And the object that we have left, if we go back to the very beginning, the object that we have left, the only object which satisfies all these conditions is in fact this pure tensor object that we had at the very beginning. So what this tells us that in, that kind of the only propagating, and this is now it's important that gravity is master's, right? So the only propagating independent degrees of freedom that we have in the metric tensor, here are really the two gravitational wave degrees of freedom. And these two degrees of freedom is essentially ones we want to study and we want to understand the dynamics. And for this it will be essentially just linearized Einstein equation is the key equation governing the dynamics of these degrees of freedom. So this is to emphasize, this is really a pure prediction of general relativity, but it's a pure classical theory. We don't need to quantize anything or talk about quantum gravity or anything like that. So we have a pure prediction just from the Einstein equation we can derive an equation of motion which we will study in the second part of the lecture, which kind of tells us how these two degrees of freedom that we have left with it at the end, how they can propagate and interact with the universe. Okay, this is a good time to take a break. I'm also happy to take any more questions. So we'll take maybe the break first and then on the, we'll come back, maybe that we have more questions after thinking for five minutes. Yeah, perfect. Okay, it's just you in five minutes. All right, are you ready to continue? Yeah, yeah. Are there any questions before we resume? Seems there are no questions. Oh, yes, I think one, I see one. Yeah, by Maria. Please go ahead, Maria. Hi, Validice, thank you for the lecture. I wanted to ask you about this sequence of steps that you were describing to talk about the symmetries. Could you briefly maybe explain the second step? I didn't get it right. Yeah, yeah, okay. So, right, so in some sense, yeah, so this, yeah. So, okay, so from the first step to the second step, right? We realize that once we go to the linearized theory, we do this time slicing and hence instead of, we only have kind of the spatial symmetry and no longer the full space time symmetry. And now the next observation is that since we are only interested in doing a linearized theory. So the object we want to describe is our H mu nu and we have restricted ourselves to the regime where the H mu nu is small compared to the background metric. So we're gonna drop all terms which are H squared or higher. And hence since we're working in a linearized theory, we can do a Fourier decomposition and then there'll be a big d3k outside of kind of every term. And we can remove this d3k everywhere and essentially only study like the equation for every individual k-mode will look the same. And in particular, two different k-modes to different momentum modes will not interact. So we can study the equation of motion for one particular momentum mode, one plane wave solution. And we don't need to know anything about the other plane wave solutions. But there will be no interaction terms. And hence I can just specify one particular direction k. So k here is the momentum mode. And then kind of study kind of, this could be a preferred direction. And I can kind of study the physics which I now, which I'll have is preferred direction. And now I can essentially look at helicity states with respect to this particular k-mode. And these helicity states now essentially tell me that in some sense what these helicity, so here I was maybe, I also did not give you very many details, right? But these, these helicity states which essentially these tell you like if you do, if you do, if I do a spatial transformation, right? So say I do an SO, I do this SO2 rotation and then some objects essentially will transform under this SO2 rotation and other objects will not transform. And essentially how these objects transform which is encoded in this M here is essentially identical to the way we associate spatial indices with them. This of course is a much more formal way of doing this, right? I mean, a mathematician will be absolutely horrified by the description of it. So the more formal way of doing it is of course to see checking kind of okay, looking at the representations under this first symmetry, seeing how they break down into representations of the second symmetry and then checking how this kind of breaks down into representations of this third symmetry and then you will find exactly these different classes. Oh, okay. Any more questions? Okay, so now let's study the properties of gravitational waves. And before we studied them, I'd like to kind of know how much you maybe already know about them. So there's another question in the chat to see, but you probably cannot answer it from kind of what I've told you so far, right? But maybe you know anyway. So we now want to talk, so I should maybe put this here just in case, so this is, oh no, I lost it. So this is in case anybody wants to participate, it doesn't have the link yet, right? So here this is the link you need to enter in order to participate in the polls. So we now want to study plane wave solutions in analogy to electromagnetism. Maybe I should put this in quotation marks. So we're looking at solutions to this equation here and trying to understand their properties. And