Transcript for:
Scattering and Memory Effects in Gravity

for the scattering part okay it's just going to be everything is going to be on scratch okay so uh the the sorry okay so what is this what is the setup so the question that bms asked and i want to say i want to sorry since questions what do you mean by that since the memory effect is the difference between incoming and outgoing stuff no the memory effect is the difference between some you do some measurements at time at on scribe plus you do some measurements at u equal to zero you do another set of measurements at u equal to 10 and then the memory measures the difference between those two measurements but both the measurements it relates it to data incoming on scribe plus square minus yes yes but that's true but what i i don't want to say anything about the matching condition if i didn't know about the matching condition it would just be a measurement i do in the far past of future null infinity and then i do another measurement in a far past of far future of future null infinity and then i'd relate those two which is what memory is and then because you don't if you send in address state yeah from scribe minus yeah uh it doesn't contribute you know if you look at like braginsky and thorn there's an assumption there which i think uh you know sasha and i sorted you know assuming that they're undressed states that are being sent in from scribe minus so i don't think the problem is even well posed you cannot answer the question what you see at u equals zero and u equals 10 without knowing uh what came in it's what came in from scribe-minded now you could make some assumptions about generic initial data or something but there's an implicit assumption yeah i'm going to make some assumptions about well it's not going to be an assumption i think the memory effect itself is physical even though as i i think as you correctly said you can't make any measurements that are physical sitting at u equal to zero just at a single unit of time that's not physical for that to make that to make that measurement physical you need information coming in from scry minus you need to know something about what's happening at square minus to make that physical but the difference between measurements at u equal to 10 and u equal to zero that difference is in itself physical and doesn't necessarily need any additional information about what's going on it's prime minus you might need some special conditions occurring at scribe minus two have a non-trivial measurement but just at the level of what the measure just doing the measurement and extracting something physical you don't really need to know a priory what's going on let me explain first and if i don't if i don't if i don't answer the question then maybe we can discuss that okay so uh what what exactly was the was the question that bms were asking and i i want to include vandenberg here whose name is basically missed in every single time um so the question they asked was that what do far away observers measure um in gr and what are the symmetry properties of these measurements that's that's roughly speaking their question what do i mean over here by far away observers so you i'm going to give a rough picture you imagine that you have some sort of static frame you know you have a bunch of galaxies in your space time and you're in a roughly speaking static frame where most of the galaxies are are relatively slow moving i mean slow moving with respect to the speed of light now what you do is that you you as an observer you boost you go to a highly boosted frame with respect to the static system and then you're going to do a measurement measurements in this highly boosted frame and those are the type of observers that bms was studying so the idea is that this is if i draw out minkowski space time here that i'm going to look at these sort of highly boosted you know so highly boosted observers look like that in a certain frame and i'm imagining that all sort of galaxies and are all moving relatively slowly and then here i am sitting on the millennium falcon which kind of looks like that and in in a highly boosted frame and then what i'm going to do as an observer is that i'm going to fly away very very very far away and once i'm sufficiently close to a scribe plus you fly away you wait for one million years with respect to your own proper time and then you turn on a detector and then you do some measurements for some finite amount of time here and then then you turn off your detector and then you go back and the idea is that you have multiple such observers each of them going out at separate angles and then after you do your measurements in a small neighborhood of scribe plus you meet back up again far away in the future and then you collate all of your results and then you can try and ask the question what information about the space time can i reconstruct okay this is that's the idea okay so i'm going to make this more precise now so since all of the measurements and everything that i'm going to be talking about is going to occur in this region of the space-time which is a tiny neighborhood of scribe plus it's obviously going to be convenient to set up a coordinate system that is that works in this neighborhood so the way we do that is is is very simple so there's sort of two ways to do it one of them i find physical but it's not the bondi gauge but i'm going to tell you the physical gauge and then i'm going to work in bondi cage because that's where all the equations were written out so scribe plus is this null surface and i'm going to define u as being the affine coordinate along the null generators of scribe plus uh and i'm going to define as xa which is z and z bar i'm working in four dimensions to be to label the generators labels null generators so this completely sets up the coordinate system on scribe plus but we now need to introduce a fourth coordinate system so that we can define a neighborhood of scrap plus and what we do there is we look at null slices a foliation of null slices each of them labeled by u equal to some constant and i'm going to define r to be the affine parameter along these null generators so r will be the affine parameter along these null slices which is different from square plus okay notice that this doesn't by itself completely fix the coordinate system because i haven't told you the origin of r so i haven't told you where r equal to 0 is set so that's a coordinate freedom that we still have which we can use later to fix so you can work out the metric in this coordinate system and it looks very nice it's simply something that looks like minus u u squared minus 2 dr okay so that's uh that's the coordinate system this i mean i'm used to calling in newman until coordinates because i first found it out found out the use of these coordinates in a paper by numerity at least in this context but there's a good chance that this these coordinate systems are introduced far before that and i'd be happy to know what the original names of the authors were okay so this is this is a good gauge to work in and all the coordinates here u r and x are somewhat physical in terms of sort of affine parameters on various geodesics but this is not the gauge that bondi worked in and i'm going to work in bondi gauge because that's the gauge in which most papers are written and it'll be you'll be able to compare equations if you do that so what bondi did is that here he instead introduced us another gauge condition which is a determinant g a b over r square is a so this is an extra gauge condition that was introduced by bondi but of course when you introduce an extra gauge condition you have to introduce another function because and this function was the this coefficient of d udr so this is the gauge that i'm going to be working this is this metric is the metric in bondi gauge excuse me i i thought you you actually restrict the determinant of g a b to be x square sine squared theta yeah that's why so i divide out by g a b r square yeah yeah and don't do that don't you fix really the number i'm that that's not going to be part of the gauge condition that's going to be part of the definition of asymptotically flat space times which i will come to later because sometimes you might require you know if you want to study super rotations i'm not going to do that in this talk but if you do want to study super rotations in fact in higher dimensions for example you don't want to be the you don't want the boundary metric to be that of the unit sphere you don't want to fix that you want the boundary metric to be fluctuating as well so that's going to come into a definition of what i call asymptotic flat which i have not yet defined okay at this point this is a four dimensional metric and in principle any space time can be written in this form you know this is this met these coordinate systems are valid in the neighborhood of any null slice of any space time okay so to make this more precise at this point i need to tell you what the definition of asymptotic flatness now here i have to say honestly that this part of the game is a bit more of an art than a science there are some principles that one have to has to follow you kind of want finite charges you know finite uh you want the equations of motions to hold at leading order so there are some guiding principles but overall there's no real uh mo there's no there's no precise algorithm to define what asymptotic charges uh what the asymptotic follow-up condition [Music] so i'm just going to tell you what the falloffs are and these are the follow-ups that i'm going to be using after that it's going to be about soft theorems they thought i can go home sorry i missed the question was there a question is there a question uh okay um yeah so yeah so so the game here is that you want the falloff conditions to be sufficiently weak so that it keeps you know all interesting physics you don't want to you want gravitational waves obviously you want black holes but you also don't want it to be too weak that you end up having pathological solutions so as i said it's more of an art so what i'm just going to do is tell you the boundary conditions so the boundary conditions i'm going to have is uh that u is 1 minus 2 mb over r then there are some leading corrections and let me be i'm going to use small o notation here beta is okay and these define asymptotic platinum so this is the definition of asymptotic flatness here i'm going to now fix as you ask this gamma a b is going to be fixed to be the unit metric on s2 this is not always what we want to do in fact this you know should not be this this this is not a good generalization to higher dimensions uh which is still an open problem right now because uh if you want to include super rotations in your face space which seems like something that we want to do then you actually don't want to fix the background metric to be that of the unit sphere the the proper definition that includes super rotations of asymptotically flat geometries and higher dimensions is not really known and the cab is a traceless is a traceless tensor so it satisfies gamma ibca what is your small small o of one yes what this is just a little low notation basically telling you that whatever appears here is just sub leading with i know i know but there is one over r square so yeah well sorry where did i do that my mistake you you mean to factorize or what no no would just be or one over r square that's a typo okay yeah just two sorry i j i'm keeping a little notation because of the log terms that you could have so if i say big o one over r square then that might be a problem the point is that you could in principle have log terms in these subleading coefficients which also have the importance as a lock mentioned some of these in his talk uh ashok also talked talked about these but to the to the order that i'll be talking about today there won't be any log terms present but but generically there are log terms in these sub-leading uh terms over here maybe maybe you meant one over r squared on the first line and one over r cubed on the third line no i'm using little or notation not big o notation just just to be oh oh oh oh okay sorry yeah that no one can then complain about the log terms that's all i'm doing because if i were to write big o one over r cube then that doesn't fall off as fast as one over r cube times log of r which we know is a term that appears at this order in the expansion of u by the way if one wants to be picky you know that you save some trouble by writing a small o but then you have to assume that derivative of so small o respect to r for instance is o small o of n plus one and the other derivatives remain the smell small smell same thing so big assumption i agree you have to you have to you have to put in those additional assumptions as well also i i don't know beta i think has to be or the r to the minus two no yes it has to be yeah it in fact we can fix it actually minus one is not enough yeah sorry i should have said because not only is it not only do we know what it is it's in fact completely fixed it's not free data this is something that some number which i don't remember uh c a b c a b plus is the thing that ashok has been talking about is that is that a log yeah the logs that appear in the larger expansion of these things will will also correspond to a logs in u because the only dimensionless log that you can actually have is log u over r so he he's talking about the log of u but if you have a log of u then you also have logs in r right yeah okay um okay so this is uh this is my definition of asymptotic flatness okay now at this point you know before talking about symmetries and whatnot i think we already have the setup necessary to simply discuss uh the observers that i was mentioning previously like what exactly are the observers that i'll be talking about so the observers that i'll be talking about are called bms observers so they travel along fixed the following fixed trajectories okay well lambda is a variable and r naught z naught and z are not are completely fixed so they are they're fixed angle uh fixed angle observers and i'm going to further assume that r naught is much much greater than lambda of course lambda is a variable so you might ask what does this really mean what i really mean is that any measurements that these observers do they will do it in a region where lambda is much much less than r naught because this is the statement that they are close to scribe scribe plus the region where lambda is much greater than r naught is close to time like infinity and that has its own problems and own sets of uh physics that has to be separately studied okay uh so these are the observers that i'm going to talk about i'll make a comment about inertial observers at the end of this talk if anyone's interested these observers so yeah these observers are not inertial so they're not free falling so these guys actually have to be on the millennium falcon but the engine has to be turned on so that they can follow this trajectory okay and what are the types of measurements that they'll be doing so uh here's here's the kind of setup that that i'm imagining each observer moves along a particular world line so let me draw this is roughly the word line of the observers uh in the region where lambda is much much less than r naught that's this this region right here some region in uh over here they're going to turn on their detectors now what is the detectors that they have well the detectors that they have is essentially they have tiny ligo measurement devices so they have a bunch of mirrors so let's say this is the point z naught z bar not where they are located and they have these a bunch of mirrors along all directions you know and and just like ligo the way the the measurements that we are going to do are the distance between these pairs of mirrors those are the measurements that we'll be doing okay so the each mirror is located at so let's say this pair of mirrors is located at z naught plus minus delta z by 2. and delta z is a variable so i have a whole set of mirrors for any value any possible value of delta z okay so they're they're at a distance coordinate distance of delta z away from each other but of course what they will measure is the geodesic distance because that's uh that's what they they'll be able to measure so you can work out the geodesic distance between these two points very very easily uh it's just literally the metric right and i'm assuming that z naught is very small as we'll ah i'll explain so what i'm going to assume is that what i'm going to assume is that uh delta z is the order one over r naught and remember r naught is very large uh so this is in fact very easy to work out so the the distance between these two mirrors at a time u for the observer r naught z naught z bar naught is simply given by g a b evaluated at u or naught z naught z bar naught delta x a delta x v this is l squared square if these observers have watches yes don't their watches become desynchronized by an amount which is not suppressed by one over r completely completely correct uh i was not going to talk about that this time because that's i wanted to keep it as simple as possible but yeah so not not only can you measure the distances as as you are mentioning what you could do is before you send those mirrors out to to the opposite directions you start the mirrors off in this central location and you see you you set clocks on both of them and you synchronize these clocks and you send the mirrors out to this z naught plus minus delta z by two distance and as long as nothing is happening that clocks will remain synchronized and when a wave passes through as we will discuss the wave induces a change but basically no the clocks will no longer be synchronized and you can measure that non-synchronicity by sending light rays and checking whether the clocks are now synchronized and in fact this change in the synchronization of clocks is another way to measure the memory effect through through through time delays as opposed to measuring distances but they're really the same thing the the cause of both of these effects is at the end of the day just but but like for the things they measure at lego the the displacement the desynchronation would be billions of times bigger right millions of times no not not uh now there's no one over but for inertial observers it will be much much larger because there's a cancellation that happens at linear or at leading order for inertial observers right and so the time delay that that inertial observers will measure will be much much more subleating will be order one over r naught squared exactly the original but the people of massive numbers are very are very different from inertial observers right if we had bms observers we would have long ago would be measuring the memory would be easier than measuring the wave right yes absolutely and i i'm using bms observers because they're the math the calculations are easier and it's conducive to a talk but i'm gonna i'm gonna give a reference in as you know in in your book in one of the problem sets of in your book was basically to explain the difference between bms observers and inertial observers and uh monica and i wrote solutions for that where we explain the differences and if all the maps that i'm doing here has been done for initial observers right uh so i'll refer you there for for those differences but you're absolutely right so if we were in fact bms observers sitting on earth we would have measured this a long time but the time delay definitely we would have measured a long time ago but unfortunately we aren't uh and inertial observers it will be much harder for them to measure the time delay i agree okay so so this is just the this is just the the the squared length and i gave you the form of the metric remember again i'm assuming that g naught that r naught is much much larger than u so this is precisely the region in which the asymptotic expansion i gave you is valid so you can use that asymptotic expansion and i'll just write down the formula for you so what we find is that the measure the distance between the mirrors as measured by such an observable is given by 2 or not okay and