 Can you hear me? Oh, yeah, welcome back. So let's continue. Let me quickly review the first episode. So we were discussing infrared structure of gravitational theory in Minkowski space in four dimensions, or more generally, theory with massless particles discusses three different topics and their relation. And then we focused on the simple toy model, which was the analog of this 2D icing, which seems to capture some of the things and would be helpful when we move now to the most general case of asymptotically flat nonlinear dynamical spacetime. So the toy model was that we discussed the Penrose diagram. It's global aspect, so we have null infinity. We have some sources, which are particles coming from, say, null infinity interacting at a point and going to future null infinity. They start in some directions specified by initial momenta and end up in some final dimension. So this is just a stress tensor, which enters in the right-hand side of Einstein equations. And then we solved for the metric, very simple exercise. And we focused on the behavior of the metric close to the null infinity, which corresponds. So here I wrote the metric of Minkowski space in retarded coordinate and an advanced coordinate retarded as the CUT minus R. So this is a constant U. Looks the time U runs here on scry from minus infinity to plus infinity. And V, which is T plus R, which also run from minus infinity here to plus infinity here. Then we found that, so this kind of source, which is a typical source when we consider gravitational scattering problem, being gravitational scattering problem, we fixed the initial data here and we look for the final data here. So we found that if we focus on the value of the metric in some observational direction labeled by unit vector n, that the expression was very simple. And moreover, we had this overall change, which is called memory. And this was related in a simple way to famous Weinberg soft factor. That was lecture one. Now let me add a couple of comments about that. And first one is that notice that if you observe the change or this metric in some direction n on the sphere, it's a very large distances. R is very large. You cannot see explicitly this other particles because they're very far away from you. They're piercing the sphere in other directions. It's like celestial sphere away from a gravitational scattering event. You cannot see what's going on. We are sitting at one point on the sphere. That's where we have our detector. Nevertheless, this metric explicitly depends on all the momentum which is piercing the sphere in all directions. So this effect in a sense is very infrared. It is sensitive to the particles, which are even far away from you. It's not local on the sphere. Second comment is that you can, as an exercise, as a very interesting exercise, you can generalize this computation with a retarded propagator to d dimensions. And you will find that in d dimensions, the picture is the following. That you have a radiation that appears at the order 1 over r d over 2 minus 1. In d equal 4, it's just 1 over r. And you will find a memory that appears at the order 1 over r d minus 3. Notice 1d is equal to 4. It's the same order. And this is related to many special things that happen in four dimensions. In high dimensions, these are different orders, and it also leads to many complications. Also, you can compare your result with a recent paper by Wald. They say that there is no memory in high dimensions. But they look at this order. So if you look at high orders, there is a memory. OK, so these are two comments. Now let us move to some other interesting aspects of this toy model, a very simple one, which is, namely, let us ask, what is the asymptotic behavior of this metric at the boundaries of SCRI? So what are the boundaries of SCRI? The SCRI is labeled by this time u and a sphere. So we can take this time, if we take u to infinity, let us denote it i plus plus. When it goes to plus minus infinity, we will denote this as i plus plus minus. Similarly, on the SCRI minus, taking the time plus minus infinity, we will denote i minus plus. So this is on the diagram we are approaching these points. This is i plus minus. This is i plus plus, i minus plus, et cetera. So notice we started with a metric, which is completely irregular on a spatial slice. You can find this metric everywhere you like. Now if you do this exercise again, remember, we have this retarded propagator, simple integral with the delta function. We can explicitly find everything. You will find a surprising result, which is, well, maybe it's surprising at first sight only, but you will find that the metric on the future, on the i plus minus, so when you approach past on the SCRI plus here, is equal to the metric when you approach from the SCRI minus, which is untypidally matched. So there is a flip on the sphere. That's a crucial thing. And what happens is that we have a completely smooth metric on a spatial slice, but when we go to SCRI plus or SCRI minus, we first take a limit r to infinity. And then we're taking the limit. This is first. And then we're taking the limit of u going to