we want to do this in vacuum. So we want to set T mu nu equals zero. So we'll be studying essentially the propagation of free gravitational waves. So we can start by making an ansatz because actually this equation here, this in particular you set the right hand side to zero, this should look pretty familiar, right? Box something equals zero, this is just a usual wave equation. So we can solve it with the usual ansatz for wave wave equations just that we have a couple of indices more. So H mu nu of X can make an ansatz that this should be solved by some amplitude which can depend on K, right? So this is essentially a plane wave solution for a transform and then sine K alpha X alpha. Then we can check, so we plug this answer into our equation, box H mu nu. So then the derivative here acts on the argument, we get K alpha, K alpha, a mu of K, this just stays. Sine K alpha X alpha, B minus sine equals zero. And now in order for this to be zero, what do we need? We need that this thing has to be zero, right? And this thing is of course nothing else than E squared minus P squared. So the fact that this has to be zero tells us that gravitational wave travel with the speed of light in vacuum. So only then there are there a solution to this equation that we got from GR. Going back to the question I was asked earlier, for example, if there was a mass term, right? Then we would have gotten a different result yet, right? So this is clearly an indication that we're dealing with massless degrees of freedom. So we can now be a bit more explicit in this, to kind of have a feeling what the gravitational wave actually looks like, we can go to the transverse traceless gauge. And I'm just going to specify that I want to put my Z axis in the direction of this K vector. And now, well, what this means is that if I now look at these conditions here, right? But I want to construct something which is symmetric of which is traceless and where essentially here these O mu components are zero. And also this here, right? This here tells me that I can have no components since my this I can probably transform, right? And this becomes a K, so I can, which I've put in that direction so I can have no components in that direction. So I already know that when I want to look at this coefficient, this A mu nu of K, I know I will have a row of zeros here. So it's the first condition from the transverse traceless gauge, I will have a row of zeros here. And now since I put the K in that direction, I will also have more zeros here. And now I can use the fact that it's traceless. So traceless tells me that the xx component has to be a minus the yy component, traceless. And this I will call h plus. So we have h plus here, minus h plus here. Now I know that it is symmetric, so I know that a xy is a yx. And this I will call h cross. So we have h cross here and h cross here. And so now you can put that essentially back in here and you now have really an explicit idea of what the gravitational wave looks like in some sense in this particular gauge. And so it's something that oscillates. It has these two degrees of freedom and they are arranged in a matrix in this way. So we already hear from this, we see A we see we're dealing with a wave in the sense that we found something which has a sign function. We found that the velocity is the speed of light. And we see again, I mean, we derived it beforehand on just counting, on just the argument of kind of how many degrees of freedom we started with and how many degrees of freedom we removed. But here we've seen it very explicitly that we only have two degrees of freedom left. And we also see that it is transverse, meaning that the directions parallel to the z axis here are zero. So now we have this gravitational wave. So this is essentially what it is to say in some sense about a gravitational wave propagating purely in vacuum. Now I want to know what is the effect of such a gravitational wave on test masses. So we want to look at the effect of this gravitational wave on test masses which are located one that won't have an x mu and the other one at x mu plus x sine mu. And now I'm going to cheat again. And I'm going to cheat because, okay, the result is important, but kind of the duration of this is not so relevant for the rest of the lecture. So I want to not waste your time writing all these indices. So what we want to know is how these test particles move when the gravitational wave comes along. So for that, we start with the geodesic equation. So the geodesic equation describes essentially how a point particle moves in term of its proper time tau. And that is essentially determined by the curvature of the metric. So this is the description of a free falling particle. Now, what we're going to do is we remember that we computed the Christoffel symbols beforehand, right, the first order in HMU. So once we have the geodesic equation, we can essentially plug in our expression for the Christoffel symbols. And then we can also see how these test particles move in a curved metric or in a flat metric in the presence of gravitational waves. And what I'm actually interested in is not really how one particular test particle moves, but what I want to know is kind of how, what is the relative motion of these two particles, right? Because there will be two free falling particles, but now the gravitational wave comes and these two particles