remember i'm taking my i'm taking delta z to be order one over one so the mirrors are very close to each other because r naught is very large in coordinate distance so what this shows you is that uh i have a leading piece this thing is a leading piece this is order one which doesn't depend on any of the data of the metric so only it's completely it's completely fixed but then we have this sub leading piece you can check because this is order one over or not so we have this sub-leading piece which you can then use to extract czz and remember delta z is variable because the observer has multiple mirrors along every possible delta z angle so what he does essentially is that he does the measurements by measuring the distances between all of these mirrors in all z directions all angular directions he measures the distances and then he uses that to extract c z z of u i'll explain how he does that just now as you and and as we'll see he can't extract c z of u he'll only be able to extract distances the differences not distance differences okay is this clear so these are the measurements that i'm thinking of can you can you show again czz how it appears in the metric ah yes yes yes uh so i'll write it for you right here gab over here was evaluated at or not was simply r naught squared gamma a b plus r naught c a b okay plus subleading okay and i'm working i should have mentioned this before i'm working in stereographic coordinates so gamma's zz gamma z bar z bar is zero uh gamma zz bar is just the metric on the unit okay two sphere and because c is a traceless tensor c z z bar is equal to zero so it only has two independent components you know the two polarizations of the graviton so c a b are the wave forms yeah so c a b is essentially the graph in fact the derivative because i'm just going to draw the news i i'm just going to mention that cab cannot be measured but differences can be measured so the derivative of cab which is what we call the new sensor is actually a measurable quantity okay i'm just going to explain that okay so this is roughly the idea so the the observer is only going to measure l but the point is that using l and then this formula he's able to extract some information about cz now let's see how how that's going to work here's what he's going to do so at some time t naught as i said as i said before let's draw the diagram again the observers are moving along this trajectory and at some initial time u equal to u naught he turns on his detector and does a measurement for the distances of of uh to measure all the distances now you might think that given that value of u naught you can extract czz but the point is that the observer doesn't know the coordinate distance between the mirrors again remember the only thing he or she can measure is the is the geodesic distance l the observer doesn't know what delta z is so in fact that initial measurement is going to be used to work out what is delta z in terms of c z z of u naught you know c z of u naught again as i mentioned before is not something you can say anything about at the initial time so we're going to use the observer is going to use the initial measurement to work out delta c now once he's done that now he knows delta z now if he measures the geodesic distance at any later time he can use that same delta z to to evaluate uh cz at that time but it's always going to be with respect to the original uh the original uh czv so here's the formula here's what he can actually measure and now i'm keeping everything physical so you can only really measure delta which is defined of u which is defined as l evaluated at u minus l evaluated at again let's be clear the initial measurement is used to work out what is delta z and then all subsequent measurements are then used to work out the difference between czs so this quantity is 1 plus z naught squared by 4 delta cz and notice that the leading term has dropped off because again i'm looking at differences so this this very leading order term has just simply cancelled out from this difference okay and this is an order one over or not effect so you can't be sitting strictly at scribe plus because then you won't measure anything you have to sit a little bit inside a little bit in the bug and delta z cz of course is just a c z z of u minus so evaluated at u minus c is easy not okay so this shows us that in fact c z z cannot be measured but differences in c z can be measured in particular because the derivative is a difference n z zz is a completely physical variable and is something that can be canon is measured it's it's squared is proportional to the energy of i mean it's roughly the amplitude of the gravitational waves as we can see here i mean the amplitude is what amplitude is is basically i have a distance between the two mirrors and if the gravitational wave passes by this uh this mirror or starts oscillating let's say i'm keeping this one fixed as a reference this mirror starts oscillating so the amplitude is by definition delta l the change in this thing and delta l is related to delta n so this this new tensor precisely captures the amplitude of the gravity waves and so it square captures the energy of the gravity waves okay uh okay this is all i had to say so these are the types of measurements that we'll be looking at yeah andy would you say you can measure the position of a particle or only the difference in positions only the difference yeah so saying that you can only measure the difference in c is is that the same as saying that you can only measure the difference in the position of a particle yes yes because we are we're measuring the lens between two mirrors so there's the diff is the distance between two objects yeah okay good so this was an interlude into the actual measurements and it'll it'll help us because it'll solve our problems when we talk about memory effects so now let's stop here and i'm going to now talk about super translations and then then in the next section after that we'll talk about how memory effect is the same thing as super translations any questions here uh hello i have a question uh like how you fix the value of the cj at value u naught you don't fix it it's i mean uh well you it's in the sense that you can fix it to be whatever you want it's completely unphysical right like part a gravitational wave measurement like how we can tell that like suppose before the suppose for a scattering process yeah like you fix that okay like before the scattering happens your space time was completely mean cos key so in a sense your cz is basically zero right no no it does not mean it's zero it means that it's dz squared of c i'm going to explain that soon okay it's not zero it can be fixed to be now this is a point where you might say that information from scry minus might be helpful because there's a matching condition for czz which tells you that cz at scribe plus minus you know it is physical if i had information about what's going on at scribe minus right because there's a matching condition which tells us that c z that's square plus minus is equal to c z z at square minus plus so if i had access to this information from the past then there's a meaningful way in which i can give some meaning to the value of c z at a particular point at a particular value but in the absence of that information given that i'm doing it trying to do everything locally at square plus you know you can't say anything about c z z at u naught uh again as i mentioned the difference is always measurable it's the same thing like the phase of an electron there's no sense in asking what is the phase of this electron but you can ask has the phase of the electron changed in a particular process that's a perfectly reasonable question in fact it's pretty much like that because super translation is an abelian group so in a billion group you can't measure charges you can only measure changes in the targets right right yeah i agree but just from the physical perspective like i can tell that like before the scattering happens the space time was completely minkowski and like after the gravitational wave passes through at late time the background changes to something else so the metric changes from eta muni to something else which is determined by the like super translations you are telling and then yeah but i just i just want to make it clear that minkowski doesn't necessarily mean cz is zero okay uh it could be cz dz square minkowski could have cz i mean if i take take minkowski space time in body coordinates and basically which is this outgoing null coordinates and do a super translation of that super translation is just a diphymorphism right so it's not going to change the riemann tensor at all right right now when i do a super translation the new cz if i if i started with the metric initially where cz was zero right the new czz will no longer be zero the new cz will take the form minus dz squared c right right yeah this is key space time you know it's still perfectly flat minkowski space time it's just that i'm now working in this different frame no no yeah i agree completely with that that after the scattering happens like it becomes digit square c but like at u equal to minus infinity like sky minus minus when there are no scattering is there so can there you can put cgg to be zero to start you can put it that's a choice and all i'm trying to say that it's a choice you will never be able to measure so here here's the idea you you you you define cz to be zero at scribe minus minus okay okay and then scattering happens c z z x square minus plus becomes whatever it is which is fixed by the constraint equations then there's a matching condition which tells you what is c z at square plus minus okay and then again there's an outgoing radiation and then c z z at square plus plus is whatever it is which is if i impose appropriate boundary conditions you perhaps no it's whatever it is so you start one of at any one of these points you have to make a definition and then everything else is fixed with respect to that definition okay thank you okay okay so let me let me now talk about super translations uh uh quickly so super translations are basically so what i told you before was that the metric that i the general metric that i wrote down had the following asymptotic structure minus du squared and let me just write down the terms that are of interest to me and then the other stuff uh well maybe right let me write one more dz cz and then subleading okay this is the asymptotic form of the metric so this is the class of space times that i'll be studying so uh here in doing so i've essentially already fixed uh bondi gauge so the question that bms has is that what are the residual set of differemorphisms that i can do act on this system okay so what are the residual set of different morphisms which i can do on this metric so as to leave the form of this metric invariant that's the crucial part and what does it mean by the form has to remain in variance so things that are identities never have to remain identically zero so which basically means that the lead derivative of g let's rr has to be the lead derivative of g this is just the bonding gauge condition also we need as we mentioned before the lead derivative of del r determined in gab over r square has to be zero this is just coming from the bondi gauge condition these are just identically true because of the gauge condition we have well the rest of it is not identically true but we have to preserve the form so if i look at the uu component of the metric the original uu component has has a structure which is of the form minus 1 plus 2 mb over r so if i look at the change in the metric i don't want to affect the minus one part of this i want to leave that in variant but i it's okay to modify this as long as it's still one over r so the boundary condition i'm going to impose is that the lead derivative of g u u is order one over r okay this is this is the idea so you can you can work it out for other coordinates as well uh so this is order r and all the other uh i don't know which other coordinates that just gua why not just write it so the derivative of gua is order one so these are all the boundary conditions that i'm going to impose so so these are coming from asymptotic flatness and this is coming from studying bondage so surprisingly this there is there exists a set of non-trivial diplomorphisms that actually satisfy this constraints and that's what bms called the super translation then there are super rotations as well i'm not going to talk about them today but this the specific class of different options which satisfy the constraints i just told you is just f w minus one over r and then sub-leading terms the sub-leading terms i don't know if they're crucial or not but perhaps an interesting feature is that they are metric dependent this will sort of play a role a little bit in in the final section of this talk so this is some very complicated diphymorphisms which depends on the background that i'm sitting in and in general obviously if i work in a general spacetime it's it's just a complete mess to work with the nice thing here is that the leading order behavior is universal there is no metric dependence in this leading order behavior and again since i'm doing everything at leading order near square plus we'll see that this universal structure of this this boxed term will really be enough to extract all the interesting results and if you now work out the action of this diffumorphism on c so the action of this diphymorphism uh i should have said f here is a function of z z bar and there's a it's called super translations because standard translations correspond to f being this function a naught plus k one so this is standard translations but the idea is that you it turns out that you can have a global you can have a even higher infinite dimensional generalization of these standard translations where f can be any function and this super has nothing to do with supersymmetry this was what this was worked out before super symmetry ever existed uh okay and so you can you can work out you can work out the transformation of cz and it's a nice do we understand now why why it's the first two orders that are universal no at the level of just this this sort of symmetry algebra right just saying from taking lead brackets or something is that is that what yeah so what i'm going to show later is that if you try and construct the the charges for these things using uh using uh covariance phase space formalism it turns out that these transformations are canonical if and only if the leading order terms are metric independent but doesn't say anything about sub-leading terms they could be whatever they want so this at least tells us that if if you want to write down charges for any of these quantities the leading order term this determine the boxed equation has to be metric independent so there's actually a reason for it to be metric independent and universal engage independent also what do you mean by gauge independent this this vector well i can't pick a lot of did all this in covariant gauge oh and gauge independent also yeah what i'm going to do when i construct the when i do the covariance space analysis i'm going to do it in a gauging variable without any reference to bondi gauge whatsoever and you get these first two terms in any gauge yes i get i get the restriction that the first two terms has to be metric independent and i get i don't i don't get any restriction on the sub leading terms and so you know the fact that they're metric dependent there was no reason for them to be metric independent at least from that perspective okay so so since i'm running short of time let me let me move on um so it's this part of the transformation of cz that's that's kind of crucial because this is telling us that if i was sitting in initially in a in a space time in which cz happened to be zero uh which could be just by choice or you know you fix one but anyway czz happens to be zero initially then after you do a super translation cz is no longer z okay so this tells us that this inhomogeneous transformation in this piece is basically telling us that this is in fact the memory effect the memory effect is measuring precisely this quantity and it should perhaps not surprise you because when we worked out the memory we haven't yet said what the memory effect is but it's going to be related to the changes in l we saw that the changes in l is precisely related to this shift in c and that's this shift over here that we are going to measure just a question that shift is you independent yes that was good that's my very next point my very next point that's a very good question let me be clear about what is the measurement what is the physical part of this measurement now this is the diffumorphism at the end of the day it's a differing office so if you try to measure anything that's diffumorphism invariant you'll obviously not get uh you obviously won't measure any differences so because this is u independent just like you said so if i were to look at the difference between c z at u equals 10 u equals 0 this extra term will simply cancel out because it's u independent the point of the memory effect is that what happens is that whenever a gravity wave passes through scribe plus it induces a super translation to the future of that gravity work but not to its past so you end up with a type of u-dependent super translations where there was no there was no super translation for u less than zero if the wave passed away if the wave passed through at u equal to zero but you have a non-trivial super translation for u greater than zero and so now if i compare the value of c z z at u equal to ten and u equal to minus ten you'll catch precisely this difference so that's what's happening so the type of super translation that we are going to be measuring is not a global super translation that's that's occurring everywhere in space time it's occurring to the future of a particular gravity wave and that's the physical part of the measurement this is like a step function it's exactly like a step function which is i think andy mentioned this in a stock it's that's how it's related to the soft theorem the soft theorem is the one over omega term and that which is the fourier transform of the step function yeah it all fits uh okay uh good that was a good that was exactly what i wanted to say next okay so now let me just go start off with the this is all you need to know about super translations let me just now discuss the memory effect so the groundwork has already been laid so there won't be much to do here uh at least many new things to do but in order to see the memory effect now it is at this point that i need to impose equations of motion you see at this so far everything i've said sort of it's just properties of asymptotically flat space i haven't imposed any equations for any of these geometries so now i'm going to impose einstein's equations i'm going to assume that my geometry satisfies einstein's equations in fact let me make a comment here because the effect that i'm studying is an effect that's only to leading order in large r it actually doesn't matter whether the action of the bug is saying hilbert action or whether it contains higher terms it could contain r squared you know riemann squared riemann or reach g to the 10th whatever it doesn't matter because the point is that because metric is asymptotically flat any of these curvature quantities they fall off at large r so if i take products of these things they will fall off even faster and so they end up affecting further sub-leading effects in fact they do affect to a certain extent the super rotation memory which i'm not going to talk about today but they will not affect the super translation memory because that's just a very leading order effect okay so this this restriction to einstein space times which satisfy einstein's equations is is not really is too restrictive you can actually generalize this beyond if you want let's just take this team union matter to include all the higher contributions from the action okay so what you end up finding is that yeah so you use this equation as a motion and you find a constraint equation it's coming from the