Einstein's Equation and Twistor Theory — Recent Developments (contd.)

0:00 Twister lines, you have the cut ruled by twister lines. That is just Ted's construction, rephrased a little differently, that's all. And I think at the bottom I just said a twister line, and also a dual twister line, and also a generator, they're all mal-Gv6. Is this connection between head space and twister space leading to a corresponding connection between quantization? Is that a twister line? I actually asked Ted. No, I'm just doing this moment, just to reformulate, I'm trying to do very something very similar to a logic. Yeah. Reformulating the... That's right. We are in terms of... We're a little human, so we have to make it together. Oh, that's it! Okay. It's way more to the afterwards. In this picture, you never really needed this, not in the past, in time, you know, say that somebody uses time together with some function which is called sigma. Oh yes, that's true. Yeah. Yeah. You're right. I mean, I could have done this without starting from this, but what I want to say in a minute, I need to do that. The main reason being that I don't otherwise have a splitting into positive and negative, into right-handed and left-handed. So the thing is that if I have a decent scry, then I have a natural splitting into the right and left-handed parts, because I can look at the on-scry, I can look at the, well, down here, I can look at the H-space, which is a complex space-time, which another way of thinking of it would be to say, take your original space-time, supposing it has a good I-plus now, and kill off all the self-dual radiation, just keep the anti-self-dual part, just construct a new space-time, starting at the top if you like,

2:30 where you've killed off all the self-dual part and just kept the anti-self-dual part. And then reconstruct that and see what space would have given you that radiation field. And that's the H space. So now we're thinking of H spaces, you start with an asymptotically flat space, try to extract from it the A sort of space-time, the complex space-time, which has the anti-self-dual radiation of the space, but where the self-dual part's been put to zero. And so that it gives you, if you like, a way of, when you start from a space-time which is neither anti-self-dual nor self-dual, gives you a way of picking and sort of splitting it into the two parts, which you might not naturally see how to do otherwise. So that's why, that's where this comes in. Is there something special about the self-dual or the endo-self-dual in the space-time? And that was the original idea. I mean, the original space-time was not meant to be anti-self-dual or self-dual. It was just irradiating space-time. You may have. Yeah, you could take out, huh? That's right. What might have as a motivation is what's mine, but at the actual construction, it just takes on yours. Yes, yes. It doesn't really depend on what it is in terms of space-time. That's correct, yes. You give yourself some initial value, quote-unquote. Yes, sure, yeah. And then you just complexify it and you obtain everything in space. Yeah, no, I quite agree with you. But you see, the motivation, I'm going back, if you like, somewhat to the earlier motivation. It's saying, you're thinking of H space as something that you construct from a space-time. You want space-time, okay, it doesn't bring you back to space-time, it gives you something else, it gives you H space. But my intention is to try and get space-time back. And that's, so it's going back to the original idea, it needs a bit more. But, yeah. Okay. Now, the twister space I've got here has simply forgotten, from what I've just been saying, it's forgotten the self-dual information. It's only got the anti-self-dual information. And what I want to try and do is to get it back. And the way to get it back is as follows. Well, A-Way, and I think it's V-Way. This is how to encode the googly, as well as the self-deal information.

5:00 Now, let me just, first of all, instead of going back to Jim's comment, then you can produce the non-projective spacetime. See, this only gives you the projective spacetime because I've got the twister lines. But if I say that my pi spinner is to be parallel propagated along, this equation here is telling us... See, this is just the spin connection in the... This is very similar to what our pi does and... That's right. That's right. That's right. I'm sure it relates to a lot of things that you do and so on. But let me just continue with this. You can say that the pi, this is, to say that this is proportional to that is to say that this is a g, well, this is to say it's a twister line, that really is the equation for a twister line. It happens also to be the equation for a GD-sick, because if I made this into a vector by putting an iota, the iota points out for generators. If I took an iota and multiplied it by the pi, because the iotas are actually propagated also, this makes, this equation tells you this is a GD-sick. So it is a GD-sick on scribe. Now, what one normally did, what I would have done, up to about a year ago, was to put 0 on the right-hand side of this equation. So that the natural thing to do is to propagate the spinner so that this is 0 here. And that gives you a scaling. I can start with a pi, which is a small or a big one, and so on, and those different scalings give us the non-projective twister space. The non-projective twister space can also be found, very naturally, by putting nought here. And that gives us the space that we had before, the leg break space, and the thing that Jim would be happy with, because he didn't talk about the construction, using the inhomogeneous, using the non-projective space, which in fact is what made me happy too.