will move one with respect to the other in some way. And I would like to understand how does that motion work? And for that, I need what is called the geodesic deviation equation. So essentially I have this equation and I have the same equation again for this other particle that I subtract the two equations. And then I get an equation for this distance the zeta mu between the two particles. And here we just see we have essentially an expansion in zeta, we got the first order in this zeta parameter. And so we have the Christoffel symbol again, we have something which looks like this term up here. And then we have one term here, which includes this deviation parameter. We can now approximate that we want to be in the non relativistic limit, which means that the derivative of the particle position with respect to proper time is small, essentially compared to the speed of light. And we can use that in one point, for example, in X mu, we can make the metric locally flat, which means that the Christoffel symbol is zero. We can however not do that globally, right? So if we do it at the point X mu, we cannot do it at the point X mu plus Xi mu. We can do it at one point. And then if you use these approximations, you plug them back in here, you have to work for a little bit. And then you get this equation here. And then when you evaluate here, the Riemann tensor, you finally, well, okay, let's start here. So this equation here again, relates the second derivative of this Xi, right? Which was the distance between the two test particles with itself via the Riemann tensor, right? The Riemann tensor is the thing that knows now about the curved background, which knows about the gravitational wave. So we can now evaluate the Riemann tensor to first order in H using the expression that I showed you on the very first page. And we can now make our life a bit easier because we've understood all these stories about degrees of freedom. So since we can now, we can already go on to transfer stressless gauge and compute because in this, this equation has to be gauge invariant, it's a physical quantity. So we can make our life easier and compute this object also kind of in the TT gauge. And then this is the final equation. So this tells us essentially how the motion of the particle is influenced by the presence of the gravitational waves. And the reason you have the two-time derivatives here, right, is that the Riemann tensor came from the Christoffel symbol. And the Christoffel symbol had all these derivatives acting on the metric. So now we can actually solve this equation. So for example, let us consider the case where this H plus is non-zero and H cross is zero just for simplicity, just to kind of study just one degree of freedom at a time. And this will at the end of the day be what is called the plus polarization. So the right hand that's here is, I'm gonna introduce A and B now A and B are just X and Y because I don't want to drag along all these zeros from above. So this is just H plus times sine. Again, we can write this a bit simpler as just sine omega t. And then we have just essentially the inner block of this big matrix because the zeros, I don't have to drag along. And this is the, so this was the H plus component, right? And it was a trace loss and hence there was one and minus one. And now we have the two test masses, right? And I want to put one test mass, I wonder what the difference between the two test masses is X zero plus delta X where this is the thing which can vary with time. And then this is Y zero plus delta Y of t. So essentially I have a test mass or the deviation of my test masses X zero and Y zero. And I want to kind of now see how these are moving and the moving will be encountered in this account of one is delta X and delta Y. And so now I just plug all of this into this equation here. So then we get, let's look at the X component first. So then we get delta X double dot on the left hand side. Now we have to take two derivatives of this guy. And so this will give a minus sign. Then we have this one half here, so minus one half and then we just get the function back from the two derivatives, we get a W squared sign omega t. And now I'm interested in small deviations, right? So delta X is much smaller than X so we can actually drop the delta X pot. We can do the same for the Y component. The difference for the Y component is that you have a different minus sign here. So you will get, I forgot the H plus, it should be an H plus. So we get exactly the same here. Just that the set of X, we get Y and we don't get the minus sign. So what does this mean? This means if we just draw a picture, this means let's say we have a coordinate plane X and Y. And let's say I have some test masses sitting at some points in this plane. Now let's consider first a case where the phase here is, so in some phase where the sign is positive, right? At some time where sign omega t is larger than zero, then we see that the things that are in that with respect to the Y direction, so things that, yeah, with the Y direction, things you have, like the sign of this is opposite with respect to the sign of this, right? Which means you can have, for example, a motion. Am I drawing this in the right way now? Yes, so here we have just an X coordinate so this will go and no Y coordinate so these will move like this. And these guys will move in the