approaching this point. In a physical space, when you're doing this order of limits, two points never come close to each other. That's why it makes a total sense that you might have thought this is completely crazy. We started with a smooth field, and now we are getting that, first of all, the limit exists. So it makes sense to talk about this limit. The field takes finite value. And second of all, we find that they are untypidally matched. So this is a, you can explicitly compute and check in this example. Also, you can, well, as an exercise or as a side comment, there is a famous, let me just write it, for electromagnetism. If you somehow feel like it, you can write the value of electric field similarly. It takes a form which satisfies the same, the same untypidal matching condition. So on i plus minus, it has a similar form, similar to this, with a soft factor. And on the other side of i naught, which is a special infinity, it's the same formula, which is, but you flip, you flip, well, let me write it just minus n. Yes, thank you very much. Please ask me about notations. I'm introducing so many of them. So ui is a four-velocity of a particle, if you remember. It's a four-velocity of a particle. Over here, where you can consider sources instead of particles with a mass, some charges, which come, again, come and go. And n is, again, observational direction. And n is a four-vector, one, and the direction in which you observe. Okay, so that's surprising, the upper. So yes, in this model, this is a result of a computation. So we start it with a source, we find metric everywhere, it's completely smooth. But then, when you take the limit first, you're going to scry, you read off the value of field and scry, and then you discover that they are typically matched. So one of the, well, as we will review, this is one of the, I think it's a main ingredient in the paper by Streminger, which is, I guess, realization, or that's a correct thing in general. So even though we did this thing in a very simple model, and as far as I know, apart from these arguments, like this kind of boundary conditions, they respect Lorentz invariance and they work, so they lead to correct predictions, there is no really mathematically rigorous derivation of antipodal matching in a generic nonlinear problem. So if you wish this is, at the moment, in our general treatment of asymptotically flat spacetime, this will be a crucial insight in all the conservations, all the conservation laws, super translation, super rotations, it will come from the fact that we match things here. So this is a really important step, and this is called antipodal matching condition. So, I'm sorry, say again? No, I think that if you will specify the some cache data here, we will find that this holds, yeah. Right, you can think about it as follows, for example. Let's take the spatial slice, there is some cache data. You can evolve it forward or backwards. So there is one initial data that is specified that specifies everything. And now, if you wish, you take this initial data and you go to square plus and square minus, and in general, these are different things. But then when you move this point further and further, you discover that what survives is just the tail, tail of the metric. What matters for this thing is tail of the metric, which controls ADM mass, ADM mass aspect. And it, first of all, it will be well-defined, and then moreover, it matched in this way. I don't know, I don't think there is a rigorous derivation in a full general framework. I would say that it's more like a correct picture that Strominger got. I don't know, maybe it is. As far as I know, that's a state of affairs. Maybe you can play this game of going from finite, I think, I wrote a paper about going from finite r, and some maybe in QED, but as far as I understand, in the full general relativity, there is no rigorous derivation. At the level of rigor, for example, as I will talk about Christa Duller-Kleinermann theorem and results which are pretty rigorous. But it seems to be completely correct picture, which actually always works and lead to correct predictions, and all the new conservation law come from realizations that this limit exists and then you have to match. So let's, this would be an important, this is a very important point. And here in this model, again, you just can compute it and verify that it's correct. Yeah, yeah, yeah, yeah, you will not see any memory. Yes, you will not see any memory, yes. That's a, so this is basically making sense of this story in high dimensions, namely how do you connect to asymptotic symmetries, et cetera, is an open problem, it's not done. But it should be all the same things, because the SOF theorem exists. So one way this way, one way this, for example, this connections work is that you find the Weinberg SOF theorem. You see that Weinberg SOF theorem exists in all dimensions, completely universal. So you know just physically that there should be memory in all dimensions. So you go to the GR literature and you see where these things go wrong and say, you see that it's subleading orders, things become subtle, but usually