 Yeah, this is the title of the conference, by the way. And I'd like to start by thanking the organizers for what I've found very stimulating and eye-opening meeting. Much of it, I have to be frank, not been aware of these rapid developments relating microscopic quantum systems to black holes. And I shall only peripherally be talking about black holes. So I want to explain that to begin with. And then I'll tell you what I'm actually going to be talking about. When the program first came out, Malcolm Perry was down to speak on the work that he and Strominger and Hawking have been doing about soft hair for black holes. And I had got interested in this subject and tried to understand it from my own point of view. And so I thought I would be giving a commentary to that. As you will see, what I'm actually going to talk about, I'm going to begin by talking about what's called carol symmetry, which has been part of what I've been doing for a few years, particularly here in France. And then I'm going to give an application of this, in some sense, to problems in black hole theory. And I'm going to be talking in a very elementary way about the topics here, gravitational memory and the notion of a soft graviton. And you should take this in the following spirit, that for me, the concept of a soft graviton, which arises from S-matrix theory, and it's very much a quantum notion, is extremely strange. And I'm trying to translate it into a language that an old fashioned and increasingly aged relativist can understand. OK. So the people I've been mostly working with over the past few years on this are Christian Duval from Marseille, Peter Horvatius from Tour, and Peng Ming Zhang, who's from Lanjou. And it came out of a stay of four years or so at LMTP in Tours, which was supported, I have to say this, of course, by an outfit called La Studio, for which I'm very grateful. And I'm also grateful for Sergei Soleduken, with whom I've also participated, but not with these particular subjects, who was the host there, the official host and extremely helpful for me. And I want to thank everyone else from Tours who was very hospitable, including Stan Nicholas. So the first part of my talk is meant to be a kind of overview of what is known as the Carroll Group. And I'm not going to assume that any of you know what Carroll Group is. And I'm trying to be pedagogic, so do interrupt me if it doesn't make any sense, et cetera, et cetera. And the second is an application to plane gravitational waves, which are particularly exciting and interesting solutions in general relativity. They are nonlinear versions of the gravitons everybody speaks about, and we'll be speaking about I understand in a few minutes after my talk. But that's the LIGO team. And so I want to extract from this theory a notion of a soft graviton. And the relationship with what is in the minds of Strominger and Perry and Hawking is that their idea is that as material goes outwards, or as collapse takes place, the Hawking process takes place, but mainly as collapse takes place, gravitational waves can be emitted from the black hole. And somehow they carry the information. That's hair that's not been taken into account by the standard no-hair theorems. So what is that hair? And can you use it to unravel or measure things about the interior of the black hole? That's their program. And I'm just going to give you a commentary on what the notion of soft hair actually is in this context by approximating an outgoing, roughly spherical wave with a plane wave. Now the contents of the first are given a quite long content group. The way I view these Carol and Galileo groups is as kinematic groups. I'll tell you what that is. These are the kind of groups you can expect to arise in physics if you have rotational invariance, if you have translational invariance of some sort that don't have to commute. And you also have some idea of passing to a moving frame. In other words, you have boosts. You try to classify all groups of that structure. And for every one of these groups, of which the Carol group is one, and the Galileo is another, you get a kind of privileged flat spacetime structure. And then as in general relativity, you can make it wiggle and get a curved version of the same structure. These are all related, or the two ones that are important for us, Galileo and Carol in this context, to a high dimensional structure, a Bergman structure. The Bergman group is that extension of the Galileo group that you need to take into account when you consider boosts because Galileo and boosts act on the wave function by a phase, as well as, of course, a shift of the arguments of the wave function. And so there's a bunch of other groups which leave invariant these structures. There are various isomorphisms, which I'll mention on passant. But the interesting point here is the BMS group is a version of the Carol group. And that's how I came to it. Now the BMS group is a group of asymptotic symmetries of spacetime and is intimately related to the kind of soft physics, if I may say so, without being pejorative, of Stromiger and Perry and Hawking and many others, of course. Then I'll give the applications. And I have in mind as applications what I've said. OK, so the idea is to look at non-Einsteinian relativity. People often think of Newton's theory as non-relativistic. Well, that's garbage. It's relativistic. It just has Galileo's principle of relativity, which isn't Einstein's. OK, so and I'm looking at it from a spacetime point of view, which Galileo and Newton didn't quite do. And in our context, the principle of relativity involves the notion of the invariance of physical laws under passing to a moving frame. And we're going to interpret this as a symmetry of some kind of spacetime structure. So that's the framework in which we're working. And we're following the path pioneered actually here in France by Bakri and Lévi-Le Blanc many years ago, who found all algebras containing some slight extra assumptions, rotations and spatial and temporal translations and boosts. And all of these may be regarded as the vignogno-contractions of the two de Sitter groups. Now, if we weren't interested in