mu nu equal to u component which you then evaluate at square plus so let me just take this equation minus equal to zero so i'm going to i'm going to take the uu component of this and then evaluate this on square plus okay so that's going to give me a constraint equation and this constraint equation is del u mb plus t u okay where this tu receives a contribution from from gravity itself of course because gravitation waves carry energy but then it also has a contribution from the matter part again evaluated on scratches so this is the equation you end up finding this is the only equation that i will need so let me just box it so this is this is really the only equation i need to explain the memory effect one more thing i would need to say which i'm going to say right now okay so let's hear the setup of the memory effect so let's assume that at u less for u less than u naught some u naught let me slowly use i and this is greater than unit remember u naught is the time when the observer starts measuring so uh before u naught anyway he doesn't measure so it doesn't matter what waves pass through over there he has no idea so let's say before u i u is less than u i the system was in an approximately short cell type geometry by which i mean that you have the bonding mass which is fixed to be the short cell mass of the system and the new tensor is zero so there's no radiation it's pure vacuum there's no gravitational radiation so news tensor equal to zero obviously means that c z z is constant and remember we can set this constant to be whatever we want because again this is one of those things which is just a reference value again i'll say it again if i had information about scry minus i could fix it but again in the absence of scry minus information you can choose it to be whatever i want the point at the end of the day is that only differences will matter okay but here's one more input that i'll say now this is going to be another input that will go into my definition of asymptotically flat space times and there is a reason i separated this out from the previous uh equations and here's the definition vacuum geometries satisfy this following constraint dz squared c z bar z bar minus which it has a simple solution which simply tells you that c z z is minus two z dz squared c now there's a reason i separated this out from the other constraints you know those other constraints were imposed by were determined by doing some sort of local analysis you locally look at a particular patch and you try to work out the charges and the follow-ups and this comes from some very complicated global analysis of stability of minkowski space time so this is the famous i mean this is this is not the famous but one of the out one of the colorado's of the famous crystal ruler you know that mammoth of a paper is this result over here so if you want to look at geometries which are sort of stable and admit interesting physics also geometries satisfy this constraint so this is a very complicated result and i have personally i have no intuition where this comes from i just know that this is true because ck showed that it's true um i mean one way to see it is that well anyway you can get some intuition but but but uh anyway this is what this is an additional condition that i'm going to impose in my definition of space times that i'm studying are you sure it's not just i mean the global results they had on the curvature uh components that uh you know sky minus miner etc this was something global and difficult to prove this is a local equation no only yeah this was you know it will be right no yeah you you might well be right i have to say my i only mention this because i haven't read through that paper in in any detail to be honest and i'm not able to distinguish what comes from global analysis what comes from local analysis i i don't really understand i think i think the equation with two derivatives of c that's a component of the vial tensor it's a component of the while tensor so some but the last equation i think chris jeweller and connor men if if you just demand finiteness of energy yes um czz the the news would only have to fall off like some power yes the new the news has to fall off like uh u to the minus three halves all right that's chris to doula and kleinerman finiteness of energy is is weaker is weaker finites of energy is one one plus delta yeah anything um isn't it just a half you just square it and integrate over no one over u or maybe it's half i think one over u is okay and that's the problem because if you if you just demand finiteness of energy c is not well defined at scribe plus minus so chris doula and kleinerman had stronger fall-offs they showed that a reasonable range of space times whatever had these stronger fall-offs on on the news near sky plus minus which allows the whole story to go through okay c could diverge logarithmically and then and then bms would make no sense yes yes and so that's where it came in okay okay dms or the matching conditions let's be clear here well i think bms doesn't make any sense without the matching condition because you then you have separate bms on scribe plus and scribe minus and that's clearly not a symmetry of anything yeah and even if you want to focus purely on even if you want to just focus purely on skype plus from that point of view the charges are not fine i think a scattering problem is not defined well yeah i'm saying even before we reach the scattering problem where we try to match the charges the charges themselves are not defined the charges are not defined yeah nothing yeah yeah and then then trying to match the charges is another problem which is also not different so for example if you would like to introduce uh super rotations into the phase space then this tells us that z c z z has to grow linearly in u because under super rotations you know c z z transforms like uh transforms like u times d z cube y z so there's this linear piece in you which is kind of the problem which has been the problem why people have not fully been able to understand uh the the proper structure of this of super rotations non-linearly uh ngr i think you get you get that even for boosted curve right for boosted curve yeah yeah yeah so sorry not standard boost remember for standard boosts this is just zero for the so the sl2c transformations this dz cube y z is just zero so it's only nonzero for the non-trivial super rotations yeah yeah you're right sorry yeah uh okay so you can ask a question yes so this condition that you write above on mass equal m and that's that because zero for is for which region of space time that's for this region this this region right here i thought this is the radiative no no so maybe maybe let me draw a diagram let me draw that i'm sorry here's what i'm imagining now so here's my here's my observed yeah i think it's best explained in the diagram honestly so here's my observer they're traveling like this so yeah i know but it's okay i guess you're in the discussion right now i'll yeah i'll be done i don't have much much more to say honestly so they turn on their detectors at you at u not so before this whatever happened the observers don't really know so they turn it on at some u naught then radiation starts entering the system at u1 so at this point uh there's some radiation that comes in and then the radiation stops at uf okay and then the then at you some u1 the observers stop measuring and then all the observers at different angles get together and they collect all the information the reason this sort of non-local information is required at some point you have to integrate over the sphere so you kind of need all the observers to get together and so they can then integrate over the entire scheme just knowing information at a single angle is not sufficient to extract the super super uh translation that is being done on the space time so those two conditions is for the region u1 for example yeah the two conditions are for this are for this region um this region where everything is vacuum and this region where everything is vacuumed mm-hmm you know there is a problem with this which is that when you have some some total mass in the space-time then you always have a tail effect yes and therefore this radiation never ends i mean you have to wait infinitely long so that you reach a stationary i mean non-radiative now okay and therefore you have to wait infinitely long so that you can observe the memory so how do you think about yeah i think the tale that you're talking about i might be wrong this is not my yeah i'm not entirely sure about this but i think the tail that you're talking about occurs at the next subleading order but not at this order no it's a one over r effect it's a one over r effect so you you have a log r over r term in the memory what type of term that should i be looking at thinking of here no it's a it appears in the bondage here but uh it's it's time dependence is um is a like uh well i think it's something like one over you but i mean you can write exact expressions when you consider for example a a binary system and you consider like the monopole of the system the monopoly is the total mass and you consider some quadrupole moment for the system and then you can see that there is an interaction between monopole and quarter pole and then this leads to a tail effect in the waveform and the problem is that this tail effect continues forever of course it's a very i mean a small effect and you can ignore it but if you want to be exact and i mean no well if i was an experimentalist i wouldn't necessarily care about exact i would put error estimates i would measure it to a certain level where i can have sufficient control over the error and then then compare that to the theoretical result and then ensure that everything is within the error part so that doesn't totally bother me as long as it's true that the effect dies off which it looks like it does uh as an experimentalist perhaps that's not that big of a problem yeah i just want to stress yeah that is n is not zero exactly all right okay and did you do but by the way this is this is what uh ashok and uh others have been the other thing and and we have been discussing that many many years ago yes this is well known that there is this one over you thing yeah i mean this these sort of memory effects have been studied by alok so he pro he's probably a better person to ask about about these the details of these things um but as i said you know if it it obviously it's good it's important to understand those effects but it's also okay to simply just say i'm going to neglect everything that comes after a certain amount of time and as long as i can capture the sort of as long as i can keep track of how large the effect is so that i can keep track of error bars that's also good enough okay so this is essentially my setup i think it's best to let me just let me just quickly go through what i'm saying over here so here uh we have czz is minus two dz squared c initial here i have the same uh set up c z z is equal to minus 2 dz squared c final and the point of super well sorry dz squared and the point of this memory effect is that c final is not equal to c initial that's really the point so if you and we can immediately check this because if i take this constraint equation over here and integrate both sides from u i to u f okay this entire equation then i'm going to find that delta mb plus integral over t u u from u y to u f is the same thing as uh minus dz squared dz bar squared delta c so this is uh this is the equation that i end up finding okay just by integrating this this equation on top and i might have missed a factor of what if i missed i missed a factor over here just because the gamma zz bar squared on this side okay so this is precisely a shift in c and remember how super translations acted so super translations acted precisely with this and if i'm writing c z z is minus two d z squared f uh minus two dz squared c the action of super translations on c is simply a shift with an angle dependent shift with a function f so this delta c that's appearing over here is precisely a super translation this is precisely delta c of some f and now we can invert these derivative operators to work out what is f so this will tell us exactly what super translation is being done and of course because this is where the angle dependence comes in because these are derivative operators if you want to invert them on the sphere you have to integrate over the sphere so then you need information about what's happening at every angle and one can do that so i'm just going i'm just going to write this final equation maybe say a few words about the charges and then just stop so the final equation is that delta f is uh i don't think there's any well let me just write delta f minus e to w bar at delta f at a point sorry not delta f f because uh its delta c is gamma w w bar g z w and then delta mb plus integral of t where this g is just the greens function which satisfies dz squared dz bar squared g is a delta function which again you can work out it's it's explicitly given i should have mentioned most of this that i'm talking about is in sasha and andy's paper which is in uh 1411.5745 anyone wants to look up the formulas that i'm mentioning the explicit form of g is given in this paper okay so that's kind of all i want to say maybe maybe to just prove sort of bring the point home i just want to show you so in i'm not really suggesting that we could have measured this at ligo but in principle this effect can pretend you know you know in principle theoretically i mean can be seen at lyco the idea is that you have this entire waveform so the initial distance between the measures between the mirrors was sitting at zero and the memory effect is essentially the fact that after this measurement has completed so if i zoom in here after the measurement has completed the mirrors end up at some non-zero point instead of zero so the original distance between the measure before the before the wave passed through was zero and so this over here is delta l this is the measurement of the memory effect of course it's super tiny so i don't it's not really going to be measured but also another interesting point is that if i do this measurement at the at the washington center and i do it at the louisiana center i should get two different answers because there are two different angles with respect to the source so this is how you can extract the angle dependent features of this memory effect okay i we already discussed the inertial observers i don't think i have anything to say about that let me just make one final comment since so i i think hopefully i have convinced you that these these sort of super translations are physical and can be measured but there's a bit more there's a there's a more formal if you would like you could you could also approach this from a bit more of a formal point of view where we say that something is physical if we can construct the charge of that transformation on face space and the charge is non-vanishing so for a for a gates symmetry any gate symmetry whether in gauge series or the femorphism you you might have guessed that because gate symmetries are unphysical that the charges are simply zero so that you did they do nothing on the face but this is of course not true so another slightly bit more formal way to approach the physicality problem is to simply construct the charges and then just check whether they're zero or not and so uh here we want to construct them on scribe plus which is a null hypersurface so we can't quite use the adm formalism which is a formalism to work with spatial slices so instead we we work with this covariant phase space formalism which is just the generalization sort of a covariantization of the adm formalism which you can then take and then apply to any any type of cauchy slice even if that's a pair of null slices or if it's half null half spatial whatever any any complicated cauchy slice you want you can apply this formula so the formalism is very very simple i'm not going to explain the formalism because that's i'm not going to explain motivate the formalism that's going to be another lecture on its own right but the idea is very simple here's what you do you take you take the variation of the action of the lagrangian and you you get two terms of course you get a bulk term which is the equations of motion and then you get a boundary term which of course depends on the variation so here this depends on the field phi and the variation delta phi then what you do is that the the symplectic form on any cauchy slice sigma is simply defined as following integrator over sigma this is the uh the unit well normal to this cauchy slice and it's defined by delta theta mu delta prime minus delta prime theta mu delta and so this is a function of two variations on the field so you should think of this guy as a two form two form in what sense it takes two arguments delta and delta prime these are vectors in the phase space and it's anti-symmetric and delta and delta prime by construction so it's an anti-symmetric two form it takes two vectors as an argument and returns a number you perform this integral you get some number and that's by definition of two form in fact this is this is this is uh up to some you know issues that one has to often worry about especially in gauge theories it's very crucial to worry about such things uh this is all you re this is the symplectic form you can then invert the simplex form you can construct the the poisson brackets or the direct brackets if you have a constraint systems and then the charge for any transformation the charges always satisfy the following properties so the charge for q c for a particular transformation c is simply given by the following equation omega delta delta c where delta c is any field variation well it's on any field variation it's a field variation which satisfies this cons this property you see this is a very non-trivial property because the left hand side is a total variation of a function the right hand side is not at all obviously at least a total variation at all so clearly this box equation couldn't possibly be true for any choice of the variations there's a special class of variations for which this is true and they're called canonical transformations that's what canonical transformations are on on any phase space so we use this method to construct the charges so if you do this if you do this for the super translation in fact if you do this for diphyomorphism at this point i'm not chosen any gauge this is not bonding cage or anything so if i work this out for uh for any diffumorphism you find the following statement so the the variation of the hamiltonian charge is the following form there's a one over four pi g integral del sigma and then there's other terms which are proportional to delta sigma sigma okay so two things to note three things to note perhaps firstly note that everything on the right hand side is a boundary term boundary on the cauchy slice this is this is essentially the statement that if i do any different if i if psi is a diphemorphism which vanishes on the boundary then then whatever that charge is that's definitely zero so those those types of diphthong diphymorphisms are definitely unphysical they have vanishing charge so if psi on the boundary is zero then this is physical so this is an unphysical diffusion but more than that if a psi is not zero now we have to consider two two cases the first case is that is delta psi let's say this is not zero then you notice that this this extra term over here is not it doesn't vanish and so the transformation is not a canonical transformation because it's not a total variation of something because this right and this side is a total variation this term is a total variation but this term is not a total variation but all such terms which are not total variations turn out if you work out the details they depend only on delta of psi so if delta of psi is non-zero then the transformations are not canonical and so they also don't have charges charges only exist for canonical transformation uh and what does it mean to have a variation of variation means that this term depends on the