then you figure out how to proceed to make things work. So remember, so we found this formula, the metric here that we found was decayed as one over r. Now in higher dimensions, you will find that radiation, if you have a gravity wave, it's decayed like one over r to d minus one, but this h here, which will depend as a function of u. If you computed integral, it will be zero. Whereas if you want to see something on zero, you have to go to this order in one over r. That's the statement. Okay, so this is the end of the toy model. Here everything is explicit, but you see many of the things. And of course there will be huge non-trivial steps of this toy model and all the asymptotics and matchings. They're supposed to work in completely general and linear asymptotical flat Minkowski space. Okay, let me introduce you to the asymptotically flat Minkowski spaces and let me raise this part because I feel that in the center everybody sees it better. Okay, of course one, if you formulate the problem that we have asymptotically flat Minkowski space, there are many subtle questions about what do we exactly mean that things is asymptotically flat and what are the fall-offs on the metric which you usually impose by hand so that you capture the correct physics. So let me first write again the metric of Minkowski space. So now I will be focusing close to, we're talking about Scribe plus and space time close to Scribe plus. So now if we will be talking about the most generic space time, of course the metric can be very different from the Minkowski space and here I introduce just coordinates on a sphere which are convenient. So Z is related to usual angle like that. The gamma is a simple fact. Okay, great. Now if we attack the full non-linear problem, of course the space time can be very different from Minkowski space but the idea of asymptotically flat space time is very easy that when you're going far away the space looks asymptotically flat and look to leading order like that, plus small corrections which become small and smaller as you take distance to be large. The way to make it a bit more precise is first let me introduce again it's, I introduce it, I don't know if it will survive next years of development but until now it was a useful gauge. So if you have a space time at any point or you can choose the coordinates of this type plus, so this is just motivated by metric above but it has nothing to do with flat space yet. It's just a gauge which you can choose at any point and it's called a boundary gauge. So here the coordinates are U, R and two coordinates on a sphere similarly as here and moreover if you fix a point on a sphere and you consider U equal constant, this again null, it's a null, null direction. So as here fixed angle and fixed U, it's a null subspace. Now to make things more interesting we ask us the following question. Okay, this is just a gauge, it's always true. So what do we call asymptotically flat space time? Well, what we have to do is we have to figure what happens when R goes to infinity. What happens to the metric? And what happens to the leading order is easy to guess. It's just you get the same formula in Kowski space. So you get DS Kowski, let me write it. But now the question what happens next? And in general to figure out correct follow-ups, well, I haven't tried, but the lore is that it's a state of art because you have to basically figure out what are the correct follow-ups which allow for non-trivial physics but also exclude seek physics like infinite energy excitations, et cetera. And in the context of say four-dimensional asymptotically flat space times, this follow-ups work out correctly in the sixties by Bondi and Saks and friends. And let me write the answer. So now here, this is just a gauge, but here's the physical insight already enters because now I'm taking the R to infinity limit and I think of R as approaching now infinity and I'm saying to you, what is the behavior of fields in this limits? And here are how the people talk about them. So first, there is a correction of this type and B over RGU square. This is called mass aspect related to a sense to mass of a space time that you will see in a second. So at this line, it will be a little bit that you, if you want to read the papers you just have to accustomed yourself with this. This ingredients, that's the objects that people discuss when they talk about asymptotically flat space time. Then there is this CZZ tensor and this will be related to radiation. So CZZ is related to radiation. It doesn't have its name, but its derivative has a name. So NZZ, which is equal to derivative with respect to U of CZZ. That has its own name and it's called news tensor. Okay, so we have this. Then there is the term of this type. And now we will discuss one more piece, which is one more subletting in R, which has this form, probably the longest formula. Last complex conjugate. Okay, so that's the expansion of the metric at large R. And here is the main player, so it's mass aspect. It's a news tensor or CZZ. And this thing here is called angular momentum aspect. Yes, please. The bracket, the bracket for