boosts, we'd be doing something that was done many years ago by Klein and Helmholtz. We'd be classifying Aristotelian spacetimes, as they're sometimes called, because the three congruence geometries are basically cosets which include rotations and translations. And if you just set yourself up to make a space of that sort, which was called the axiom of free mobility by Helmholtz, you land up with the three congruence geometries we use in Robertson-Walker. So now we just add the time direction and see what we get. OK, well, here's a diagram which explains this log. So they found basically eight of these groups. Actually, they confounded these or conflated these two. They didn't distinguish between ADS and S, but I have done. So this box, these vertices, show you the big daddy up here and arrows going away. And the arrows are contractions. So for example, you can let the velocity of light go to infinity, and you get the Newton-Hawk group. More familiarity, you can go along this direction and get the Poincare group by allowing the leg scale in the ADS group to go to infinity. And once you've got the Poincare group, well, if you let the velocity of light go to infinity, you get the Galilean group. So most of those you're familiar with, there's a sort of reflection in this plane here, because this is negative lambda and this is positive lambda. But what about going this direction? Well, the one that I'm going to be talking about is the Carroll group, and that's when you let the velocity of light go to zero. And that could have many applications. OK, so that's written here in terms of what you need to do. Newton-Hawk, you have to be a bit clever about taking lambda to zero, but you also take c goes to infinity. And so you get the time, and that time is related to the period of an oscillator, the oscillator being a regular oscillator in the positive lambda case, sorry, in the negative lambda case, and an upside down oscillator. So these are how it goes. And actually, there's a certain duality between the Galileo and Carroll groups. You can think of it this way, we all know that we have a light cone in Einstein's version of relativity. What you can do is open it up like this. This is Galilean physics, where you can travel as fast as you like, or you can close it down, shrink it to a line element, and that means you can't travel anywhere, actually. You've got absolute space so you can't move. Here you've got absolute time. Now, each of these has a spacetime structure. And there is a duality which is really inverting, as we will see, passing from the physical space to the tangent space, or from tangent space to cotangent space. So if you like the wave particle duality discovered by de Broglie, it's simply that when you go to the opposite space, you get the same light cone. That's how he invented it, in fact, by he had his eye on relativity, as far as I can tell from what's written. Now, this thing happens frequently in applications when you have ultra-local theories. You throw away spatial gradients in your theory and turn them into quantum mechanics or particle on a line or something. So actually, the Karel group should have an application to much of what has been talked about in this meeting, but I'm not going to discuss that in detail. OK. And all of these kinematic groups have a flat invariant model spacetime, and it allows curving. Now, for Galileo, this is well known and known since Cartan's time. And it's a spacetime, so it has the same number of coordinates as what we think of as Vinkowski spacetime, but it has a degenerate co-metric, whose kernel are these normals. They are the one forms that give these constant time surfaces. OK. Karelian spacetime has a degenerate metric, whose kernel is a vector, and that vector is the direction that we go in. OK. I'd like to characterize this whole Karelian business to quote the much not lamented Mrs. Sacha. She used to say whenever she had a policy, Mrs. May, if you follow English politics, has got some kind of, she's trying to use the same slogan, Tina, there is no alternative. OK. Well, I hope that's intuitively clear. And the name comes from Louis Carroll from this famous quotation. Well, in our country, said Alice, still panting a little, you'd generally get somewhere else if you run very fast for a long time, as we've been doing. A slow sort of country, said the queen. Now here you see it takes all the running you can do to keep in the same place. If you want to get somewhere else, you must run at least twice as fast. OK. And the key point is the boosts act differently. This is how Galilean boosts act and Karelian boosts act. They change the time, but not the x. That's why x is absolute. I mean, there are translations, but. And I'd like to introduce two different times. T is Newtonian time, or Galilean time, and S is Karelian time. And that will make, I hope, some things rather manifest and clear. In one plus one dimension, there's no difference. So this is Lewis-Karel, who's sort of off the slide. This is Galileo. You can interchange them. So in the string theory literature, people talk about Galilean algebras, conformal Galilean algebras, but they could as well talk about Karel algebras. The classification will be the same. Now, to see what's going on here, let's consider the contravariant metric on the contravariant tensor, which is the inverse of the metric in Minkowski space. You see, you've got 1 over c squared here, and you can take c to infinity with impunity. It just kills the first term, and that gives you the definition, really, of a Newton-Cartan spacetime. It's some manifold, some degenerate covariant, twice symmetric tensor, with a one-dimensional kernel, a connection, which would be the analog of the Levy-Chavita, but this is all you could do, which leaves invariant the kernel. And it's a deep as one-dimensional manifold, and we'll find that there are lots of significant examples shortly. On the other hand, if you look at the metric itself, it's got c squared here. It's no good taking that to infinity, but you can take it to 0, and then you just lose this slot here. And that motivates the definition of a Corallian spacetime as a quadruple with