opposite direction. Yeah, like this, right? So if you have a positive Y, the delta Y will be positive. If you have a negative Y, the delta Y will be negative and it's the opposite of way around for the X component, right? So if you started, hypothetically, you started, say not just with four points but you started with a circle of points, then essentially in this phase of your oscillation, your circle will be deformed into an ellipse of this type. And now if you go to the other half of the phase, right? If you go to sign omega t smaller than zero, then everything happens just in the opposite direction. So essentially you get deformed like this, right? So then this is what I'm drawing again, and this is now you understand why I'm calling this plus polarization the whole time, right? So this is with some fantasy, you can see now across in this plane. So what has happened is you imagine you have a circle of gravitational waves and as your gravitational wave, the gravitational wave is coming along the Z axis, right? So the gravitational wave is essentially coming at us, right? From behind the plane. And now all the particles on this surface, all these particles on this plane, they now start to oscillate in such a way that the ellipse always kind of deformed first in one direction then deformed in the other direction. However, such that the surface inside will always be the same because the surface is a scalar and we're really just looking at tens of polarizations. Let me just briefly draw the second polarization and then we can do, then we have plenty of time for questions. So this is kind of what it's referred to as the plus polarization, right? And then you can also do the cross polarization. So to do that, you consider the case where H plus is zero, H cross is non-zero, so the opposite case and you go through the exact same steps, right? Just that now you have the off diagonal elements instead, the symmetric off diagonal elements and then what you will find is maybe a good exercise to do to kind of convince yourself that you're following this. And so we have the XY plane and now essentially what happens if we start with our circle, it will be deformed in one phase of the oscillation where we kind of be deformed into an ellipse like this and in the second phase, it will be deformed into an ellipse of this type, right? And this is precisely why we call it the cross polarization. Okay, yes, there's a question. Go ahead. Yeah, so is this sort of similar to what happens when a gravitational wave passes through a planetary object? So this applies to any test masses that you could consider. Okay, so like this difference in sort of stretching compression instead of what we measure like laser interferometer. Exactly, exactly. So what you would exactly, so if you would build a, if you would build a gravitational wave detector, right? And you would, for example, let's say we want to measure this gravitational wave, right? We would build an interferometer ideally, which is aligned precisely in this way, right? And we measure the interferometer measures this distance, right? With respect to this distance, that that is the measurement that the interferometer makes. And now as the gravitational wave comes through, right? This distance is now shortened, right? And this distance is stretched. And that's precisely what the interferometer will measure. Okay, thank you. There was, by the way, there was, historically there was a long debate if this is actually something that could be conceptually measured, because kind of the question is, well, but what if, so you're trying to measure a distance, right? But you're actually perturbing the metric. So in some sense, you're changing the distance, but you're also changing the ruler with which you're measuring the distance, right? And hence you have to be really, and this is essentially where all this discussion about the gauge freedom and what is actually physical and what our gauge artifacts where this enters. You have to be kind of very careful in what you're doing to really make sure that you actually are convinced yourself that this is really something physical that you can conceptually measure. And people were essentially, I mean, the computation was there, right? But people were still a bit confused. And the reason and why people, the argument which apparently in the end really convinced people that it was a physical effect was kind of the argument which is called the sticky bead argument that you have, you imagine you have an infinite rod and you have a bead which is on this rod and can move up and down, but the rod is sticky so that there's some friction when the bead moves. And but since now the rod is infinite, right? There's no question of does the ruler change or does the ruler not change, right? The gravitational wave would now move this bead up and down the sticky rod and then the friction force will be something where the heat from the friction will be something which you can objectively measure independent of the question, I mean, in theory, right? Which you can measure independent of the fact if kind of now the rod is stretched or the rod is not stretched. So this, yeah, apparently this is the, historically this was one of the arguments that convinced people that this was real but now that we are kind of enough that there's consensus on this, I think now this kind of this picture that