CZ, ah, close here. There is no bracket, sorry. Oh, there is a bracket, yes, yes, yes, yes. It's here and it close here, yeah. So it just refers to CZ that you have to take complex conjugate of CZZ, yeah. Thanks. Okay, so sorry, this is a derivative with respect to U. And important thing is that because it's an expansion at large R, all these fields, they adjust the function of all these fields. Let me write them phi, a function of U, Z and Z bar. So they only depends on the point on the sphere and this time. Okay, very good, so, yeah, very, yes, thank you. So now we have, we have, this is a sphere. So we're playing with the metric on the sphere. So these calligraphic Ds, they refer to covariant derivative on a two-sphere. What is this metric? Now we do the following thing. So let's now write the equations of motion. Or, okay, let me maybe do another thing. So far I haven't used the equation, Einstein equations, I haven't wrote. Now let me, let me write, yeah, let me write equations of motion so that we get a bit of an intuition about what's going on. Just to understand a little bit, to have some physical intuition about what are these objects, what do they mean, it's, let us consider, for example, these equations of motion, Einstein tensor, some matter, UU component, and we expand it at large r, and we get the following equation, UU of mb is equal to one quarter, of plus t2, z bar, and z bar, z bar. This one? Yeah, notice that the sphere, sphere has r square in front of it, in the Minkowski space. And here, this is the equation, and here the stress tensor tUU is defined as a, is equal to one, four, nz, z, nz, z. Plus the limit of the matter stress tensor. Okay, what does it mean? Well, tUU, you can think of tUU as a flux of radiation through null infinity. It has two pieces, if you have some massless particles, you have a flux of massless particles, which is equal to this. And nz, z square is a flux due to gravity waves. So this is a, this is a gravity waves. That's why this new, this new tensor is related to gravity wave, and it has its own name. Now you can, there is a famous, famous equation, which is, if you integrate this formula on the sphere, let's integrate it over the sphere. We get dU integral, d2z, mz, z bar, mass aspect. And this is just total divergence. It integrates to zero. And you get the formula that this is, and notice that tUU, if the energy, so this is the energy of the matter, which is positive. And this is a square of, also, it's an energy of gravity waves, so tUU is positive definite. We get that the change of the integral of mb is negative. Along the u. So it's monotonic, along scry. And this formula called Bondi-Mass loss formula. If you wish, the picture is that you have a space time with some mass, which is adm mass. The integral of Bondi-Mass aspect here at minus infinity is equal to adm mass. And as soon as you evolve forward in the future, your space time loses its energy, because it radiates it away. And the change of your mass is negative definite, because radiation carries energy away. That's a picture. So you wish, so mass aspect is a density of this integral. It is not positive definite anymore. And it has this energy flux part. And in the discussions of soft theorems, et cetera, this part is called hard part, because you can really measure it. And this piece called usually soft part. Because it integrates to zero, and it will be related to the subtle memory-like effects, which we discussed in the lecture. Now that's very, very simple. And now let's get the first, say, non-trivial conditions or set of conservation laws. And by noticing the following. Yeah, so it's an excellent question. If you write Einstein equations, we will find that there are relations between those. And so NZZ is completely unrestricted. You can take it whatever you like. And then for, there are extra data, which is integration constants. And there are integration constants for mass aspects. There is an integration constant for CZZ. And there is an integration constant for angular momentum aspect. So I will come to that. And there is a similar equation for, there is a similar equation for DU and Z. Let me write it. So that, again, you can do it yourself if you plug this ansatz and Einstein equations. You can check this formulas. There are also some leading terms, which are important. This is sometimes confusing in the papers because in the papers it's written this plus dot dot dot. But actually to get things right, you have to write some little bit of dot dot dots to add these components. Sorry, okay, so let me finish this. It's a question, again, similar to the one we wrote before. TUZ, it's, again, the same thing. It's related to angular momentum flux. You can show that TUZ measures the flux of angular momentum. Again, you can write it. There is, again, the soft part. There is also derivative of MB, but importantly, okay, you have this evolution equation for NZ. So what is the data after you plug all these equations? What is the data that specifies the solution? Well, let me write it. It's, you take N, NZZ as a function of U, Z and Z bar. And you take integration