c, this degenerate metric now, a vector field, and a connection. And you'd probably want the whole lot to be left invariant. OK. So those are the basic structures we'll be thinking about. The new one is Corallian. The standard flat case is pretty clear, and it's easy to check that this is part of the isometry group, which is infinite-dimensional. That is an arbitrary function of the spatial variables. We'll still leave nothing invariant. And if you want to get a finite-dimensional group, you can restrict this, but then you've got to demand that it preserves the connection. So the standard translation in affine space is preserved. That restricts f, and you get this finite-dimensional group sitting up here, which was invented by Levy-Leblond actually before the classification. There are other guys who are active at the time. I think there was somebody called Sen, and it was part of the fact that mostly what people did in the 60s was group theory and particle physics. Anyhow, so that's the picture here. And all kinematic groups have a description in terms of Lorentzian geometry in four-plus-one dimensions. And that's what's kind of interesting. And that's what's going to come in with the gravitational waves. Now, everybody knows about Kaluca Klein theory, so you get ordinary Lorentz geometry going down on a translation. Now, Newton-Cartan spacetime arises from a reduction of a null translation, which we'll see in a bit more detail. This was first seen by Christian Duval and colleagues who were trying to understand Galileo invariance in a nice way and has come to be used in the supergravity theory in more recent times. Now, you can think of reduction as push forward, if you know any geometry. And the other thing you can do in geometry is pull back. So if you want to make a cheap reputation, you start pulling back and pushing forward, and you get something new. Kuralian spacetime arises as the pullback to a null hyperplane. And in fact, what I'm going to say is given any null surface, for example, future null infinity, Kuralian structures come into play. And that's why a lot of people have been doing Kuralian physics without calling it that. OK, let's see. OK, at this level, are there any questions about the basic idea? OK, everybody happy or asleep? Sorry. Actually, you were saying that you're replacing C by 1 by C. No, no, no. I'm replacing C by 0. No, no, no. For the Kuralian, it is C goes to 0. And for the Galilean, it is C goes to infinity. Ah, exactly. So C goes to infinity. Somehow it is invertible. Is there some inverse? Well, it's not quite invertible. You see this duality, it's just reversing the arrow. Yeah, just that is what I'm asking is some subgroup of SL2Z actually acts on this. It goes to C, it goes to 1 by C. OK, let me answer that now in the next few minutes. I'll tell you how it looks like, perhaps in more precise ways. Any other? If pushing forward and pull back is going to be important in the future of your talk, maybe you should re-explain non-experts what it means really. You are not going to discuss it. OK, OK. It looks like going out. Yeah, yeah, no, no. OK, OK. The other basic point is that, suppose you've got a curve here and you've got a map between these guys, you can take the curve like that. And you can do that for any tensor. And that's because the Jacobian you're looking at, you need powers of this guy and then multiply it. Doesn't have to be invertible, OK? On the other hand, if you've got some gadget here, you can pull it back here. But what you need is this thing the upside down way. And that means that you can pull back covectors. OK, and for example, if you have a map from a manifold here into some other manifold, let's say this is spacetime and this is some sub-manifold, you can now, because the metric on this gadget is covariant, you can pull it back to here. OK, and so the induced metric on a sub-manifold is to pull back onto the sub-manifold of the metric in the big embedding space, OK? And to say it's a null surface is to say that when you've pulled it back, it has less than maximal rank, one less than maximal rank. It's got a light cone structure. So let's take a null hyperplane. A null hyperplane looks something like this. And there's a definite null direction on that. And that's the zero. That's the guy along which there's no alternative. If you want to stay there, you've got to travel at the velocity of light. Is that OK with everybody now? But you can, of course, in a language of a modern homological algebra or something, thinking of just reversing algebra and arrows. We think of covariant and contravariant. Pencils. Yeah, yeah. That's exactly that. Yeah, I know exactly. It's just one way of thinking about it. Why in this language, Carolians say it's time to put back to a null, that's why. Well, I'll show you how that works in detail in something like the next overhead. Oh, wait a minute. I won't go into this. This is something technical about it. OK. Now, I want to give you Minkowski's space in these light-cone coordinates. So these will be x plus, x minus. And if you write down all of these generators of isometries, you've got some boosts, you've got some null translations, et cetera. And now the point is that the various groups we shall be interested in are the groups that I've listed here. Forget these two because they have to do with very special relativity. I'm not going to talk about that. The Carolian group is the span of these translations, these rotations, not all of the rotations necessarily. Well, they're all the two-dimensional rotations. And then a null translation and some boosts. Now, these all leave invariant the surface u equals constant. That's how I picked them out. That's the null hyperplane. And in this picture, the way you think of it, you look sideways on. And the axes are basically, I'm going to get this right, u and v. So this hyperplane, u equals constant, is the one which the Carol group leaves invariant. OK. So if you're in five dimensions and leave it invariant, this is a four-dimensional hyperplane. It's ruled by these null generators. And those are the absolute time directions. OK. This kind of picture will be important in what