you're just kind of stretching one arm with respect to the other is perfectly fine and accurate. So in conclusion, what we learned from this discussion of this plane waves, we learned that we have just to summarize we learned that we have two degrees of freedom. So when we linearize generativity, we get two degrees of freedom we can describe which corresponds to two linearly independent solutions which are plus and cross. And now, by the way, I also think I understand better someone was asking before and if the 10 degrees of freedom were really linearly independent and yeah, I mean, in fact, I think my answer was not quite correct, right? I mean, this is in the end the correct answer, right? Once you take into account all the gauge degrees of freedom and so on, right? Then in the end, you really just have two degrees of freedom that you can discuss. They are propagating transverse waves which deform a two-dimensional plane. So this is different than electromagnetism, right? Electromagnetism, you also have transverse waves but they are the entire thing, like here you really treat a mention, right? One dimension is the one where the gravitational wave is moving and then the plane orthogonal to this is the formed whereas for an electromagnetic wave of course you could go the entire thing in one dimension less, right? Because essentially you are the formation is only in one dimension not in two dimensions. This is precisely why did essentially the photon is a vector particle, right? The one-dimensional, it's been one particle and here we have a spin two particle. Yeah, so it's important to note that this is a spin two particle that we're discussing meaning for, in other words, this is a tensor and it's important to stress this is an inherent prediction of generativity. So there's no quantum gravity involved, no PSM physics, business or something which kind of has to be there if generativity is correct. And maybe one more note. So here we've talked about the plus and the cross basis. Sometimes people also talk about the left and the right basis. So H, L, R and they, this is just in case you ever come across this. These are related, this is again very similar to electromagnetism. These are related to these two guys by in this way, plus minus I H cross. So there's just two different linear, linear combinations, but I mean at the end of the day you have two degrees of freedom that you want to describe. Okay, are there any more questions? I have one last polling question which we can do in parallel. Yes, Maro. Thank you for taking my question. Can you elaborate a little bit more about this stretching of the metric and that it is actually physically affecting not that we can measure? Right, so you have- Because they're my first occasion I feel like it's not measurable. Right, right. And I can say that you're in good company, right? That this was actually a long discussion. So in my jurors textbook, right? So in this textbook here, wait the beginning, here, this first one, there's actually two chapters dealing just with this question. And essentially there are two frames which would make sense to discuss things. One is kind of what was called the free falling frame. So in the free falling frame, kind of your test masses are free falling, right? And then you kind of look at the effect of the gravitational wave on the trajectories. And you can, this is essentially where this argument with the gravitational deviation equation or geodesic deviation equation comes from. So you essentially really look asking kind of do trajectories deviate from each other. So that's one frame to look at. And the other frame to look at is what is referred to the proper detector frame. But the detector is not, because the detector is not a free falling object, right? It's fixed, it's fixed to Earth. It is not on a free falling trajectory. And in both the cases, you can, if the danger arises when you mix essentially in two situations, right? Then you can confuse yourself about what is physical and what is not physical. But if you kind of carefully stick to one frame or to the other frame, then you can convince yourself in both frames that it is really a physical effect. And essentially in one frame, and now don't ask me which one, in one frame essentially you can understand the gravitational wave as really a change in length scales. This is to me a bit more the intuitive understanding. And this is in the detector frame, right? Because this is really like how an interferometer works. And in the other frame, in the free falling frame, you instead have to think of it essentially as a shift in time instead of in space. So essentially the travel time of the objects changes. So the gravitational wave perturbs to travel the proper time of the moving objects. And again, this is a measurable physical effect. But yeah, you have to, it is quite easy to, yeah, but this all kind of boils down to doing kind of the gauge fixing in a proper way. And for example, this transfer stressless gauge which I use today and which I will use in most of the lectures and which actually most of the people use all the time. And this is really a gauge which only works in vacuum, right? So it's actually not a gauge that you can use really when talking about detectors. But people do it anyway and most of the time it's fine. So it's one of these cases where you can do a really good proper