constants for, say if you wish, CZZ at I plus minus, MB at I plus minus, and Z at I plus minus. So these are first order equations. We have three first order equations. This one, this one, and this one. They're all specified in terms of NZZ, but they have integration constants. So these are these three integration constants, and this is a new tensor which specifies the radiation. That's a phase space. If you go to higher orders, I think it's an open problem, again, as far as I know in relativity to figure out if it's all the data, if everything is subleading terms related to that, or there are some new tensors. I think it's, maybe it's believed that it's all you need. If you go to higher orders in R. Now, here comes, by the way, I'm leading you with not the usual path. The usual path is that we have a symmetries, and then we derive the conservation laws. But in this, in this subject, it happens that it's easier actually to first find the charges in conservation laws and then figure out what the symmetries are. And that's what we will, we are one step away from this infinite number of new conservation equations. Yep. I wouldn't say so. So this is some, this is some data, this function in a sphere. It's not a number, actually. It's not a number, yes, sorry. This is a, because we are on the boundary, it's not a function of you, but it's a function of a sphere. So it has these three functions. Oh, it's crime minus one. Well, that's, I think that's the current picture. I don't know if it's not a theorem as far as I know. Yeah, it's a boundary, and then you integrate it with the news. So, well, it's actually far from my zero, because this, but yeah. On a picture, yeah, exactly. So you specify the three integration constants, and then you integrate along with the NZZ and the constraint equations. Okay, now we will do the step, the glorious step of getting infinite number of new conservation laws. So, and this step is as we, as again, I tried to explain a little bit in this toy model. Now you can ask, okay, we have our Penrose diagram. We identified the fields. The NZZ is a radiation. What happens, what are the asymptotic properties of the fields as they approach boundaries of scry? If I ask you if I give you some system, nonlinear equations with some matter, it's a non-trivial question. And, well, in particular in Strominger's paper, it seems to re-occurring result which pops out in this discussion about by Kristadul and Kleinermann. It's, I know nothing about it, but so this is some rigorous theorem about nonlinear stability of Minkowski space. The idea is that if you take some Kashi data which is finitely but small, close to vacuum, it all disperses to now infinity and you don't form a black hole. But also they figure out, from that, the fall-offs of these fields. In particular, you can ask, how do we know that if I take the limit of these fields on a scry, when u goes to zero, it is at limits exists. Maybe it blows up, maybe these fields blow up as we take u to infinity. In a paper by Strominger, he referred to this paper, this result from this theorem where they show that, say, a new tensor, it falls fast enough that you can integrate it. And this limits exist. As far as I know, this is only, say, you can refer to this Kristadul and Kleinermann only to show that this guy exists, this CZ guy. Even this thing about mass aspect is just a conjecture again because you see that it works. You assume that it exists, then you do some trick and you get some relations which happen to be correct when you check them. So it suggests that this is a correct picture. Moreover, then another stretch is, okay, Kristadul and Kleinermann is for spacetime without black holes, but again, this picture seems to be correct even in the presence of black holes. So maybe this Kristadul and Kleinermann, I don't know what's its role at the end. Maybe that's just a correct thing, correct way to think about scattering and it's extremely hard to prove it with some mathematical rigor. Anyhow, so, with a small digression about this famous result, we get this data. Moreover, we believe that this limits exist and now the idea is that, well, if we have a scattering problem, we believe that say there are some symmetries here, there are some symmetries there, there are generators of symmetries, there should be some match along this point I note. There should be some match of the fields and the idea that the correct matching is this antipodal matching, let's cry. As we write before, now we write the following things. Mass aspect, Z bar at i plus minus is equal to i minus plus where important points at this coordinates which I choose on the sphere are antipodal. So if you take point Z here, if you take point on the sphere in the north pole on this part of the equation, the same point Z and Z bar in this coordinate on scri minus corresponds to the south pole. So you identify fields with a flip. Is that clear? What are we imposing? We are matching this integration constant here and here with a flip, which is very non-intuitive. And you can ask, okay, how is it possible? We have a completely smooth date on spatial slides. How is