fields otherwise you can't vary them this is i think the answer to andy's question that the leading order terms notice that only the leading order term because only terms appearing on delta mat the leading order terms must be field independent otherwise the transformation is just not canonical but now finally we can take the final step which is that if delta psi on del sigma is zero then it's canonical and everything is great we can even construct the chart then obviously just this is the charge the thing in the box is the charge okay so these are the three sort of possibilities for the entire system this in fact should perhaps maybe you've seen it before this is a generalization of the komar integral but it turns out to work out exactly in this very nice form it's it's actually exactly the komar integral so it's a bit surprising that the the exact same form the qumar integral of course applies to stationary space times with a global time like killing vector uh this is this this assumes absolutely nothing about the space time itself okay i think so basically i think i've shown that this this this uh discharge is nonzero and so it's really physical from this point of view and i think i'll just end that i just really gonna for too long thanks thank you very much so maybe we should uh we have 10 minutes for discussions but first maybe we should clap for a nice lecture thank you okay we have 10 minutes of discussion sorry can i make a comment yes um so you said you mentioned that this memory is a super tiny effect and uh but i want to i mean for example if you look at uh some papers by mark favata uh he shows that the the the value of the memory is like five percent of the peak of the waveform it's not really super tiny but the problem is that is this is a low frequency effect where you have a lot of noise and so on so the problem with the measurement is that you have the noise but the effect is not really super tiny it's tiny in powers of or not one over or not so it depends on how far away from the system you are well the gravitational wave is always one over r so and memory is i mean it's small in the same way i mean both of them are one over r effects right my understanding is their latest i mean they're hoping to measure it at lego yeah so the way i understood they would be measuring it because one nice thing i don't fully understand this because i think it's because the if you look at the delta c the thing that appears over here now i'm not really sure about this term but but the thing that appears over here is the stress tensor the tu component of the stress tensor and this stress tensor satisfies some positivity properties so i from what i've heard from some other people is that because of these positivity properties even if it's true that a single gravitational wave that passes by the effect is small but if you have multiple gravitational wave events that pass through because of the positivity properties of this tu the net effect always grows so everything just adds up you don't have cancellations and that's why if you if you if you study hundred events for example then you might be able to actually imagine this is what i heard from some some experimenters that i was talking to uh they basically said that if you collect like a hundred such black hole merger events together and then add up the add up all of their contributions that will definitely be miserable but a single event i'm not quite sure can i make a slight comment on that yes um there is actually a sign degeneracy that you have to break so you have to accurately resolve the event to a high enough degree where you can break this sign degeneracy of the memory okay um and then if you have that accuracy then you can add the memory signals but i believe the most recent estimate for lego is order 2000 events to confidently detect memory uh but it is just a one over r effect it just shows up pretty naturally in the waveform okay i'll show some examples of that on wednesday okay or thursday sorry thank you okay any other questions we still have andy has his hand up yeah so um so there's a difference between the classical story and the quantum story i know this is based mostly the classical story here and in the classical um well this is something i guess we we've just discussed a lot but just um in the classical story you can always set um c z z to zero it for c to zero it's scribe minus minus in the beginning of time or whatever yeah and that's just like saying if you want to compute the trajectory of a particle you can set its initial position to be the origin yes now but you know in a quantum theory you know you can have a state which is a superposition of a particle at x equals zero and x equals one and there's no sense then in the quantum theory in which you can you know those states are orthogonal and the superposition is is different so um so we wouldn't want to say that you know it's it acts as a coordinate on the face space we wouldn't want to say that it's unphysical yeah i mean and you know one way in the in a quantum theory bms we could take a an initial vacuum which was not an eigenstate of c superposition of eigenstates of c and and take a superposition of that and there's then there's no way in the quantum theory we can take it to be zero just like no i agree i agree so the analog of this and the quantum system is roughly that the one point function of c z perhaps has some ambiguity but you know quantum theory you have super power you can do endpoint correlation functions c is easy right that and you can extract far more information you know you know people always you know anyway i just i just learned that this the whole classical definition of the no hair theorem is you know you can't really very naively just generalize that to quantum mechanics because in quantum mechanics you have access to far more possible measurement than you can you in in a classical three you can only measure one point function and a quantum theory strictly you can measure hundred point functions and there's a lot more information that you can actually extract from just a single operator so uh yeah i don't know the maybe since you mentioned it i i kind of wanted to briefly mention it i don't know if you what do you think about busso's paper you remember so he he he has a paper where he says that these charges in a quantum theory cannot be measured in a finite amount of time and roughly speaking he shows the following thing he shows that if i try to evaluate the two point function of let's say mb i mean he does it for various operators he does it for cz as well but the equation i remember off the top of my head is that if you try to measure the two point function of m b squared roughly sitting at z and z plus delta z so you can show that this thing is proportional to one over delta u times r times square root of r naught where delta u is the amount of time for which the observers are performing the measurement so let's say i'm limiting the observers to only perform measurements between so in our case in in the example that we suggested delta u is just u 1 minus u naught so there's a finite amount of time for which any measurement is allowed but you notice that this thing diverges as r naught goes to infinity so this actually goes to infinity if delta u is finite sorry this is a two point function in a quantum state in some quantum state and in fact i think he took it to be the vacuum state okay yeah the two point function of the mb operator in the in in in a vacuum state and he showed that it's proportional to this on square root of r naught divided by delta u and i think his argument basically was based on this so he was he was arguing that if i would if i would as an observer if i'm only allowed to make finite time measurements so that delta u is always bounded above then i will never be able to measure any of these quantities because mb essentially this equation tells us that mb is a bad observer at scribe plus mb is just not a good observable to measure so you can't i wouldn't agree with that i wouldn't agree with that i mean in quantum mechanics we assume yeah that we can specify the initial state with arbitrary accuracy and measure the outgoing state with arbitrary accuracy we do exponentially many repeated experiments too yeah yeah but but you determine it you might be however accurate you are if you're only restricted to finite time lens the the point that that buso was trying to make i mean the whole point of what i discussed for that at least classical and i think uh this point was raised by by ali was that you have these tail effects but the idea is that these tail effects we can put an error bound with these tail effects and then still say nothing physical about them but in a quantum theory these these tail effects they have not bounded at all they're completely unbounded and so in reality in a quantum theory this this this memory effect is probably if i'm only allowed to do finite time measurements it's just not not non-measurable because you know even one of them sorry yeah so uh maybe we should but should we wrap up now because we have another lecture sorry um so yeah maybe we can continue this uh in the next discussion this time left yeah so so let's thank prahar once again for a very nice lecture um and uh we should uh do you want to share your screen yes sir yeah yeah i think you have to make me co-host or something like that only then i can i i don't know so you should be cost now okay no somehow uh i'm not being able to say her it is showing that host disabled participants screen says which one is still not up for yeah yeah he's still not a co-host okay now it should work yeah yeah yeah i think now it will work thank you so can you now see my screen yes yes okay so we are very happy to have vishwajeet from epfl who is going to tell us about uh a systematic approach to understand all orders of theorems and tail memory so we should take it all okay so first of all let me thank you for inviting me to give a talk so and i'm apologizing for like not giving a presentation writing in my ipad because i'm still not comfortable with it so i just made a presentation and yeah and another thing is that like the title of my talk is like too much optimistic though we have not reached that goal but i will ah try to tell that systematically how we can approach this goal and how much we have made success to it so yeah and i think uh like prague has given like very nice talk to how to measure uh gravitational memory so that will be helpful to understand ah like my work so let me start by giving the outline which i am going to tell so maybe i should stop okay so the outline of my talk is basically i will talk about first soft theorem in space time dimension greater than four then i will talk about soft theorem in space time dimension four then i will talk about how to take classical limit of soft theorem and then i will talk about like from this classical limit one can extract the gravitational wave form and then i will explicitly derive independently how to derive this long wavelength gravitational wave form and i will show how it matches with the classical limit then i will talk about uh what are the implications of this derivations and like uh in terms of gravitational tail memory so i'll also i should also tell that like ah all these results which i am going to tell all there is always electromagnetic counterpart in terms of soft photon theorem or electromagnetic waveform but i will not talk about that thing today though the electromagnetic path is better understood and like the theme of the workshop is like uh just to mention i think prahar has already mentioned that where we can measure this gravitational memory like for example if we consider like black hole merger problem we know that when they are far apart in spiraling they are we usually use post newtonian analysis to get the waveform and then like when they are very close to each other in the merger phase then one have to use numerical relativity because of the strong gravitational force and then after that when they form a single black hole so they don't like to understand the ring down phase one uses black hole perturbation theory and then when all the ring down phases happen quasi normal modes died out after that if we wait for long time then we can measure the soft theorem so soft theorem only helps to understand the gravitational wave form in this phase ah in the uh like in the gravitational wave form diagram so after some time so in the late time only we can use soft theorem to understand the gravitational memory so let me okay so i will basically uh talk on mostly on these papers so uh like uh which i worked with uh osoksen or no prio saha and currently i am also working on one problem of determining spin dependent tail memory which will appear with devotee logos so let me start with telling soft graviton theorem in space time dimension greater than 4 so what is soft theorem so basically consider an amplitude with which has arbitrary number of external finite energy particles which can have arbitrary mass and can have arbitrary spin which you will call to be hard particles and also this amplitude contains some small energy gravitons which we call soft gravitons then we start with this amplitude and then we try to relate this amplitude with the amplitude which does not contains any of the soft gravitons and the relation is basically via expansion in power of soft moment soft momenta is the momentum of the soft gravitons like for example if we consider for the single subgravitan case suppose consider scattering where n number of hard particles are involved and one soft graviton with polarization epsilon momenta k and the hard particles have moment of pa and spin sigma then we can see that this n plus one point amplitude we can relate to a n point amplitude in terms of some operator operating on this end point amplitude so like i divide i have written down this uh terms as s 0 s 1 s 2 as a power series expansion in omega and this s 0 goes like 1 over the energy of the sub graviton s 1 is energy to the power 0 order one quantity and s2 is order omega and possibly uh also higher order expansions so i'll just explain that this results or this uh expansion power series expansion since omega is true in d equal to four only at tree level and for d greater than 4 those are valid for all loop orders so let me proceed how one can derive these soft factors so there are many prescriptions but the prescriptions which i am going to ah explain is developed by osoxin which is called the covariantization prescription so suppose consider the theories described by a general coordinate invariant one particularly reducible effective action and since it is a one pair accent so just tree level amplitude computed from this uh action gives the full loop amplitude result so in a sense like if we want to compute gamma n plus 1 then we have to compute such kind of a diagram where n number of final energy legs are there and one soft graviton is attached with momenta k and this amplitude we can divide into subset of two diagrams in one kind of diagrams where basically the soft graviton it at attach to finite energy lake in the external lake like in the external line in a sense like there we will have a propagator with momentum p a plus k and the other set of diagrams where basically this soft graviton is not attached to any external legs that set of diagram we consider as gamma tilde n plus 1 and then basically what we have to do we have to find out what are the what is this three point vertex gamma three and what is this gamma tilde and that can be shown that up to some order in expansion of omega can be derived just by coherentizing the one particle irreducible effective action in the soft graviton background but for that one have to assume that this gamma 3 and gamma tilde should not contain any power of soft momentum in the denominator in a sense 1 over omega those cannot contribute in a sense this is also we assumed for a local ah quantum field theory and but this may not be true for a general ah like one particular irreducible effective x action and if we assume this thing then we can see from the first kind of diagram since there is a propagator 1 over p a plus k with momentum 1 over p a plus k so by on cell condition that contributes 1 over p a dot k so this is 1 over omega so this first kind of diagrams can contribute from the leading order like the s0 1 over omega but the second kind of diagrams only can contribute omega to the power 0. so like with this understanding we can see that the assumptions which we made that the vertices gamma 3 or gamma tilde kind of amputated green's function cannot contribute at 1 over omega this is actually not true in 4 space time dimensions and that can be seen just from this simple example like consider say four scalars phi for theory uh coupled to gravity and then one graviton when running in the loop with momenta l you can see when the loop momentum is in order of the external graviton with momenta k it turns out that this kind of five point amplitude or uh like five point amputated green's function can contribute one over k so this assumptions is not valid in d equal to four that is why this prescription cannot be generalized to d equal to 4 for loop amplitudes so now with this prescription one finds the soft graviton theorem up to sub sub leading order so i am just giving here the results so this is this gamma n plus one point amplitude which is related to this gamma n amplitude if you just stripped out the ah polarization of the hard particles so you can identify the first factor is basically the leading soft factor which is determined in terms of the momentum of the incoming and outgoing hard particles and the information of the soft graviton that is basically polarization and its momentum similarly the subleading soft factor which is basically you can see this is like in given in terms of the total angular momentum of the hard particles which involves the spin of the ah external particles as well similarly if one try to analyze this thing up to sub subleading order it turns out that in the sub subleading order there is a universal piece but with this one non-universal piece which means this is theory dependent piece appears and this is very easy to understand like what happens is that for like you can couple the like or you can when you covariantize this action of the finite energy particles in the graphs of graviton background since riemann tensor is gauge invariant so you can always couple demand tensor in arbitrary way to the finite energy particles and since the remain tensor uh involves two derivative over the metric so that gives it extra two power of soft momentum that's why corresponding to leading soft uh theorem it appears at sub sub leading order and this is like theory dependent piece and like if one try to analyze that okay like how up to how much order we can do this soft expansion like one hits problem at uh sub cube leading order and it is not quite known that for a generic theory of quantum gravity whether we can ah like ah whether we can even show that thus like this m gamma n plus one can be com factorizable at fully factorizable at order omega square which is the sub cube leading order which measure yourself a question yeah but the the sub that term that the uh the third on the third line it's not like completely arbitrary right there's some finite finite number of numbers uh finite number of operators that can correct this term right right it has actually only two kind of uh things it here contribution can appears once you fix the non-minimal coupling with the riemann tensor that is one kind of term can contribute to this order and another kind of term contribute is basically uh the three point function which i showed the gamma three which is basically two finite energy particles and one soft graviton is attached so if you just give to it these two informations then this term is completely fixed no arbitrariness is there beyond that and then beyond that we don't really understand how much ambiguity there is because yeah yeah yeah yeah also um and isn't it also true that you know despite this the fact that it looks like this structure is uh depends