computation once you know the results. You can kind of do a shorter computation and reproduce the result. But I strongly encourage you to look into, but it's in section one or two, if there's a longer discussion on proper detector frame versus a free falling frame. Thank you very much. All right, next is Ivan. Yes, I wanted to ask a question in different sessions and different book. I heard the term harmonic gauge and harmonic coordinates and I wanted to ask if they are related to the gauge which was used today. Honestly, I don't know. I don't know what is the harmonic gauge? Okay, I'm asking because the derivation looks very similar to a book which I'm reading, but they're using harmonic gauge. So I'm wondering if it's the same or if they didn't, but I guess you don't know, okay. No, I don't know the harmonic gauge. I mean, I know that you can use different gauges, right? I mean, here we use Dorian's gauge and it is obviously different choices. It's very similar to the story in electromagnetism. You can also do something like Coulomb gauge and so on. Yeah, but not this specific, I don't know. Okay, thank you anyway. So I encourage you to do the last poll question here because this is kind of telling me if the poll is the speed of the lecture like and this is crucial for what you will be facing tomorrow. So this is your chance to have influence. Any more questions? Adriana? Sorry, Adriana. Yeah, so I know that there is a lot of complexity and discussion on how to define the energy of gravitational wave. So I'm thinking when we write the linearized Einstein equation, how should I interpret this energy and momentum tensor just like only Newtonian mother in there? Okay, so we'll touch a bit on that tomorrow. So as long as, okay. So essentially what we did today, right is that we took a flat space time and we perturbed it with the gravitational wave. And in that case, it is kind of clear what is more or less clear what is a gravitational wave. The complication arises when you have curved space time. And in fact, as soon as we have a non-zero energy momentum tensor here on the right hand side, it essentially means we have curved space time. But when you have curved space time, this means that your space time is space and time dependent. Well, at the very least it is space dependent. So then the question becomes how do you actually separate the background part from the perturbation part, right? Because if the background part now also has a space dependence, then how do I know what to put in the background or what to put in the perturbation? And that is kind of where the difficulties come from. And we'll talk about that a bit tomorrow. But I mean, I can tell you that the overall strategy is that you need a separation of scales. So you need to essentially say, I can do some sort of averaging procedure at some scale. And I need essentially that the background is say homogeneous at these scales and the gravitational wave is not, right? So if I want to, for example, measure gravitational wave with LIGO, then we use the fact that, of course, LIGO is not in vacuum, right? And so there is the gravitational potential of the Earth and all sorts of stuff. So what we use is the fact is that the gravitational waves that we see with LIGO oscillate at a frequency of hertz. But the gravitational wave backgrounds that we have from whatever, you know, the Earth and whatnot, they only vary very, very slowly. So if we do a time averaging procedure on a timescale which is much longer than the oscillation time of the gravitational wave, but much slower than the oscillation after the gravitational wave, but much faster than anything where the background would change, then we can in this way separate the gravitational wave from the background. But in fact, it doesn't make sense. Like in curved spacetime, so vacuum everything is easy, in curved spacetime, it doesn't make sense to talk about the gravitational wave without kind of specifying how you have even defined what is your background. Thank you. And so related to this, whatever is sourcing the gravitational wave, is it contained in the, like classical? I don't know how to express it. So sorry to say that again. So the gravitational wave is sourced by something. So I'm wondering if this something is also problematic or if this is just in the stress energy tensor. This is just in the stress energy tensor. Yeah, yeah, yeah, yeah, yeah. Exactly, so in practice, in practice how this will work is that you have some event which is localized in spacetime. That creates a bunch of gravitational waves. And then they just propagate to the vacuum and we can describe them just fine, right? And then when we detect them, we have to again be a bit careful. Yeah, and we'll talk about in the last lecture, we'll talk about sources. I mean, the most obvious source is essentially the one that the detection at LIGO or detect several, many detections as LIGO has already made, right? From two black holes colliding with each other, right? That kind of disturbs the metric so much that kind of we get a radiating signal. This is the gravitational waves that LIGO has measured. And yeah, there you would essentially say, okay, there was one event in spacetime is the merging of the gravitational waves that generated the bunch of, generated this gravitational wave. That's in fact a very complicated process to model that really to