it possible we have a flip? Well, it's possible that these points are really not close to each other. You should not think of them as close to each other. You have some smooth data, you take some limits and you get this picture. So as soon as you have this equation, we get an infinite number of new of conservation loss, as you will see now. And the second equation that we will write is related to the same type of identity for all other three constants. So we have the same thing, we have for NZ, again with a flip and the same hip for CZZ. There are some minus signs involved, but I choose a convention such that we get the flipping. Let's see what are the consequences of this identification. To get to it, it's very simple. Let me introduce some hints. The conservation charge. Two set of conservation charges. One conservation charge, QF, corresponds to integral over the sphere with some function, arbitrary function, labeled by function. And here we have mass aspect ZZ bar at I plus minus. And this conservation charges will be super translation conservation charges. Super translation. I haven't explained why so far, just we are following the road of charges. And the second set is the same thing but with angular momentum aspect. So you take some vector field and you integrate gamma ZZ bar, angular momentum aspect. So this is another set of charges. Why do I call them conservation charges? Well, let's integrate this equation with some function F and with some vector field Y. Now notice that we can use integration by parts and write, simply, we can write, for example, here, we can rewrite it as an integral over scry plus of dU dU mB, the minus sign. So here was a minus mB i minus minus. What I did here is simply integrated by parts, if you wish. I rewrite, this is the integral of total derivative. It's equal to the difference of its boundary values. This is a one boundary value and the second boundary value I just subtract. Start with these charges and then I rewrote the total derivative. Okay, now we are getting, and now for this total derivative we can use Einstein equations. Exactly this. And for this term, imagine that we start, we consider space time where it starts in a vacuum. So mB is zero and it ends up in a vacuum where mB is zero. Consider space time like that. This term is then zero. Now we do the same thing on the left-hand side and we use again equations of motion for mB. How much time do I have? This is, if you do this for these equations and for these equations, this is what is called this infinite number of conservations law for super translations, this is this. And this is called super rotations. What is the meaning of this conservation laws? Well, let me first discuss super translations. So we're doing this procedure. We get the following identity. We get that the integral of this right-hand side, no, it's absolutely arbitrary. Yes. We get that the integral of this thing on the square plus is equal to the same integral with, again, it's a typically matched point on the sphere on the square minus. So this is a crucial point. Now what is the meaning of this thing? Let me explain. Let me write first the same thing here. So this is the flux of energy in y minus and y plus. If you choose f, if you choose f to be constant, this is just the conservation of total energy. This term integrates to zero and then we get that the total flux of energy through a square minus is equal to total energy flux of square plus, yeah? Oh, thank you. Yes, I'm sorry. Absolutely crucial. Yes, it's... Yes, there is dv. So now this fields, the functions of u and z. And if you take f to be its lower harmonics on the sphere, say, if you take a constant, it's just the conservation of total energy and total momentum. This piece drops out and you get that total energy is equal to total momentum. Now the idea is that what we have arrived is that it's a new set of identities which express some conservation, but for any function of f. Sometimes this thing is called... For example, let me choose f to be a delta function on a sphere. Then sometimes this is called energy conservation at every angle. This might be confusing because how can energy be conserved at every angle, you would say? Consider scattering of two particles like that, which end up being like that, and measure the energy flux somewhere here or somewhere here. Clearly energy flux is not conserved. So this part is different from the point on the sphere. However, the point is that this integral of N, notice that for this part we can do this integral over U and it will become just Czz. So it will become the soft mode because remember that we can plug for Nzz du Czz and then we can just do this integral and we will get this soft mode. So the idea is that the change in the energy flux at every angle in the antipodal image spheres is exactly compensated with this change in the columnic field which you can measure with a memory. Find useful to think about it is that you say that energy... energy flux is not conserved at every angle but it's remembered. If you take a flux plus memory in the past, it's equal to flux plus memory in the future along