on the type of terms they appear in the lagrange it actually can the kinematic form of this extra term completely is universal the only thing you have freedom to change is just the the the complete yeah the coefficient i agree i agree like you can see this term is like written as r mu nu so i have not explained it is written as like r mu new sigma and this is basically the fourier transform of the riemann tensor which is like in terms of epsilon mu nu and k ro k sigma you can write it down and like so this structure is completely fixed that only like riemann tensor component or mu neurosigma like fourier transform and that coupled to a factor in mu neuro sigma and also you can write this in in terms of the given non-minimal coupling ah in the theory and that's at this order and also contributions of this like three point function which is gamma three which i have so this n new neural sigma is that is that a fixed like kinematic form that yeah it has a fixed kinematic form it is like so it is like epsilon mu nu k ok sigma and then use the like anti-symmetry and symmetry properties of this riemann tensor to fix the other components okay thank you yeah possibly i should write it down anyway sorry are you saying there's only one tensor structure that's allowed there and there's only one constant or there are a couple of okay so only one tensor structure is allowed there but there are two kind of possible coupling constant could be there one is from the non-minimal coupling and another is how this gamma three looks like in the two derivative order the vertex so only these two three parameters are there but both the parameters contributes to the same structure army are mural new sigma yeah i think this is like the this is derived by alloc and osen in their paper the full structure okay so okay so let me proceed so like now the question could be okay like fine for a generic theory where non-minimal couplings are there the matter particles can have arbitrary spin it may be very complicated to extend this soft factorization to higher orders but can we consider a very simple theory say a scalar coupled to gravity minimally and can we extend this factorization procedure there so people have tried like there are two papers from 2018 where they tried that okay let us use just gauge in variance and using gauge invariance for this theory at least at tree level can we ah so that this factorization ah can be completely determined to arbitrary order and like let me draw again the same diagrams but now just i remove this one pi vertex by just a minimal coupling like graviton minimally coupled to scalar and since they are spinless so good so then okay let's try to write it down that okay say gamma n plus 1 can be written [Music] pa plus k so now this is endpoint function but one of the momenta is pa plus k and rest of this gamma tilde n plus 1 we don't know so this is the unknown function but now suppose we assume that like like the taylor series expansion holds and we try to expand in power of omega which is valid indeed greater than four but and d equal to four it is tree level true so just expand it and just we expand it in power of k and just write it down in terms of unknown coefficient here which coming from this gamma tilde like calling r r and then from this i know what will be the coefficient because this is just the derivative operating on this same comma and just expanded in a taylor series now demand that okay this amplitude is gauge invariant that is stripped of epsilon mu nu and like contact with k mu and k nu and demand that this gamma n plus 1 vanishes what you will find if you do that you will recover exactly the up to sub sub leading order all the terms just from the gauge invariants like you will recover this first term is 0 in the s1 since there is no spin so you will recover this orbital angular momentum operating on this gamma part and here also the without this pin this two orbital momentum operating on this gamma part and there is no non-minimal coupling other thing so this is zero so that you can recover up to sub subleading or just demanding gauge in variance and using the structure that which i just mentioned that two kind of diagrams with this kind of structure so gadget means enough to fix up to sub sub leading order for this simple theory but okay we are not going to stop at sub sub leading order can we extend this to sub to the power n order for n greater than or equal to 3 if we try to do that and just by gauge in variance we find that a part of it can be completely fixed which is i have written here which you can identify is basically sub sub leading soft factor and then like k alpha del del p alpha whole to the power n minus 2 n minus 2 number of this factor and you can identify this factor easily because this will come just from the power series expansion of this gamma n and the problem is that this part is gauging gauging variant fully so you cannot like now you can add to this term with any kind of term which is like self gauging variant and this term cannot be fixed even it is not clear whether this can be written in terms of gamma n that is in a fully factorized form but you can just fix what will be the properties of this r and the just from demanding gauge in variance and that will be just if you like ah make antis like transform mu and any of the alpha is your new and any of the alpha if you just ah change exchange them then it will it have to be anti-symmetric just to ah just to make the gamma n plus one gauge invariant but you cannot fix this r completely so this is the status uh of like in the soft theorem at tree level in d equal to four or d greater than for loop label up to now so up to now is there any question and you have a question yes can we understand the part that is fixed is somehow um the higher end and momentum components or how do we characterize which part of the of the you know say the n greater than equals three expansion which part is fixed and which isn't oh the n greater equal to three the first line is fixed completely and this r term in the second line so this part i understand that that first part is fixed but what what distinguishes it from the second line is that okay okay okay that's how higher angular momentum or no i know how it came out of the calculation but how do i just say which part of the amplitude am i is determined by the soft serum at order omega squared or order what is this yeah omega cube starting from omega cubed yeah okay sorry starting from order omega square yeah so what happens is that this kind of contribution appears when the graviton is not connected to any external line like it is connected to some internal lake in the tree level diagram like if you compute just simple for a simple theory say phi cube theory coupled to gravity and you can show that like suppose the graviton is connected to the propagated part then you can show that like it is not always possible to like like find out this factor r like in terms of gamma n like factorizable so yeah so this is like from the s matrix perspective how the uh how it happens uh only i can tell the question too i cannot raise my head um i mean usually the soft theorems are formulated in terms of the full amplitude and not of the uh you know irreducible gamma right is the passage straight forward from one to the other okay it looks that you get also some extra emission from the internal line right because right yeah so how does it go is it straightforward to go from gamma to the amplitude that's the state the question means starting from gamma n to gamma n plus one so no i mean gamma is one particle irreducible right right uh and you use that i think okay okay the full amplitude is constructed out of gamma from three diagrams right right so which have internal propagators and indicates that your recipe includes emission from the internal propagators right so so that it should cancel out somehow because in the total amplitude you only attach to the external lines not to internal propagators no no like ah like when there is some soft graviton is emitting like it can suppose you consider a phi q theory right where you have and there is another propagator uh for phi propagator and suppose in the internal uh like internal scalar propagator one graviton is emitting from there so that diagram if you compute you can show that like so at the subleading order so let me identify it this so this term you can see you will get this term just from this contribution when it is connected to the internal line okay of the subleading order so it all works uh yeah so past term you will get when it is connected to the external lake but the second term of the angular momentum you will get when it is connected to the internal leg so you have to insert or this subgraviton in all possible ways in the diagrams and then only you will find the full factorization okay but the leading the leading term cancels out because i mean what is internal for the amplitude external for gamma that was my point okay i i am talking about like this one pi effective action and then the diagrams are amputated green's function like we already make this uh external lines to go on let me make a trivial example you have your phy cube theory and you have a two to two scattering process right exchange one you telling me i take gamma gamma is a three-point function okay and to make it to add the soft graviton attach your graviton to each one of these three legs right now you sew it back you put it back in there you end up with this graviton which can be emitted from the internal line right so that contribution should cancel in the leading one over omega sorry i'm like okay yeah i am not being able to understand but if the graviton is inserted in the internal leg that cannot contribute at one over omega right exactly but it seems to do so at the gum at the level of your gum that is my puzzle i agree with the final result and i was wondering if you can see easily you see when you have your gamma you have two three point functions right can you see which say which gamma gamma 3 or gamma n okay i was making an example in which gamma n is happens to be gamma 3 also okay okay and as you see in your diagram you have you have a gamma 3 times another gamma 3 right right okay now you constructed the four point function out of two gamma threes right right that's the recipe to go from gamma to the amplitude right pleased and now you end up with a diagram for for the four point function in which the graviton is emitted from an internal leg right so apparently this is a leading term in one over omega but it cannot be there that was my question but okay i don't want to take too much of your time we can discuss it on slack if you want okay okay i'm possibly not able to follow like the diagrams what you are telling okay one has to draw the diagrams and i cannot do it here okay sorry sorry go ahead okay but i mean if you look at the two paper that you mentioned amada and the other one right right they do exactly that starting from the amplitude from the amplitude exactly right oh so yeah so they started from the amplitude this is like in our language which i described earlier i was just in this language i just tried to uh like explore in this language but yeah i agree yeah yeah i mean usually i always saw it done as paulo says with amplitudes not with gamma but maybe there are some interesting advantages on using uh you know the one particle irreducible action although i see this little problem but probably can be we can ask ashok by the way ashok had an another meeting today so he can only be there starting tomorrow so to bed he's not there we can ask him tomorrow okay see you sir i'm not sure if you can do from the the gamma because nobody has done for the dilator the tilaton can be done in the other case in this and nobody has done the dilation with the gamma okay okay so like for a moment you can think this is like done for only three level amplitudes like this is like at least i want to mention like like so i just tried to boldly generalize for gamma but like for tree level amplitudes like these are the results are derived ah for the minimally couple scalar theory okay okay so maybe yeah okay i just tried to generalize the earlier idea okay uh okay now like so this is the status in d greater than four uh now like like can we do the similar thing try to derive soft theorem in d equal to four so that in d equal to four the main problem is that the s matrix is i infrared divergent and this is the main problem then how to like if it is divergent quantity then can we relate to divergent quantity or how to derive soft theorem or how whether those really make sense because two relating to infinities but the thing is that since optimum is a relation between two s matrices and if we can show that like both the s matrices contain same i i divergent factor like both gamma n plus 1 and gamma n then possibly just taking the ratio or like ah like ah cancelling the same divergent factor in the soft theorem relation from both side in terms of the ir finite part of the amplitude we still can make sense of soft theorem and this is what i am going to tell and like so like what we are going to do is basically we start with a n plus one point amplitude and we will try to like exponentiate the full ir divergent piece from this amplitude and try to write in terms of some higher finite parts of part of this amplitude which we will denote by gamma n plus one g and this g will be clear later and then also like try to another exponentiate the ir divergent piece from the gamma in amplitude and in terms of the ir finite piece and like this uh can be done at le this can be done which i am going to tell like but at least in the when the loop momentum are smaller than the finite energy particles moment in the ir limit but what happens is that for electrodynamics case like both gamma n plus one and gamma in contains same i a divergent piece so it is very easy to so but for the gravitational case when you are trying to derive soft graviton theorem it turns out that gamma n plus one contains extra divergent piece relative to gamma n and this is like very easy to understand like what happens is this gum when like when the virtual ah loop or the when the virtual graviton is connected to a finite energy particles and the soft gravitons that also contributes to extra ah iron divergent piece which is not present in gamma n so like the so we have to i regulate this extra k phase factor which i am calling k phase and the kgr is the known uh this iconal factor when in the ir limit and then we can cancel from both side of gamma n plus 1 and gamma and this e to the power kg r and try to relate what will be the uh how what is the relation between this n plus one point finite piece of the amplitude and n point finite piece of the amplitude and then we can try to find out what will be the soft factor s 0 s 1 s 2 all those terms so this is like uh like what i am going to tell so what do we like so basically what we have to do we have to first factor out the ir divergent piece from the ir finite piece and that thing like for quantum electrodynamics case actually is developed by grammar and e in 1973 and we basically like generalize this idea for the case of gravity and which i am going to explain actually you can ask a question sure uh can you just go back i just want to so when you say it's k phase do you mean that k is imaginary or is it just e to the minus something that's oh it okay like uh k phase i just name it uh phase because like in the gravitational memory i will show that this is related to some phase part okay actually i just named it but you can think this k is basically nothing but the same iconal factor the kgr is the iconal factor when the virtual graviton is running between two finite energy particles and k phase is the iconal factor when the graviton is running between a hard particle and the soft graviton but but this k phase contains a real part and an imaginary part right right right it contains both real and imaginary piece i see and you're you're dealing with that part as well the imagination yeah i am dealing with here in this matrix both real and imaginary piece and i will show that like what happens in the classical limit only the or in the classical soft factor only the imaginary piece remains and that's why i am calling this thing for a moment phase okay okay but this is just a nomenclature but ah yeah the meaning is possibly clear sorry can i ask you also one question i mean so basically what you're describing is that the the ir divergences of gravitational amplitudes to all loop orders are just given by the exponentiation of the one loop amplitude or the so there's actually some there's some ambiguity of what you're exponentiating you can choose to either exponentially adjust the ir divergent piece of the one loop amplitude but you can also choose to exponentiate some finite parts as well exactly what so what uh so what scheme are you using for to define your gamma g this this is exactly i'm going to explain next like always i can put some ir finite piece to the ir divergent piece and i can exponentiate so uh like what i am telling here is that if you take like the ir limit then at least this k phase and kgrp should contains the ir divergent piece but it can contain more so i am going to explain the prescription uh how to do this so the strategy is that what we will do is that we will split the full graviton propagator with momentum say l which is flowing from particle a to particle b and we will split this into two parts and one part will called k graviton and the other part uh we will call g graviton and this is you can see the ah like um uh the propagator for the graviton indeed under case and we are splitting this just uh k in terms of two terms one is k and another is g and how to choose this k and g so first one is that like the demand is that if we compute any loop diagram with the k graviton which is the k graviton propagator then that suit contains the full ir divergent piece but it can contain some more something more also but at least it should contain the full ir divergent piece and if we compute a one loop diagram with the g graviton so that means this is completely higher finite and this is one of the demand and with this also i like to like con how much this fact k contains extra piece to fix that i will demand that if we compute a one loop diagram with no external ah soft graviton or no external graviton is there only finite energy particle say for scalar theory only scalar particles are there are no gravitons for this one loop diagram with gigabit propagator will completely vanish so in that way i can put some ir finite piece also with this ir divergent piece in the definition of k this is how i will show you what is this k and g in the next page and another thing is that so that this quad identity kind of relation valid so that we make this virtual k graviton to be choose to be such that it will be like a pure gauge kind of thing so it will be like proportional to l momentum of this graviton such that if we insert this graviton to a scalar line we can write it in terms of two diagrams where this ah k graviton is connected to extreme left and extreme right with this we also can check that if we choose properly this k we can also check that even this like even if there is a external graviton is there if we just ah choose this k such that it goes to the extreme left of this k extreme right of this k and connected via this ah like two scalar two graviton vertex then this diagram vanishes so this will help to even like if we insert a k graviton it will help to propagate to the for the diagram to the extreme left and the extreme right so if we demand all these conditions we can construct what will be this k and g and this is the cons result of the construction i have written here so this k basically looks like you can see this is proportional to l mu l new kind of factor l royal sigma which will help to do the fast process when we insert a k graviton it will split into two part and the other thing is that we choose this c in such a way so that it contains the full ir divergent piece and also for demanding sorry also demanding the fact that if we compare to