high accuracy. But then once they're kind of propagating in vacuum, they are very easy to describe. I should maybe you also mention I probably should have said this at the very beginning and they catch your attention. So the reason why gravitational waves are such a unique way to probe particle physics and to probe the universe is that in, like when we use photons to probe the cosmic history, we can always only probe back until the decoupling of the cosmic microwave background because at early times, the universe was like, there was a plasma that was charged particles and light didn't propagate freely. So it's very, very hard to probe any times before the C and B decoupling. But the gravitational waves really don't care at all if the particles are charged or not charged, right? Because they don't know about charge. They just know about masses. So gravitational waves in principle can probe the entire cosmic history up to cosmic inflation and in principle retain the information. And if we can detect them, right? Then we can decode all this information. Then the problem is of course that the gravitational wave, it doesn't care about all the plasma in the universe. It also doesn't care a lot about architecture, right? They tend to just, they're so weakly coupled that they're extremely difficult to detect. But that is kind of the kind of the, that is the underlying reason why this is potentially a very promising tool for part of the physics. I see a little bit of a box. Yeah, hi. So I have a question related to that, what you just said. So you said that in this propagation region, one usually assumes that there's just a vacuum and therefore we can really even know what gravitational waves are. So, but there are some situations which people are already thinking about or really are physically, where maybe this approximation could be problematic. Yeah. Well, yeah, so in the gravitational wave interact extremely weakly with matter and stuff, right? So, and then the universe is pretty empty, right? So for most of the time, it's really fine. I mean, did the gravitational wave, I mean, sure, I mean, if you have whatever, if you have some massive objects in the path of the gravitational wave, then the gravitational wave will be lensed just in the same way that light is lensed, right? But this is something we understand, right? Essentially, this just kind of means that the trajectory of the gravitational wave changes. And this is in fact something that people have looked at. There was, for example, at some point, there was a case that LIGO had measured two black hole merger events, which kind of looked, were kind of close together in time and looked maybe somewhat similar and people were speculating if maybe one event was the lensed image of the other event, right? So, one event, but then kind of two trajectories around some very massive object, then both kind of being detected at Earth. So these are the type of things that people have looked at. But apart from that, the gravitational wave doesn't care much if there's what I've been up, there's a cloud of hydrogen or something in the way it just doesn't care. That one makes it so powerful to probe the very early universe. So that's basically the only region which could be problematic would be if the region would be exactly the one which also produces gravitational waves like some binary system closely before the merger or something like that. Right, right, right. Because there also we have the problem that the perturbation is not small, right? Because here we're always, I'm always working in the linearized theory, right? So I'm always saying that H has to be very much smaller than kind of the metric itself. And now if you're very close to a black hole, that is no longer the case, right? These black holes are merging and you have to deal with strong gravity and that is difficult. And then you need to do that properly, you need numerical simulations. Okay, thank you very much. I see no more questions. There was a question in the chat. Francesco, can you read by yourself or should I read? So Francesco, ask it whether the reference that you gave by Maggiore contains explanation of the SVT, the composition. Okay, so the Maggiore textbook contains the discussion about how you do the individual gauge fixing steps, right? To then arrive at the two degrees of freedom. So he talks about crew on gauge and these things and the transverse tracers gauge. Now this story here is kind of the very general picture of cosmological, it goes under the name of cosmological perturbation theory. And I honestly haven't find the perfect reference but a good place to start reading about this would be the Tazi lectures by Daniel Beaumont. So Daniel Beaumont, I think there's two ends, they may not be, I'm not sure, Tazi lectures. And here's one on inflation and one on, I don't know, called cosmology or something like that. And they both in the appendix of these lectures, I found those useful. All right. And I see another question in the chat, no? When Antonio? Yeah, could you tell us a little bit about rotational waves as a test for particle production during reheating? Can I help us to understand if there's been a preheating or not? Yeah, that's a very good question. We will come a little bit to it in the third lecture. I can give you a spoiler and that is that, so in principle, yes. So preheating is a very, so in