follows. OK, I hope that's OK. Now, there's something called a Bergman manifold. Because what we're going to be doing is saying we're going to take this space and you look at everything. I said you get the Carol group by everything that leaves invariant, u equals constant. You get the Galilean group. So this is the original discussion. You get the Galilean group by taking a null vector, like this guy here, and asking everything that commutes with it. Because that's to push you down on a null direction onto a quotient space. And everything that commutes with that, well, you actually want something which normalizes this vector because it will commute with a projection. It will consist either of moving up the v direction or something which can be thought of as living on the quotient. So that is an extra generator, which you don't see. Because when you're looking on the quotient, you've factored it out. But that is the central extension of Bergman. So all this looks a bit complicated. So here is the very easy way to see it. I'll come back to that definition. Suppose you want to solve the massless wave equation in five dimensions with one time. Well, the massless wave equation is just this operator here. This operator, this is just d2 by dt squared plus a spatial thing. It's just the way you write. And this guy is just the regular Laplace operator in three dimensions. Are there two phi? Typo. All right, I hadn't noticed that. It's actually on a function phi. I beg your pardon. OK, now, if you Fourier transform in this v direction, which I'm now calling s. So s is my Corollian time. Then I get a psi of x, t and x. It's a triviality to see it satisfies the Schrodinger equation. We all know the Schrodinger equation commutes with Galilean transformations up to a phase. That's the Bergman group. If you throw away the phase, then you get the Galilean group. So these three lines with this typo corrected are all you really need to know about these sophisticated ideas. And this is why Newton never saw the Bergman group, or neither did Galileo. They wouldn't even think about it. It's a phase that's invisible unless you're in quantum mechanics, and it's not really there. OK. Now, so the standard Newton-Cartan structure, which is to have time and space, is obtained by pushing forward the flat Bergman structure to the quotient, sometimes called the light-like shadow. So you think of all of physics in this five-dimensional world just projected down by light rays onto some screen. Now, the screen has to be thought of in terms of a vector space quotient, because it's a projection. And the Bergman group consists of the isometries which preserve this vector field, which is pushing you down. You can also obtain the central extension of the conformal Schrodinger group, which is the symmetry of the free Schrodinger equation, which people are interested in, by considering conformal transformations of this Bergman structure, which commute with a projection. So it's a nice language for discussing these sorts of things. And you can make it wiggly by defining a Bergman manifold as a triple, bg and xi. And basically, a Bergman manifold is a manifold with Lorentzian metric and invertible, that means a null vector field, which is covariant constant or parallel, as they say, with respect to the Levy-Chavita connection. And the standard Bergman manifold is just flat. And here you see I've got my two times. Instead of u and v, I've called them T and s. And this is basically Galilean time, and this is Coralian time, if you like. And the standard one is what I've just said. You just take the flat one. But if you want a curved one, you get. So the standard, to say it again, the carol structure is obtained by pulling back the flat Bergman structure to a null hypersurface. OK. Now, there's one distinction between the carol group. As you well know, you could discover the central extension of the Galilean group. Purely algebraically, you just add something to the algebra and get it to close and check your Jacobi identities. The same kind of arguments will tell you that there is no central extension to the carol group. It's more or less what, through your details. Yeah, I think so. Yeah. Although, of course, you did it in such a language that no human being can understand. No, it's quite close to what you do. Yeah, yeah, yeah, OK. OK. Now, you can make non-standard Coralian structures simply by taking the product of some Romanian manifold with the reels. You give the metric value 0 on that guy. And you give the, this you choose any metric you like, which is Romanian. And that will give you what you want. And now, you have to choose a connection. And one way would be to choose the Levy-Civita collection associated to this space. So those give you some interesting examples. And here are, or you can also define a conformal group by defining automorphism, which leaves a variant, this degenerate metric, and leaves up to a scale, the vector field. And this is like a conformal weight. In fact, it's related in simple cases to the conformal weight you use in conformal mechanics or whatever. And the main point is that the killing vector, a flat killing vector structure has this enormous family of vector fields. And this guy is called the super translation, because it depends on many. It's like a translation, but it depends upon many variables. And it's moving you up this direction in new space, in this case. OK, so you've met these before, actually, but not known them. It's like, what was the man's name? Mr. Kipal Pro? Jordan, yeah? OK. So anything to you? You can take your pick, but it's just, you know, conformal weights can be anything you like. I caught it in the integer because that's convenient. I'll give you a couple of examples where you've seen these. So the first one is take S1 with its angle d theta squared. And then we get diff S1, which is with the semi-direct product of super translations. So this is a standard virocyl or wit algebra that you get. So that's rather trivial. The next one is much more interesting. Take the two-sphere, n equals 2, and you take its conformal group, which is PSL2C, unless you have some defects in the sphere. The