every angle. So that's what the super translation conservation is about. Any questions? So far you should be... Here's a charger. You see it's co-dimensioned too in the quantum field theory the question is in the quantum field theory the charge is defined in co-dimension 1 here we have the sphere whereas the charge is given by the value of the field on the sphere or the integral on the sphere which is co-dimension 2 but this is because it comes from gauge theory. The simplest example of this type which you can have in mind is consider a particle which is freely propagating through flat space. Minus infinity to plus infinity. In this case this change in the memory will be trivial as it will be zero and then you see that the flux of energy on untypedally matched points on the sphere because if you start with some point if you start with the south pole and you then propagate forward you will end up in the north pole so then this becomes trivial. So that's... consider your favorite example and see that it works. Great that I confused everybody. Please ask me questions if you... Yeah, so if you... remember we... we discussed in the first lecture this model where it was H so this... to connect to the previous lecture just think of Czz as being H and then you integrate Nzz you get delta Czz which is delta H so it's a memory and this is the centipidal matching that we discussed for H now you can write exactly the same... exactly the same identity for this second set of charges using this equation and again making... you will get something which looks like relation between fluxes of angular momentum this type of expression it has hard part which is this which is really if you put a colorimeter and you measure the flux you will measure this and it has a soft part which is something that will be measured with some type of residual deformation in your detector for example this... for this guy you can measure it with some clock desynchronization but there are many other ways anyhow so this... the philosophy here is always that this new conservation laws they generalize the existing ones like energy conservation momentum conservation by including this soft piece now... that's a picture notice that I haven't... I call them super translation and super rotations and I call them charges and I call them symmetries but I haven't explained why is that for the sake... so actually with a fund that's more pedagogical but... I shouldn't have raised that the idea would be now that if you... if we call them charges so what kind of symmetries they generate that's the next question I would like to ask so we identify infinite set of quantities which can serve we can use them to constrain the data given if you know something if you know something on SCRI- you know something on SCRI+, and the next step which I was going to explain is, ok, miraculously in hindsight identify the charges what are the symmetries that they correspond to and we will find that these are indeed the charges and they do indeed generate the symmetric symmetries that's... that's the idea and moreover if you assume well... if you assume that these charges commute with S-metrics if you imagine now some quantum theory scattering and you assume that they commute again maybe for for super translation you can try to relate to some ADM Hamiltonian and... argue for this but for super rotation basically I think this is an assumption again that these things commute so if you assume that if you consider this as the charges and you assume that they commute and then you sandwich these identities into some in and out between some in and out states you will recover soft theorems that's a connection to soft theorems and this way you get these charges which are conserved they have a flux of energy which is familiar they have this memory part if you sandwich them between the S-metrics they generate soft theorems and this is a notice that's completely full in on linear analysis of Minkowski space we don't... so we jumped from the linear simple problem to full in on linear problem there is no approximation here if there are no questions I would I would like to... in a five or... ah no if I just I lost the track of time ah we started at 2.30 so yeah I should... I will finish in a minute ok so now in principle I would have to explain I would like to explain two more things and I will explain them then partially in the next lecture is that one thing is that this a bit unusual unusual thing from charges to symmetries from charges to asymptotic symmetries so we will find that we introduce the notion of asymptotic symmetries and then try to find the charges that generate them we will find that this are they this are those and then ah from another second piece is that from these charges to familiar soft theorems because you if you think about scattering amplitudes you are used to think in momentum space here everything in coordinate space and so to go from this charges in coordinate space to usual soft theorems there is this sort of LSE procedure Lehmann-Symanchik Zimmermann in coordinate space so this is this are the things I will explain next time thank you