one loop diagram with g graviton it vanishes we can completely fix all the terms uh in that way and we can like we can show that at least at one loop order we can write the amplitude in terms of this in terms of a higher divergent factor which can contains also somewhere finite piece times a completely higher finite amplitude so this is the construction i have given but just let me give a disclaimer that this full exponentiation of this eco iconal factor analyzing all loop waters has not yet been proved with this construction which is which is there for qed case like this kg decomposition the original paper has shown that for uh like electromagnetism case like it is possible to sow the exponential analyzing all loop orders but for gravity since this multiple self gravitating interaction could be there so it is little harder to prove it so this is not yet done but since we have done analysis only at up to one loop order so up to one loop order we have checked that this helps to factor out the ir divergent piece from the ir finite piece and so and this will be enough to derive the subleading soft graviton theorem which i am going to describe and now it happens that at subleading order the soft factor is not omega to the power 0 like d greater than 4 but it goes like log omega in small omega expansion so what we have to do so now what we have done is that okay so in the loop graviton we split it into two parts k graviton and j gravity g graviton and the k graviton is the ir divergent piece so cancels from both side so in the first kind of diagrams this is the one loop diagrams and k is the k is the like this is the basically the external graviton with momenta k and now this kind of diagrams only we have to evaluate with the extern with the loop with the virtual gravitation which is the g graviton and then the rest kind of diagrams where this splitting of this kg has not been done because that contains some extra divergent piece which is not there in gamma n so that part we have to evaluate with the full g graviton propagator and if we do this thing and try to and put a ir cutter for this kind of diagram since this contains extra ir divergent factor and try to analyze we can see that now these diagrams can contribute to log omega when the loop momentum is greater than the momentum of the external gravitons in that range if we approximate we can find that it goes like log omega or it contributes to log omega and this kind of diagrams contributes to that region also but with the region when the loop momentum is smaller than omega and greater than the ir cut off there also it contributes to log omega and that basically gives this phase contribution which i am i was telling and so let me show the results and i can explain from where it come from so it happened so at subleading order we find that it goes like log omega and the coefficient of log omega i have written here explicitly so here you can see that those are determinants in terms of the momentum of the incoming and outgoing particles and the moment and polarization of this sub graviton and one thing i want to mention the first two line so i'm using the convention eta is plus one if the particle is outgoing minus one is incoming so in the so you can see this log omega coefficient has a double sum over the particles not like the tree level diagrams are only the single sum so there is a pair of sum of particles and the first two kind of terms is you can see there is a constraint over the sum over a b particles where eta eta b equal to one which is basically these are non vanishing only both the particles are outgoing or both the particles are incoming on the other hand the last two kind of terms there there is no constant or incoming or outgoing so if so this basically even if a particle is incoming and b particle is outgoing still these terms are non vanishing and this is the full logarithmic contribution and now i'm going to tell like like i can write down this result in a different way and this gives a very good observational point usually there is a question yeah where's the i'm sorry i'm not saying where it's log omega i don't see a lot where's logo on the first i just have written the coefficient of log omega term so in each term there is a log omega so this is the coefficient of the log omega oh so this is all multiplied by log omega yeah yeah and and um it and is this all um is this all vanished if there are no no massive particles no stable message for massless particles these are non-venison if all the particles are massless still this log omega is present sorry if they're all massless it's still there yeah if all the hard particles are massless still this is not vanishing for electromagnetism it vanishes but for gravity like even if they are massless still it is there i say and and i the way you described it it sounded like it had to do with infrared divergences but i don't see an infrared cut off in here is it locked so let me explain like what happens is that so this kind of term the second line and the fourth line yeah these are basically coming from these two kind of diagrams when the loop momentum is smaller than omega and greater than the ir cut off so this the like the log appears in the second and fourth line these are basically not log omega by log of omega r if you call this r to be the length of the higher cutoff okay so so if you use some kind of coherence state formalism which would be relevant to a classical scattering problem in which the infrared divergences are all absent would this effect still be there yeah yeah still be there yeah like i'm going to tell like if we just write you could have you could have derived it without ever mentioning infrared divergences yeah like what i am telling is so this r is like you can see like suppose some gravitational radiation process is happening so what can happen is that if you put a detector at distance r you really cannot measure a wavelength which is greater than r because it will pass through the detector so you have a natural cutoff for your process and this basically gives you the higher cut so if you wait a long time you can still measure i mean that's like the memory effect right yeah but if the omega or the lambda which is basically the way wavelength of the gravitational wave form if this is greater than r r is you are not suppose sitting at infinity at this infinite distance but it finite distance r right then if you are sitting at a finite distance obviously you can like you you cannot if you sit at infinity then in the gravitational waveform that contributes to one over r so it will like it like r zero what prahar was telling if you take strictly r zero going to infinity it will vanish but the finite r0 if you see it then you already have a natural for lambda that lambda cannot be greater than or not otherwise it like you cannot observe that waveform and that is the ir cutoff which physically tells you to put to regulate the ir divergences thank you thank you so let me proceed so what i am going to tell is that this result you can write in a form from which that you can see that possibly even without doing all this analysis you can expect this result could be derived so this is this slide so this sort factor which i have written there this log omega term now i am including the log omega factor also this is i am calling the subleading soft factor this can be written as a whatever i was showing this k phase term which was ah the iconal factor when the graviton is propagating from a finite energy particles to soft graviton if you multiply this phase if if you evaluate this phase factor in a regulated range of integration which is from higher cut up to the energy of the soft graviton and you multiply to the leading soft factor and similarly this is the subtree level subleading soft factor operating on a iconal factor which is when the graviton is like propagating from two finite energy particles but now regulated in a range when the loop momentum is greater than omega n less than the energy of the final energy particles then also you can see the sub leading soft factor can be just written in this way in terms of the three levels of factors so what this is telling like so just what i want to mention is this kg are regulated is basically you can see this is the iconal factor when the loop momentum is greater than omega and this k phase is the semiconductor factor but the blue point is smaller than omega so what this is suggesting if you just naively assume that okay tree levels of theorem is valid even at loop level or the d greater than force of theorem is valid even in d equal to four and write down naively this amplitude now you see we have told that this gamma n plus one basically written as exponential of some k phase and exponential of kgr and this gamma n is exponential of kgr right and now suppose you regulate this k phase k g r in the integration range which i mentioned and just substitute in this expression and then commute through this tree level uh soft factors then you can see if you like this different cell operator operating on this kg are regulated will con will receive a contribution and this kgr regulated goes like log omega and this k phase goes like omega log omega and you can see that since the gamma n plus 1 has this extra factor you just if you just expand it it will multiply to the one over omega piece and also can contribute to log omega and this is exactly the factor which i have written in the earlier slide that means like if we just consider the knife travel soft factors are valid for the full amplitude but with the fact that the integrals are now regulated in the specified region i mentioned and then just you commute through this amplitude commute through these three levels of factors you still can recover the result which i mentioned here and explicitly the expressions are written earlier so this is just one observation which i wanted to mention and what this observation is telling is that though we have derived this result at one loop order but since this exponential if the exponentialization is true which is not yet proved rigorously then this is like valid to all loop order that is this log omega correction will not receive any contribution from higher loop order because the higher loop water contribution is just the one loop exponentiation of the kgr so this is the point which i wanted to make ah for the d equal to force of theorem now like if we suppose now we observe this thing and try to okay like if we don't evaluate the sub leading sub sub leading order try to generalize this observation and we can conjecture that what will be the sub sub leading sub factor will look like and there if we just generalize this idea and assume the sub subleading soft graviton factor which we know ah from higher dimensions or tree level in ah tree level in d equal to four and just generalize this idea of exponentializing this k phase and k g r we can conjecture what will be this sub sub leading soft factor will look like and it is you can see this is the phase square times the leading soft factor because phase term will go like omega log omega so this is omega log omega square times one over omega so it will like omega log omega square at the leading non analytic order and obviously there is a omega log omega correction which we cannot determine because of the fact that this log omega has a scale like log omega is not a dimensionless parameter so like there is a scale of this omega and this scale like if you choose like two times r or four times r that will give a con that will correct this omega log omega contribution so only we can fix omega log omega square contribution which is the leading non-analytic piece which you can derive analyzing two loop amplitudes and like which we have now derived classic like classical anal from the classical analysis which we have proved the classical part of it which i am going to describe uh in a minute this is the conjecture and which you can prove analyzing two loop amplitudes which is not yet proved so i think andy has i'm not sure his hand is raised okay so let me proceed then so possibly then we can try to see that okay like can we uh do this even higher order says up to the power n leading order and we can try to write it down but since they are still this r factor which i shown earlier even at tree level it's undetermined so like without knowing this r we cannot fully specify what will be the form but from the rest of the terms we can conjecture how this structure look like i am not going to tell too much about it only what i can tell is the leading non-analytic piece from the end loop will turn out to be omega to the power n minus 1 times log omega to the power n factor so this is just uh like extending our observation for the end loop case so here i am giving a table like from which order how we will get contributions so from tree level we have seen that the leading soft factor goes like 1 over omega the sub leading goes like omega to the power 0 sub subleading order omega so you will have a taylor series expansion in omega from one loop the leading non-negative piece turns out to be log omega then you will have obviously omega to the power 0 and then omega log omega omega omega square log omega all these or higher order terms and just i want to mention this omega log omega term turns out to be spin dependent piece because you know like how spin couple to find energy particles this is coupled via the spin connection term and the spin connection term basically turns out to be derivative over the bl bind ah which is basically kind of matrix component so that contains the extra factor of omega soft moment that's why relative to the minimal coupling part or the momentum dependent part the spin dependent term comes with a factor of omega and also then you can analyze two loop and it turns out to be leading like leading non-analytic piece turns out to be omega log omega square and then like there are other kind of terms which uh like cannot be fixed because of the ah like scale of scale like which fixes the omega log like log of omega what so you can also generalize to n loop and you can predict the structure which i mentioned earlier and the leading nonitic piece we expect to be completely fixed and factorizable and like it is uh it is expected to be universal just like from the understanding which i mentioned earlier so this is like too much optimistic viewpoint like how much we can expect just observing ah what we have done so this ends my uh like soft theorem derivation from s matrix so be sure you have six minutes but okay i mean i think there's also half in a discussion so you can take some of obviously that just wanted to mention that yeah yeah yes yes sir thank you so now uh like i'll discuss about the classical limit of soft theorem so for that i think this is already discussed uh in the last week bios oxen so let me just briefly discuss so like say consider like start from d greater than four and what is classical limit means is that like only single soft graviton emitting that you cannot take to be considered as a gravitational wave in the classical limit only at a particular frequency range a large number of graviton like large number of coherent state of gravitons emitted and then you can consider this to be a gravity wave so like for that you have to start with the multiple subgraviton theorem and that also we have the result and you take that okay so for a multiple subgrading theorem we take a limit that when large number of graviton should emit and that if you demand it turns out that the scattered object have to have mass which is large in the unit of planck mass and also you want that the radiation frequency to be small or the gravitational wave frequency to be small so the total radiation energy you can assume that this is small ah like compared to the total you know like total uh total radius energy has to be less than the energy of the finite energy particles and this if you uh demand then like you have to scatter in a large impact parameter regime or a proof scatter limit when one of the object is very heavy and ah like the pro particle is scattering in the background of that heavy object but like like this can be relaxed and later it it can be relaxed ah and one can still derive the soft theorem so what this tells us if we just assume these two conditions in the classical limit then it will tell us that okay you have to replace the orbital angular momentum operator by its classical angular moment and the spin operator to be by the classical spin of the object in the soft factors and also you have to drop the contact terms which i have not shown in the multiple sub factor and then like what happens is that if you do this and then independently compute from true like if independently if you know h mu nu which is the matrix fluctuation you can compute what will be the radius and energy on the other hand from the soft theorem you know how the soft factorization happens from there you also compute the radius and energy of the graviton and compare these two things and what you will get is basically the gravitational wave form this alpha beta is basically the trace reverse matrix fluctuation and its fourier transform in time variable and if you contact with the polarization it is turns out to be completely proportional to the soft factor in the classical limit that means you just replace uh the or like the operators orbital momentum operators by the classical momentums and the spins by the classical spins and like which tells us this waveform is like proportional to the soft factors and this is since the it is derived from the energy so there will be some phase ambiguity up to some phase this relation is correct and also this normalization is fixed so now so here like i just want to tell for d equal to 4 if you try to specify this thing this relation will look like this that this is the fourier transform in the time variable of the ah trace reverse matrix fluctuation contracted with the polarization turns out to be this normalization with some rd turbine phase psi times the soft factor where the soft factor is written here which is the lead you can up to sub reading order i have just written it down this is the leading soft factor and this is the sub leading software where this j rho sigma is not now the derivative of the momentum but you have to replace the classical which is the r cross p term for the orbital piece and rest of the thing which i want to mention is that like ah so this e mu nu like where the x vector where we are detecting the gravitational wave form that is a distance that basically expected which is set up at distance r along the direction cosine denoted by n hat and the momentum k mu is basically omega which is the energy of the subgraviton and the which is this and the n mu and n mu has the like zeroth components one and the unit vector along the direction of the detector from the scattering center so if we do this thing and try to see that how it will look like for like d equal to 4 it turns out that in d equal to 4 the sub leading soft factor diverges because of the fact that due to long range gravitational interaction the trajectories are not like they are moving at straight line trajectory at infinity but it receives a log sigma correction where sigma is the proper time and that is so that means like it is not even at like large time it is not moving at constant velocity but it is accelerating and this acceleration term one can determine just by solving geodesic equation but more than that what happens is that the classical angular momentum which is like r in terms of the trajectories of the particle we just substitute here it turns out to be goes like log sigma which if you take strictly sigma going to infinity it diverges but the from the physical ground we know that if we want to determine the gravitational waveform with frequency omega then after the time of the trajectory which is sigma uh in order of omega inverse we cannot uh like this cannot contribute to gravitational waveform with frequency omega so there is a natural cutoff for the sigma and this is like the omega inverse and then if we just substitute this thing then we can uh like find out what will be the soft factor