defects correspond to what people call superrotations. And you take this semi-direct product, and you get where the T's are thought of as half-denses. And you get what's called the Bondi-Metzner-Sachs group. So that was discovered quite independently of these considerations by considering asymptotically flat manifolds in general relativity and containing radiation. The asymptotic symmetry is much larger than the Poincare group, which was anticipated. And so you can think of the theory of asymptotic gravitational radiation as basically the theory of the BMS group. The super translations act on the null generators. So you should think of, we've seen this picture in this conference. This is Scribe plus. It's a half cone. Here are the null generators. You can think of this as a line bundle over S2. And the thing is, to coordinate it, you need a section, but there's no global section because these ends, in general, are singular. If you're actually talking about a real physical space time, this is a singular end, so there's no way of defining what the origin is of these generators, i.e. what the zero section is. And equally well, you cannot make a correlation between the generators here. So the asymptotic symmetry group of a general space time would be expected to be BMS in the future and BMS in the past. Now, our friend's strome is for four-dimensional. For four-dimensional, yeah. High-dimensional is no such thing. The asymptotics are different. But I live in four-dimensions for this talk. Yeah, I mean, that's a problem with, I think, the Hawking-Strominger and Perry proposal. It's one of many. One, another one is they believe that there is a rule for making this and this isomorphism between them. I don't understand the rule, but they refer to the god of the subject, a Demetrius Christodoulos. And if anyone understands him, talk to me quickly. Does that exist in the normal dimension? You can define it formally, and I once did. And you could define it superversion in all dimensions. But the trouble is that to make it really work, you've got to know that there are solutions of the Einstein equations which have sufficient regularity that this is a legitimate symmetry. And it's been established more or less incontestably that this is not generically the case in higher dimensions. That means actually, which symmetry are you? BMS symmetry. So the S matrix in principle should be BMS times BMS, right? It is believed by some people that you can go to the diagonal group. But this is only in four dimensions. You can formally define the BMS group in any dimension using the procedure ideas I gave you. But the boundary conditions and the asymptotics of realistic metrics forbid these actions to be smooth. So the S matrix, if it exists, who knows? But it certainly isn't BMS. It carries BMS. And can you comment on the extension of SL2C to the conformal group? What do you think of that? Well, it's what you need to get the formulae to work out in a nice way. You want to be saying, there's a certain notion of a conformal group of scry, which is preserving a conformal structure of scry, which is performed. Which are singular on this asymptotic sphere. Then you have the infinite dimensional conformal group. Oh, yeah, you can do that. But that is what they call super rotations. That's valid if for some reason you're not asymptotically flat, but you've got some punctures. So if you had a flux tube coming through, or you might have what is rather like tab nut, an interpretation of tab nut is a null strut with a string that's running through it. You can formally construct them. And they would be valid, of course, if you chose to work on the cylinder, which is what you do in many conformal field theories. So the mathematical structures exist. So if you want to take account of the full conformal group in a sense of string theory, because you're doing field theory on a circle, then the BMS group would contain the full conformal group. We go beyond SL2C. I read in some places these are like large gauge transformations. I'm going to come to that. Can I come to that later? But they are in some sense. You see, what they're doing is taking an asymptotically flat space, and they are isometries of that space in some sense. But of course, since they act non-trivially at infinity, they're called large gauge transformations. That's not, I think, because of any topological content. It's simply they don't fall off at infinity. And I'm going to come back to that very point on my last transparency. So they could take a configuration with finite action and make it infinite action? Well, this is all Lorenzian. And actually, strictly speaking, the action is garbage if you're in Lorenzian space. So it wouldn't, you might regulate it or something. But yes, it would leave invariant the metric. What Bondi and Sacks, well, Bondi started off and Sacks completed. You take a power series expansion of the metric at infinity. He was the first to demonstrate, well, Troutman was first, but anyhow he's called the person that's first to demonstrate mass loss because there's a formula for the flux. And what they discovered is that everything they did was invariant as an asymptotic expansion under this bigger group. And they couldn't fix. There was no way of fixing the total energy. Sorry, not the total energy. You couldn't fix the super translations. If you want to get the Poincare subgroup, you restrict the super translations to just the lowest dimensional harmonics on the sphere. And they will give you the translation part of the Poincare group. Because you want Lorentz, semi-direct product translations, right? So the translations are a finite dimensional commutative subgroup. Well, all super translations commute. But you want a finite four dimensional subgroup. So you take the constant plus the three lowest harmonics. Just carry it out, use super rotation. Well, they kind of mix it. I don't know. The latest paper mentions this. And in fact, I have spoken with Stravinger about this in the past. And he's basically thinking of wires at infinity. He didn't know what they were. Incidentally, the reason, this is a slightly cheeky thing. The reason, I think, that Malcolm