order to source a gravitational wave, right? What you need is this energy momentum tensor. And in particular, since we know now we need to solve a tensor perturbation, we need an energy momentum tensor which actually has a tensor contribution, which essentially means we need an anisotropic energy momentum tensor. So typically if you have some very violent process which is not symmetric, symmetric meaning there's no radio or spherical symmetry, then you generically source gravitational waves and the more violent or more energetic your processes, the more gravitational waves you get. So preheating is very violent, right? It's very high energies. It's not at all symmetric, right? You may have bubbles which collide and all sorts of things going on. So it's a strong source of gravitational waves. However, the problem is that preheating happens typically at a very high, very early in cosmic history, which means that the gravitation waves get diluted a lot until they reach us. And also kind of the length scale of the gravitational wave involved depends on the typical size of the process. And then of course you have the redshift to today. But if you kind of look at the typical mass scale of a preheating particle or an implant particle, this is very high mass scale, which means it's a very small length scale, which means it translates to extremely high frequencies today. So the highest frequency that we can probe would lie over something like hertz kilohertz or so. And in order to really probe like broad classes of preheating models, you would need to be able to do megahertz or gigahertz or something. And there are, there's currently, there's several ideas out there, but there's no clear path forward on how to detect gravitational waves at those frequencies. Okay, it's great. Next question by Antonio. And maybe this was the same one. I gave you the same one. Yeah, okay. So then Artemis, please go ahead. Yeah, can you hear me? Yes. Okay, I was just wondering, you mentioned that what we did today works for a massless graviton. So massless gravity. What would change if we had a massive particle? Yeah, so I'm not an expert on this, but I mean, if we look at just kind of the final properties, right? So one thing that would change is it would no longer propagate up the speed of light, right? But it would kind of propagate at whatever speed is dictated by its mass. We would also have more than two degrees of freedom. And you can kind of understand that by the analogy with electromagnetism, right? So the massless photon just has two degrees of freedom, the two transverse degrees of freedom. But if we give it a math, we in addition also have the longer two degree of freedom. And the same is true for gravity. So in models where you make the graviton or you add a mass term to your gravitational waves, then you don't only have these two degrees of freedom, the plus and the cross, but you have additional longitudinal modes and additional scalar modes. And in fact, one of the things that LIGO has already done is they have checked, for example, if they have in their data, right? They've seen these modes, right? From the blackboard merges. And they have, for example, checked. If they have, and this is what you expect, right? From GR, but they have also checked in their data if they have any indication of detecting other modes. But so far they have, they have not, right? Otherwise we would have heard. Right, my question was more like, I understand, of course, it's gonna be a difference at the end, like the solution would get will be different. But I'm not sure I see, well, I haven't seen actually how it would work. What would be the equation that would start it from? Cause we started from linearizing Einstein's equation and we have no particles in there. So like, I guess we would have to do a QFT approach or something like that to have like, because we have no particle. Like we end up with a box operator without a mass, but... Well, we would have to modify GR, right? Because in GR, it's a prediction that the gravitation waves are massless, right? So what we would need to do is we would need to snuff. So our starting point was Einstein equation, right? So general relativity. If we want to somehow arrive at a theory which has massive gravitational waves, we need to start with a different theory. So essentially we need to modify and sometimes we need to, well, I don't even know if it's entirely correct to say we need to modify the left hand side, but we need to modify this equation. We need to have some more complicated form of this equation, which allows for extra degrees of freedom. So really the starting point changes, right? And then if this changes, then of course this equation changes, right? And then of course, this equation changes, right? And you will no longer just have box H on the left hand side, but it will have something more complicated. Okay, okay, thank you. Do you have more material to cover today or? No, no, I was done. I was done. I took it seriously that you said two times 35 minutes. Okay, 45 I said. Well, it was, yeah, it's fine, it's fine. Perfect. Yeah, no, it's fine. We could start with the next one. No, I think this is a good point to start. Perfect, so let's stop the recording. Okay, great. And there were many questions, but there is more time for more. So now the recording is not on, so don't be afraid to ask any question.