but before that just i want to mention one observation here the we can just solve the trajectory of the equation and find out the acceleration term which is the coefficient of the log sigma in the trajectory and what we see is that if we just replace log sigma with log omega inverse then this i times c times log omega inverse turns out to be the momentum derivative of the same iconal factor but when we evaluate the iconal factor not with the findman propagator but with the propagator is there a question i can see okay so what i want to mention is that the acceleration term can be identified with the derivative with respect to the same iconal factor but when the iconic factor is evaluated with the propagator not with the fineman propagator so what this tells us that like if we like what this tells us that if we just now replace this c log omega inverse with respect to del del pmu of the same iconal factor regulatory iconal factor but with propagator we can get the gravitational waveform up to subleading order and if also one have to take care of the back scattering effect uh like due to the long range interaction of the gravitational wrong interaction on the gravitational wave by the matter particles and if you take care of all these things we can show that this subleading order gravitational waveform which goes like log omega in the frequency space that has this kind of behavior this is you can identify the soft factors which i have written down only the difference is that this k has now a superscript cl which is the classical not the regulated piece which means that we have to evaluate the same regulated phase factor or regulated kgr but with a moment but with the final propagator replaced by the radar rate propagator so in that way like we can predict what will be the long range gravitational wave form just starting from the classical limit of the soft gravitation theorem in d equal to four so yeah if there is any question just i can try to answer okay so one can then like again generalize this observation to predict the spin dependent part of the gravitational wave form starting from the sub subleading uh soft factor and like here just i have written it down just like use the sub sub leading soft factor which we have uh mentioned in d equal to four or we conjectured in d equal to four and just replaced all the regulator level by the classical level and i just mentioned the meaning of this classical and then expand in power of omega and you can also extract the spin dependent gravitational uh like wave form and it turns out to be at order omega log omega like the which i also mentioned in the table uh and this one can predict but this is not fully ah this is not the full contribution because like it will receive contribution from order g cube which are not like which is like telling that to determine the spin dependent gravitational wave form like you like it is not just universal or completely determined at g square order but it will receive correction from higher orders but like so you have to do both a post mean expansion in power of g as well as in power of omega to get the like log omega to get the spin dependent omega log omega piece but the other cases which i mentioned the log omega or the omega omega whole square which i perceive the leading on analytic piece those are completely determined at that order of g you don't have to go to like you don't like these are exact at that order of g don't receive corrections from higher g g is the gravitational constant so is there any question up to now so we showed you justice to clarify the omega log omega spin dependent piece is universal no no it's not universal in a sense you can see like only we can determine the ordered g square piece but not order g cube piece and order g cube piece uh also has contributions because if there is omega log omega whole square then this log omega scale can contribute to omega log omega so this is not completely fixed and also you can see like this depends on this ra rb which are basically in the orbital angular momentum these are basically the impact parameters so not only you have to give the data like of the incoming and outgoing momentum but also the impact parameters right right right you said two loop there is omega log omega that's the non-universal contribution yeah yeah but in one loop there's a universal contribution from this pin yeah at one loop there is your universal contribution okay yeah now so there is no question okay thank you yes okay so now we can try to give a systematic like up to now we just have taken classical limit of soft theorem and try to tell how the waveform look like but we can independently without going through s matrix just from the classical uh like scattering problem we can also derive the low frequency gravitational wave form and we can check that whether our predictions from the soft limit of the s matrices are still true or not and like for that we consider kind of a scattering process where suppose m number of particles with momentum p one prime to p m prime are coming in and goes through some complicated interactions and then going n number of particles are going out and suppose this complicated interaction is happening in some finite region of space of size l and then like and then what we want to determine is basically what will be the gravitational wave from with frequency omega which is much smaller than l inverse in a sense the wavelength is greater than the length scale l or in like if we do a fourier transform we want to determine the late time or or early time gravitational wave form with time u which is much much greater than this length scale l so these two things are related by uh fourier transform so this is our goal and the reason r we will choose such that all the complicated strong forces or interactions are happening inside region are and outside only long range gravitational force is present and this can be like uh like choose like for a special case like for say consider for a black hole merger problem we can choose that after the time when the kinetic energy uh when the kinetic energy is greater than the potential energy that sets up the scale that is why then we can consider the interaction strength to be small and that can set up the length scale l so then what we have to do is basically uh we have to basically solve the einstein's equation and and the geodesic equation to alternatively uh iteratively to find out each order in g how the gravitational wave form will behave in small frequency limit so for that here just i defined the metric fluctuation h mu nu to be the deviation from on the item u nu and the trace reverse metric fluctuation here and in redundance this einstein's linear resistance equation takes the following form and in the right hand side this the mu nu has both matter part and also the gravitational energy momentum piece which is basically uh just from the einstein tensor if you just subtract the linearized piece and then if you try to determine the time like time fourier transform of this waveform in small omega limit it turns out that this can be related to the full fourier transform of the energy momentum tensor times 2g upon r so the goal will be to determine then the fourier transform of the energy momentum tensor ah in in small omega expansion and that will determine the itilda mu nu so i am due to time i am just keeping the details and just let me tell the strategy why this iterative expansion works because in ah c equal to one unit it turns out that g m omega m is some mass parameter which is the momentum of the particles of the mass of the particles which are objects which are scattering and g sigma omega square g m r omega square all these are the dimensionless parameter so it tells us that like expanding in power of g is a good parameter because that can take care of the corresponding like m omega piece or the spin dependent piece and all these orders but like we have to ah start with the energy momentum tensor and we know that the energy momentum tensor has a like full derivative expansions up to subleading order which is the spin dependent piece it is like completely uh like fixed but after that order like the structure of the objects will come in in terms of like the multiple moments and the tidal effects and other terms so but those will be import those will not be important to our analysis because if we are interested to the few uh like non and few leading non-analytic piece since those involves more derivative of our delta function in fourier transform that contributes more power of momentum so it will be subdominant so up to the order which we are interested we can just throw them out and like we can just uh take these two terms and we can try to analyze and similar thing happens for uh like gravitational energy momentum tensor also you can add higher curvature terms up to the for four r square all these terms but since they all again involves more derivative over matrix so that contains gives more power of omega so it will contribute more subleading order in omega expansion and so the strategy will be basically we have to solve the einstein's equation to get corrected metric and zero c equation to get corrected trajectory and we have to do this iterative order in g and finally we have to compute the fourier transform of the energy momentum tensor in omega into zero limit and try to extract the non-analytic pieces and i am skipping the details and try to tell you like if we what if we try to derive it exactly matches with the expectation which we just mentioned uh like from the classical limit of the soft theorem and here i am giving a uh like list so this omega to the power minus one from like zero loop and this is iterative order g it appears that contributes to memory at heavy side theta function of u which discussed that you have to take the change of the matrix fluctuation from you going to minus infinity going to plus infinity and that difference it will give the leading memory which is known and but like tree level the sub from starting from the sub leading order which are in power of omega only those contributes to delta u delta prime u and all these things so they are like only valid near u equal to 0 those cannot contribute to the difference between the the gravitational change of the gravitational wave from u equal to minus infinity n equal to plus infinity so those cannot contribute to any displacement kind of memory but like like there are some proposals of other kind of memory not displacement kind they are these terms could be important then like at one loop order ah like it appears to be log omega and that gives a one over u tail memory and similarly i am just focusing on the leading non-analytic pieces then at two loop order it goes like omega log omega square and that contributes to a memory u to the power minus two log u and an n loop order which is go like g to the power n plus one in classical iterative expansion and which contributes to u to the power minus n log u to the power n minus 1 memory so you can see that there is like tail of tail of tail memory so like there will be lot of tail memory and all of this will be like important if you are like not at strictly going to infinity but some finite u and you have to choose this u such that the like all the strong interactions already happens and only long range interaction is there after that time you consider this u and you can extract all these coefficients systematically and so these are the expressions explicit expressions of these terms which i am skipping in the discussion if someone asked then i can only tell and just i want to mention that what prague is telling that whether this memory can be observable or not so for some kind of observations like for core cool of supernova hyper velocity stars neutron stars neutron star merger all these processes taking data uh we try to evaluate that what will be the change of the detector position with respect to the original position which is known as the gravitational shear that turns out to be in the order of 10 to the power minus 22 which is like in the edge of the current uh like ligo resolution ligo detector resolution so in principle like those could be observed uh so we are hoping to observe and also just want to mention this all these tail effects which i am mentioning like those are also in the same order like 10 to the power minus 2 if you are not taking u to be too large but what happens for a black hole merger process where when only the final state is one massive black hole and all are gravitational massless particles all this u to the power minus 1 e to the power minus 2 log u times vanishing like because like you have only one massive object and all other massless particles that can be observed just from the expression if you if we go to take this limit and this is one of the prediction this is vanishing of this uh gravitational tale memory will be a test of the general relativity so let me thank you and finish here i am over time sorry for that thank thanks we showed you for a very nice lecture thanks all thanks we still have 15 minutes for discussion well i have a uh sorry i have a question right and concerning the vanishing of the u to the power minus one tail in the case of uh binary black hole merger this is due to consider it takes into account the non-linear uh memory what is the proof that the coefficient of u minus one is zero oh like okay so let me so so this is like the coefficient so and i want to mention that all these tail memories are all one over r order like in rx function which the distance from the center to detector is one over r and so and this one over u term you can see like what happens is that there is two kind of contribution one is this fast line which i was telling the kind of phase term multiplied to the leading memory so this is also people have earlier explored this thing this is like this tail term is already known earlier but with this due to the long range interaction between the finite energy the objects which you are scattering this is the extra tail part is there and what happens if we take the like in the finite uh like in the finite energy particle sum if we consider one of the particle is a massive and all the rest are massless which is true for black hole scat black hole merger problem then there is that answer to be cancellation between this first term and the last two terms and it vanishes so this is just observation what is the physical understanding or why this is vanishing that is still not clear to us but are you if the scattering there are two massive bodies coming in if i send two massive black holes and they capture uh to merge right then i have in the exit i have only one massive body and mass less are you still saying the coefficient of one over u vanishes in the tail yes yes yes even okay you are saying this yeah yeah thank you andy and uh um i suppose this is a question for uh for for tebow you you made some comment um during the last talk that these tales had appeared previously from some other kind of analysis you had some other could you could you clarify for us what i'm not sure i understood you correctly but how is this related to um yeah okay so you mean the one over you tell here yeah yeah yeah so for instance um yes actually we're in karja snow at the time when uh ashok uh first in 2018 know presented this in our 1992 paper with luke blanche where we had analyzed in appearance sense uh detail uh including the we okay both actually the so-called memory and the memory detail and then we had noticed but without doing the exact thing in a pm sense that the the large u limit of the the tail integral in the radiative in the wave form h i j t t would be one over u with d k like one over you so so i guess in the recent work there's some formula for the coefficient yeah yeah exactly and post me in post me costume yeah we had the multiple expansion and tn expanded yes i see thank you and there are a couple of other hands up uh first so i just want to confirm the scales that i'm looking at here so would you say that these formulas are valid when r is much greater than u yeah r is uh much yeah r is like u is in between the length scale l and r l i introduced is basically the scattering region where the strong interaction is happening right so these tail effects are really tiny because uh they're of second order like roughly order one over r squared right because no no no no so like this you can also you can choose this u to be much closer to l like what i am telling how to choose this u so you you have to choose when the interaction strength is becoming weak you don't have to take to be infinitely weak like so and this that means like when the suppose to between two part consider one particle which is moving it has some kinetic energy and it is feeling a long range force potential energy due to other particles so when the kinetic energy of the particles is greater than the like the potential energy due to the long range force just at that scale from that time scale you can trust our formula and that sets the value of the u uh and from and if you try to determine with that u then you can see that this tail term is same order to the leading gravitational memory whatever u you take r definitely has to be larger than that yeah r has to be definitely larger than so r is both the term has one over r term you see that this first line is the gravitational memory this is gravitational dc memory this is the tail memory so both are one over r now this one by u you you have a set of like you can see with respect to this this term has a factor of g in g and a power of momentum in the numerator which basically g times m upon u is the dimensionless quantity so this term is the dimensionless quantity and it turns out that this u basically you can trash this formula when u is in order of g times m so this term is order one that means leading and the tail term are in the same order so both are like in the same order and like these are not tiny with respect to the gravitational memory but if you go to strictly you're going to in the order of r when u is in order of r up to that large time then obviously this is subdominant relative to the leading memory and also just one question about the g newton that's appearing here that's the normalized g newton what's the renormalization oh this is okay so completely a classical result so this is just classical g newton the observation g newton is oh yeah of course of course this is the classical limit of the so in the soft theorem sorry so then maybe as like in the soft terms when you wrote down all of those formulas those were all the re-normalized g newton at some scale what is the how should i make sense of those uh okay uh like i i i think i should consider this as like in effective field theory language uh like where this whatever like i started with the like say scalar coupled to gravity so i can think these are valid in the energy range like okay possibly yes it it will be it will be right which would it will be normalized yeah yeah in the effective increase yeah so this is the g at the uv scale so now now you you need to have a uv scale and you need to have an ir scale so the g that's appearing in your soft theorems which scale are you evaluating that g newton at okay uh i think it's at zero zero yeah zero yeah at the end really the low energy effective action of this year yeah yes yeah yeah so strictly g newton in that limit g newton should go to zero right because in the ir given the dimension effectively dimensional parameter with positive mass dimension and i well it is but that's why i don't think it it has the usual running of a gauge coupling you know you you just can fix it it's your momentum through an experiment okay through mutual's experiment yeah okay yeah yeah just uh just to make sure that i understand we wanted to go back to this question of uh the memory effects bunching when you have a just a single final state right like if instead you consider really a two to two scattering in that case you do you do get an entrepreneur yes yes yes right like for have for what happens for neutron star merger case like there are like finite like in the final state not a single particle but multiple and