is not here today is that Stravinger is giving a talk. He's obviously visiting Cambridge. I think he prefers the company of Stravinger, unfortunately, to that at IHS. But anyhow, any more questions? OK. Yeah. Yeah. Yeah, there are various ways of weakening this. There's something called the Pneumonanti group. But it's a bit technical. I think I want to pass on. I'll just comment that these symmetries arise in a wide variety of circumstances. We've seen BMS. They arise in tachyon condensates. Because in a tachyon condensate, particularly if you're taking strong coupling, it means that the brain is getting very close to being light-like, if you think of the tachyon as a coordinate of the brain. And so strong coupling emerges as a corollion limit of the world brain theory. This is something I did ages ago with these people, Hashimoto and Ye. You can construct corollion theories, invariant theories using, in fact, Surio's method. We used that, and we got various interesting things, including the corollion massive particle. And we also got out of it at some stage, the null string or short string. So that was past work. And I don't want to go too much on it. I want now to go to the next application. OK, so what is a plane gravitational wave? There are various definitions, one of which is these are vacuum metrics with a covariately constant null vector field, and they admit a five-dimensional isometry group, which is the same dimension as the isometry group or the symmetry group of a plane electromagnetic wave. One approach to this is to say the analog of a photon, well, what is a photon in classical theory? It's a plane electromagnetic wave, we say arbitrary profile. And that, you easily check, has a five-dimensional symmetry group, so then you say what spacetime can have the same. And that was done by Bondi and people, Bondi, Pirani, and Robinson, all three of which are now sadly deceased. In the 50s, there was a major step in understanding gravitational waves, because you could do this at the exact level. It wasn't perturbative. Now what we've recently realized, that started it off on this game, but then we realized a lot more things we could do, is that isometry group, it's almost obvious from this, is in fact a subgroup of the six-dimensional carol group. And the reason it's a sub in the relevant dimension, the reason it's a subgroup is that gravitational waves are polarized, so you lose the SO2 generator. If you were just talking about scalar waves, then you'd regain that rotation, and the isometry group of scalar plane waves, or the symmetry group, is in fact a full carol group. Now that's quite useful, because if you've got a group, you can start calculating. I'll just remind some of you of the importance of these gadgets. They admit a covariantly constant null vector and spinner. They're BPS. All invariants vanish. They're exact solutions with no quantum corrections of any theory you can more or less think of, but certainly string theories. They've been used in a wide variety of contexts in the literature. Now, they admit two useful coordinate systems, one of which is global and one of which is not. And this is the key to the discussion. OK, the global one was found in 1922 by Brinkman, I think it was 1922. And here it is. This is, I is running from one to two. Here are our null coordinates. I've changed my convention slightly. I'll explain that in a moment. Here is an extra profile function. This is a 2 by 2 traceless matrix. And you can give it arbitrarily. To solve the Einstein equations, you just give two arbitrary functions of this retarded time view. And moreover, you can superpose solutions, which is very remarkable and only true in this coordinate system. Now, this coordinate system is global and harmonic. But the one that's adapted to the symmetry group, you obviously have one killing vector because you have the null translation. But there's a three-dimensional abelian subgroup to the carol group. So you can use these coordinates, which are called Rosen coordinates, despite the fact that a better job was done by Baldwin and Jeffrey 25 years before them also. So this is an exact solution of the Einstein equations. Here is the coordinate transformation. But now you've got to solve a very nonlinear equation. If B is a to the minus 1 a dot a is a matrix 2 by 2, then this is k, which is related to the Riemann tensor. And you've got to solve this equation. And that's tricky to make that trace free. So another disadvantage of these coordinate systems is that they always break down after a finite time. They focus because they're based on a set of null rays, which are well-known to focus to make experiments in the field. But they do exhibit the symmetry manifestly of three translations, two transverse and, of course, the V translation. So they're useful to use. And I'll come to that later. The fact that in these coordinates, which is what Einstein played about with, misled him into think that there was no such thing as a gravitational wave. And that took a long time to sort out. Eventually, people said, well, no, this is the Cauchy coordinate, and you're using the non-Cauchy coordinate. So in Brinkman coordinates, the field equations are trivially satisfied. But in Baldwin, Geoffrey, and Rosen coordinates, everything is rather non-trivial. But you can solve for geodesics exactly. It's an elementary undergraduate problem. Now, I want to consider what are called sandwich waves. So the filling of the sandwich is curvature. There's a before zone and an after zone. And these are flat. OK, this is our view of a gravitational wave arriving at LIGO. LIGO is the detector is flat. And the particles, we're going to use freely falling particles as a detector. They're at rest. The wave comes through. And then it gets to the after. The particles then find themselves in the after zone. And they, in general, will be moving. And we're going to study that motion. Now, for technical reasons having to do with previous papers, my V is in the opposite direction to what it has been on overheads. But it doesn't matter. V shifts you up and down here. OK, so the amplitudes here are the cross and plus amplitudes. And they vanish outside a finite interval. That's the definition of a sandwich wave. And what we're going to look at is what happens. Now, it turns out by a trivial application of Nertis theorem that in the Baldwin-Jeffrey-Rosen coordinates, if x are constant before the wave, they're constant for all time. It's just the most elementary result. It has to do with the translation invariance. And so supposing that Aij was delta ij before. So you use the same coordinate system, a global coordinate system for one half of Minkowski space. In general, Aij will not be constant afterwards, even though it's flat. If you impose the condition of the metric as flat, there's a two parameter, well, it depends on two matrices, but there's a two matrix collection of solutions, most of which actually focus to a singularity afterwards, which has to do with the focusing effect of the wave. So afterwards, the particles will have a non-trivial time dependence. And that's how you could deduce retrospectively what the wave was. That's the important point. It's related to what's called the memory effect, and I've got a few more slides about that. But if everybody in LIGO went to sleep and then woke up and said, by golly, that wave must have passed, our measurements were off. As long as they'd left a few free particles, then you could reconstruct what had happened. Reconstruct some things. OK. No, no, no. In practice, this is on the verge. As a matter of fact, with Lisa, they may see these effects. It's all very debatable, and it's all at the limits of technology. The description I'm giving you is very simplified. OK, so that's, broadly speaking, the gravitational memory effect. Now, different authors give it different names. A particular case which started this off was given by the postage stamp man, Zeldovish, and his colleague, Polnarev. I realized the other day he's actually lives quite close to us in Cambridge. He's down at Queen Mary College. I don't know where he actually is, houses, but anyhow. This was explicitly pointed out by Zeldovish and Polnarev in this paper here. And now the trouble is here that they use the words permanent separation. That's a special case, as it turns out. And I want to do a little bit of self-advertisement, but I excuse it by advertising my supervisor. This is very clear in linear theory. And one of the things that he invigorated me into doing is working on gravitational wave detectors, thus endangering the remote prospect of my ever getting a PhD. Indeed, I got a shift of the whole thing soon, saying that they will not find gravitational waves until I retire, or after I retire. And that's one of the few predictions in physics that I've made, which is absolutely correct. I retired before the, what is it, September 14. OK, so basically they use linear theory. And then it's kind of trivial because they used, basically linear theory says Aij is delta ij plus Hij. And in a gravitational wave, the formula for Hij afterwards is equal to 0. And so you can have this. In fact, we did something else, which I want to advertise just as it's very similar to that. What we actually said was that the real formula for Hij is something like Hij, well, what do I want to say? The one we wanted, r0i0j, sorry, r0i0j, which is basically Kij up to a factor, is something like the fourth derivative of the dipole moment and a 1 over r factor. So if you integrate this three times, and the quadrupole moment has gone from constant to constant, you get three integrals of the Riemann tensor. So the memory, so basically what happens afterwards since the equation for the particle is this famous Jacobi equation, xj xi equals 0. This is the equation we heard in the lecture earlier. Yeah, sorry. You see that by looking at this particle, you can deduce this. I mean, and it involves integrals. And so we were interested in this relation. And it's quite clear from this equation that the generic motion, once this is finished, is linear in time. Now, this is all perturbation theory, but we can reproduce this absolutely precisely in the formalism that one has. Incidentally, there are some people who call this a permanent change of spacetime, and this comes back to this question about big-gauge transformations. I don't think it's a change of permanent definition, a change of spacetime, because it's flat afterwards and it's flat before. But there is, perhaps you want to refine your notion of what a spacetime is and use a privileged coordinate system. OK, you could do this with optics. I won't spend too much time on this, but it's OK. The cases when you have, at the end, the particles in relative rest. Yes? That requires fine-tuning. It does, it does. And the wave profile. Exactly, yeah. And there's a story there. We've done a lot of numerical computations, or at least our colleague Zhang, in Landjew. And the trouble is that, for the ones we picked, they were always moving. Now, I think the answer to this is that the notion is somehow that that gets damped out. Zeldovich and later Broginski and Kristjok seem to think those were not so very important. But that gets very silo energy transfer. Yeah, yeah, exactly. Energy and then they give it back. Yeah, yeah, yeah, yeah, yeah. Yeah, no, it's a very abstract question. I haven't time to show you now, but if you read our paper, there are a lot of diagrams of how the geodesics move and so forth, which can be done very easily for certain assumptions. I make the remark here that you can think of photons moving through a gravitational field as moving through a permeable medium. It has a permittivity and a permeability. This has worked out many years ago by Igor Tam and is now used in transformation optics. And you can actually find what the epsidons and muses are. They've got to be equal, and they're written by this. So if Aij is not delta ij, but nevertheless flat, you have a refractive medium regarding, if you think of it, as in these BJR coordinates we call them, the Rosen coordinates. OK, so it's a bit coordinate dependent. But you could say that it polarizes the vacuum after it's passed. So there are some subtleties here, and they make a relativist rather uncomfortable. Now, after the wave, although