for that case we determined the order is like gravitational memory is 10 to the power minus 22 yeah i actually had in mind even a simpler kind of more duncan experiment right instead of having kind of matter ejected away just think about a unbound orbit right really a scattering right and then suppose that it's super energetic and so it creates enough gravitational wave that we can detect right right and then in the final case you have the two states yeah it is not anything yeah it is non-vanishing okay yeah thanks yeah it is a non-balancing so basically maybe i can just ask something this is actually regarding what andy uh also asked him so the initial uh is it correct to say that the new results you guys have the the main the one important thing is that the initial and final moment are assumed to be uh you know given independent and then you derive the universal terms in terms of initial and final momentum and this is why the source multiple moments never really appear in the computation because you are not determining the final momentum in terms of the exactly exactly so the so so again that is the new i mean the universal result is really that right assuming the asymptote yeah yeah yeah so what happens is that like say even for the leading or this one over u piece you can determine this final moment of pa in terms of the initial momentum p pa prime by evaluating the trajectory using geodesic equation but the corrections will be like if you start this order it will come at order g square so ordered theta will vanish in that case and it will start only contributing in order g square and all order g contribution will be there so only like this is exactly the order g in a sense when both the initial and the final momentos are given yes yeah thanks thank you and and just one one maybe if there's no other question i can ask one more i think is still up so i don't know if he has uh oh no sorry okay so this this is related to actually gabrielle's question so i i was a bit confused that the i mean the 1pi effective action that you used is the amputated with the amputated uh one pi effective action right right right so why is it not the same as effectively working with pss matrix i mean that amputated green's function yeah is it not same as this matrix i okay yeah this is my confusion also so amputated is fine but yeah if one particle irreducible i actually you know suppose i i was thinking also about another example suppose i have a three to three scattering okay in three to three scattering you can have an intermediate particle which is perfectly on shell you exchange a particle you know you have you have two two to two scatterings exchanging a particle which is perfectly on shell now in that case do you have to put uh to attach your graviton also to that guy which is exchange and on shell why the exchange particle is on cell like suppose three particles uh scattering is happening p one p two p three momentum the extent particle have momentum p one plus p two plus p three so uh and then propagated is like one over p one plus p two plus p three whole square plus m square and say all the particles are identical so this is not a one cell right it's propagator no okay you have say particle one and two which give three and four this particle four which is the spell yeah by particle five and gives six and seven yeah it's a well-known process which occurs i mean it cannot occur but that in more in general i i okay my more general question is the following the soft theorems do they apply genetically or always generically means for non-exceptional momentum so supposing process and the at t equals zero and you exchange a particle which is uh on shell do the soft theorem apply equally well to this exceptional kinematic goal but i i feel that like since you have to conserve the momentum like all the sum over the incoming and outgoing particles movement plus soft graviton's momentum and if you demand this thing then can you really the condition you are telling that the like a change particle is or the intermediate particle is one cell you can choose such a momentum configuration just consistent with the total momentum conserving delta function uh this is not clear to me possible or not yeah what i'm saying for instance in that three to three process you can since you fix the external momentum you can fix and in some way that three particle exchange with the other three particles in one shell particle that that is perfectly allowed kinematically okay so in that case is the recipe of attaching external gravitons only to the external legs justified or do you have to add extra diagrams this is a little bit related to this question about one particle irreducible and i thought at the beginning when you started with gamma you wanted to avoid those situations by talking about one particle irreducible vertices then you never encountered that situation if you talk about the full amplitude that full amplitude can have intermediate states on shell it doesn't happen in two to two but it happens in three two three four to four four to five i'm sure it's once okay okay okay like yeah i have not thought so i have to think and maybe like in the next discussion when austin will be here yeah that's right the point and it's just a curiosity i'm sure there is a simple answer okay okay yeah but i think burn at all's proof vishwaji from gauging variance applies generically right so the at least that from that proof we know the leadings of factor probably any any configuration of momentarily yeah yeah from the gauge in variance argument yeah but i mean you can view the the sub leading terms as being emissions from you know part of it to be emission from internal legs right yes that is certainly the stop leading right the leading ones it should be like uh yeah it is only yes yes only only yeah should be should be correct so i i was wondering if you start with effective with effective action one particle irreducible you seem to get some extra diagrams when you when you put back together the gammas to make the full amp but they must cancel out perhaps the perhaps the reason is that you know in one blob the particle is outgoing and the other blob the particle is incoming and maybe there is a sign and and that cancels i don't know i have to check okay okay okay maybe that's it okay i'll think about it and maybe in the next discussion tomorrow in this in this limit that you're talking about don't you have to isn't there like again an order of limits question like whether like how far on shell like how how far away from unshelness you are with this intermediate exchange state versus like how soft your graviton is like there are some virtuality cues there's some virtuality q squared how far you're away from the pole of this intermediate exchange state yeah and then like depending on what omega is like it like kicks you on like completely unshelled or not right so there's like we are absolutely right in the case of two to two i mean i i was looking at your three to no i was looking at your three two three scale right because like yeah if especially if as as this says you fix the kinematics the initial and final momenta you can make them such that you are exactly on shell yeah yeah i agree that's and that's it in that case i don't know in that case but then if you are slightly off shell then maybe you're right then yeah yeah no but i think yeah i i don't know if this yeah i don't know if they if these soft terms are supposed to hold for extremely special kinematics exactly i mean you could also i mean you i think you can even ask the same thing like in in more and more like if you make certain licks more and more collinear like linear to one yeah yeah you know some yeah some theories may apply to generic momentum where you exclude those those situations but in two to two scatting at high energy for instance in the work with the martian trafaloni we found occasionally that depending on on the omega emission from the in internal legs are also important well we know they are important at finite omega and as you say if you are at small omega then it's clearly a competition between how offshore the guy you exchange is and how big is omega there is some kind of competition between the two and as you know very well for the for the uh for the total energy emitted gravitons it's important right to keep the emission from the internal legs you know the age diagram that you also computed gives a contribution to the total hadith damage i think if you go too special like from a diagrammatic perspective if you go to a special kinematic regions you have you have more regions of in your you have more like loop variable regions to consider in general yeah right like this i guess you're right there's no there's another if you have another scale like there is like this like particular scale that you're now introducing so now you have to analyze your loop your loop integrals more carefully and like more scales can show up not just like soft not just the soft scale that you're usually considering for like fixed angle scattering of the hard particles yeah if you are not at fixed angle yeah look in the in the gravitational scattering problem the the momentum transfer is is very small right your t is very small your q square is very small is actually h over b right h bar over b so it's it's really you know quantum so in that case how important is the emission of soft graviton from the coulombic graviton that you exchange from what i remember i think the soft urine derived from gauge invariant are valid for any moment yeah yeah provide a provider that you have enough external particle i mean on the four point or on the three point it's not easy i mean it's complicated to the right but if you go to five point and there is no problem and on the problem of trip in three if the original if you are on top of the pole you're on the top of the pole in both sides before and so doesn't matter i think there is a cancellation i mean yeah you don't have to include an emission from the internal on shell particle no i don't think so yeah that's i believe also to be true but okay it's not obvious when you discuss things in terms of gamma because when you're discussing types of gamma in each gamma you do have an emission from the particle we find you finally used to join the two blobs you see what i mean i don't know if the question that one works with gamma yeah it's a problem to get for instance if you work with the amplitude i mean to derive a soft tier and for graviton for dylatone you do the same calculation but probably from the effective action uh is not trivial actually in the in the sub sub leading case in uh we did show that the the contribution from the electron i mean when you look at the soft theorems in string amplitude at some sub leading order i think that we could derive it even from the one pi effective action in higher than four dimensions so it matched with the result that were obtained yeah yeah in fact in the diletton case there seems to be completely universality at the sub-sub-leading level all the string theory have the same behavior which is not true for graviton i mean yes yes so we we could get that from the one pier effective action also right yes so it must work somehow yes okay any other questions one one very last one um can you go back in a moment on this question that was asked i think by the chairman about the phase i mean whether all these logs come with the face come with an eye come with without an eye okay okay yeah so i can so maybe the expression for the face drum because you wrote phase and then you commented that finally only the the real contributes but i don't see it yeah so here here i have written down the waveform this is basically the phase term which i was maintaining okay so this is like order g and this is pb.k so omega log omega piece there is also a log r piece which basically gives the time delay effect so that in omega going to zero limit we can forget for a moment but that basically tells us that what is the cut off and so this is so this is the classical phase piece and what i am going to show that if we just evaluate this k phase piece with greens function then only we get this term and this actually matches with the classical explicit computation but this phase piece con if we compute with a fineman propagator then this contains also a real piece which i am going to show so with this imaginary there is a real piece also but that does not contribute to gravitational waveform that appears in the soft factor so this is okay uh so let me show so this is the expression for the phase term and the first part is which i just shown like contributes to the classical waveform as a phase but there is another term which comes from the like uh like in the if you use feynman propagator for the graviton then this graviton has a like one over l square plus i epsilon and in the contour deformation in the l zero if we want to deform l zero that piece actually gives extra contribution and you can see this is this i and this i multiplies and this is basically the l part i see so both both are present both are present in the s matrix computation but at the classical limit the second part is not coming so which one i'm supposed to believe is the quantum piece suppressed yeah quantum piece is suppressed like if you like what happens is that if you consider the like energy of the graph like the energy of the gravitational radiation is small compared to the finite energy particles which you can achieve in large impact parameter regime then you can see that this piece like if you write is suppress relative to this part oh it is i see it's not obvious but it is so it so like if we just write down the full soft factor uh so i don't know uh so in the this slide so here i have written down so this part actually the first two lines appear in the classical waveform but not the last two line but in s matrix computation in the soft com computation both all the terms are there but if you take the momentum transfer is small or the energy of the gravitational radiation is low then you can see that first two terms are dominant relative to the last two terms oh okay so this is goes with a long distance or yes small momentum transfer limit yeah in the low momentum transfer limit yeah but but we should be sure if they manage only leading order right i mean there it will be contribution yeah it's not exactly zero like like if like like ah so the point is that if you just substitute the final momentum of the particles is initial momentum plus some correction and the correction you know in order of g if you can show that at leading order when the correction is not present these two lines are vanishing and but like in like corrections you can find out the correction but the correction is more suppressed because when the momentum transfer which is t you are calling is small this term is like suppressed in power of t but in principle it's present i mean it's present yeah in principle in the soft like in the soft factor quantum soft factor expression it is present yeah but so it boils down to the fact that the momentum transfer is it has an extra h-bar yeah so like so somehow like when you're dealing with everything else of the impact parameter then it's h-bar over b so yeah maybe this explains why it is there only at the corner but uh uh i i mean if you take the final moment of initial momentum but plus and equal so impulse is also classical then then these terms uh uh these terms are present when impulse is taken into account so i i think right which is that correct yeah but uh what i'm telling like it possibly like if you take the impulse to be small then at least yeah but the impulse is classical quantity so including impulse then this term is not zero it's not going at h bar i mean it's not a h by vanishing yeah but actually i look like yeah i don't know like if you have a like particular expressions for the impulse and if you substitute here and try to check all the orders like whether it is really in higher order in small impulse or zero non-zero yeah i don't know i can uh like that possibly needed the particular form the impulse to substitute here yeah right that's true in the two-two scattering you can show that if you go to leading order in impulse there is a cancellation for two yeah that is what you have done actually like yeah i think in your recent paper yeah thank you okay any other question so i have one final question uh also for enrique and gabrielle maybe that in the recent work uh in deriving classical quantities from the amplitudes there's this method of regions technique which is employed where you divide the loop momentum into potential and radiation and stop region right but in the case of a large impact parameter the radiation region is where the exchange moment is as gabriel was saying of the order of impulse inverse impact parameter inverse but the contribution you get for the tails is from the loop moment which is much larger than the frequency radiation frequency yeah yeah like what i can tell in this asymmetric computation the first two terms actually comes from the potential region like if the lang in the language potential and radiation region and the last two terms comes from the radiation region when you take l0 in order of l where l is the momentum of the graviton but yeah yeah that was my doubt that i thought you said l the integration which is l much larger than omega isn't it right right l is much greater than omega but omega is the external energy what i am telling is when l 0 is in order of l vector which is which means you are taking the pole from the graviton propagator ah okay okay it's known to be the radiation region right but this is the radiation region and this is the last two lines are the radiation region contribution which are suppressed relative to the potential region going to use basically the massive particle poles contributes propagators of the massive particles so that means the quantum piece is the radiation region complete contribution exactly exactly so that's why we interpreted that as some radiation back reaction effect though we don't have any like classical or other understanding like without this derivation so so you expecting classical limit these terms to be not there so you expect since like we have given an independent classical derivation and we have these two terms are not appealing that means classical in classical limit there's two terms should not be there but how to take from this quantum and classical without taking small momentum transfer that is not clear thank you yes so let's thank vishwajeet once again for an excellent talk thank you and now sasha gabriel can i don't think we have any special communication to make uh dimitri just that tomorrow we start uh half an hour earlier right but the meeting is not there yeah that's that's right oh i i'm here right yeah he wasn't thinking oh you are here okay is that half an our earlier now in the program i didn't see um did we write that there is a discussion also tomorrow from 4 to 4 30 sorry from um let's see no no tomorrow we finish from 5 to 5 30 probably goodbye to 5 30. yeah oh yeah that's it it's not written in the program for some reason but it's there okay you have to to go and read the schedule to have um the complete uh timetable of the com the the workshop yeah but i okay okay but there is always a the half an hour discussion after the second yes after excuse me 17 so after 5 pm european time yes actually i have a small announcement if any now we will close the zoom connection and but if anybody is uh willing to continue discussing uh on these topics there is also a gather town space i wrote the link on the chat actually parallel discussion can take place simultaneously so if anybody is willing to use this space either now or whenever you want we have also this opportunity yeah you can have fun with gather you can cry for fun there are also blackboards in some room so interactive blackboards so maybe we can zoom okay it's up to you if you are used to or if you want to try yeah please feel free to go to gather town okay and if not to all the others we see you tomorrow then thank you thank you thank you bye-bye bye bye bye bye bye stop closer calling