Aij is not delta ij, there is a coordinate transformation, which we calculate explicitly, which brings the metric after the wave has passed to canonical flat form. And it's not difficult to find it. Now, the way you could look at this, because that doesn't tend to the identity at large spatial distances, it turns out, you could say that this flat plane wave metric, or the flat plane wave metrics in Baldwin, Jeffrey Rosen coordinates, can be thought of as soft gravitons left after the passage of a wave pulse. The pulse itself is non-vanishing curvature, but it leaves a memory of itself if you could track these soft gravitons. Now, from a formal point of view, these soft gravitons are just solutions of the equation for a gravitational wave. It just happens to be that they're flat. Now, we found the diffeomorphisms. That seems to be the necessary assumption to make. It's certainly consistent with standard assumptions in gauge theories to give life to soft gravitons. Are there differences in higher dimensions here? Everything I've done here will work in all dimensions. The complications that talked about by, for example, Hollands and others, who is it? It's Ishibashi and Wald, are related to the fall-off of the gravitational field, which is not just power law, but in some cases even square roots and logarithms and stuff like that. So that's a general issue in higher dimensions and would affect any discussion which is not strictly planar. But we don't see that here at all. I want to emphasize that we've said nothing about the sources. I mean, at linear level, that was said in old paper and lots of people at the same time. We know exactly formally for the quadrupole moment and the relation. Outside of linear theory, there's a great deal of work which has been done, not least by Tebow. And I just haven't looked into it personally with sufficient precision to make a reliable statement. So I just want to conclude by saying that this body work is here, so if you want to look up any of it on the web, there it is. Thank you for your attention. So if you say that the same effect appeared in a higher dimension, whereas the self-graviton effect is only relevant in four because of the log divergent, how can it be related? Well, strictly speaking, messing around with the BMS group is not necessary for this. You could consider ways. BMS, I'm just talking about this memory effect. No, no, no, no. But the memory effect is where you measure it. You could measure it anywhere. I mean, supposing you're sitting here, some bomb goes off, there's gravitational waves at finite distance. Everything I've said will go through, but you'll have to take into account the spherical geometry. You would not notice singularities in the fall-off unless you go to infinity. So it will leave an effect not described precisely by plane waves. It will leave this disturbed Minkowski space metric. And so in principle, you could resurrect some information about the wave. Now, I'm not a great advocate for the program. I've just been slowly working through trying to see if I understand the concepts. I think there's a much more serious question, which is very much the subject of this has been the subject of this meeting. These soft gravitons are detectable. That's clear, just as you can detect soft photons, because if you had charged particles, the charged particles will get accelerated as electromagnetic wave goes through. But you're talking about things with zero energy by anybody's definition. And then there's an issue of how much on-gime information can be carried by something with zero energy. And that's a very debatable thing. Personally, I've never seen a convincing argument either way. But that's what they really have to face up to. Their calculations so far are entirely classical. And I think. Greenberg and Choprition tells you the opposite. That is, whatever is really soft is somehow in your soul. And you just have to define an air-safe observable subject. All those things are not observable. Well, that's what I would have said. I mean, that's the standard block-naughted statement. And that's been repeated by Parati recently, in forcible terms. So I don't have enough expertise, really, to claim I understand it. It's not at all obvious. Maybe I don't remember whether. That's not in the recent paper by Tom Hanger. So that's why he was moving into super-rotation. Well, I'm not sure what his motivation is. But I wouldn't mess with a super-rotation. Yeah, that's right. I think they don't correspond. Andy has a thing about the conformal group. The way to put in something. Yeah, yeah. I mean, Andy has a thing about the conformal group. And SL2C is too small for him. So he needs some. Yeah. I mean, the memory, if you are talking about it, I think it's related to this time delay that I was talking about. Yeah, yeah. No, exactly. If you just go to the impulse limit for your sandwich waste, in which everything is concentrated, you send the sandwich to zero size, but still finite energy. That's right. Then you get this time delay. I mean, that's in some sense. Then you can go to the title of the sex accord, whereas I think for a fact, weight, you cannot. No, no, no. In fact, I should say that we have some work in progress on impulsive ways. And in fact, part of this began with a paper of Surios. So Christian Duvall was reading this paper. It comes from the 70s. And Surio considers impulsive waves. And then we got into a nightmare, because the theory of impulsive waves, although often used when you get down to it, is a big mess. And all of our formulae were being paradoxical, because everything is non-continuous, et cetera, et cetera. So we still haven't sorted that out. But we were precisely trying to get impulsive waves in the Brinkman coordinates are quite straightforward. You just put in a delta function, and it goes through formulae. But what Surio had done was to use the BJR coordinates, as we call them. Then the metric is not smooth, and the relationship between them has a discontinuity. I didn't expand on the formulae, but something called P, which multiplies the coordinates to get from one to the other. And we haven't got to home base yet on that.