7:30 But now, what we do is the following. We take this crazy looking equation here. It may look a little crazy, but it's, in fact, there's very little you can do, which is invariant in all the ways that are required. This psi is the... Well, you see, to make it make sense on scry, you have to do the scaling. This is the self-dual vial curvature written in spiniform, and then, oh, which one is? This is the self-dual vial curvature written in spiniform, and then you define this object by taking the conformal factor, you need a conformal factor, of course, to make scry finite, and so on. You take a suitable conformal factor to make square finite, and the bar curvature goes to zero, but if you take omega to the minus one times it, you get something finite, and the reason for taking omega to the minus one times it is that this entity now satisfies the massless field equations in vacuum. So, that's conformally invariant. This is the the way that the massless field equations would transform to preserve the... I don't know if I said that very well. The way I like to think of it is that the big capital psi, this thing here, is actually the Weill curvature, or the self-dual part of the Weill curvature. The little psi here is a massless spin-two field, which happens to equal the Weill curvature when you choose the omega being one. But if you choose a general conformal factor, they just scale away from each other. But this one satisfies the master's field equations. This one doesn't. But that's a useful philosophy to have when you're talking about scry and so on, because this is the thing that remains finite on scry, as it should, because scry is a...conformally you don't care. I mean, master's field equations don't particularly care with scry, they're just shooting through And that was one of the initial reasons for thinking that scry was a good idea. And anyway, here we have the scaling, so, okay, this is basically the component of the Weyl curvature in the direction of the generators, but you have to bring the scaling in, because the Weyl curvature is zero. Okay, and this is the derivative in that direction, and then, to make everything balanced, I put

10:00 the pi 1 to the minus 5 there, so you'll count the number of 1 primes, you see you've got 4 of them there, 5 of them there, you've got to balance them all off. So there's no choice about that, if you want it to be scaling there. Well, I played around with all sorts of things you might try here, not this one first, but nothing else worked. Are these broken into the second derivative of the shear? Yes, this is already the second derivative ... It's already the first derivative ... This is the second derivative of the shear. No, would that be ? Não, Si4. Yes, Si4. It's going to be the second derivative of the shear. That's right. That's right. Second derivative of the shear. I'll tell you why it's actually not an unnatural thing in a minute, but that's just the definition for the moment. And I said I did try all sorts of different things, and the only thing which balanced was this, which gave you everything you need to balance all sorts of things, and that's that. So, although it looks a little funny, it is, in fact, really quite a natural thing. Pi-1 is just this component here, and all these things, I think, are also explanatory. Sorry? You could say it's a shell of unscripted. Well, it might be, but it does tie in with other things, you see. That's the thing about it. Yeah, I should make this comment, or should I make another one first? What's the time? Oh, I've got my watch and I think they've done them my transferences. Oh, I see. Okay, yep. Right. So we'll try and tidy things up by then. Solve the Einstein equations, complete generality and that sort of thing. Yes. I don't know, well, let me what is actually a useful thing to think of is not simply to do this at scry, as I've done here, but to do it also on a finite light cone.

12:30 In fact, most of the things I was saying will work just as well on the finite light cone, with certain slight differences. This is the finite light cone, complexified, so it seems analytic and all that. instead of iotas, this is notation, pointing out the generators. The equation I had before is now to be replaced by this one, where this is the same old psi that we have, the Masters-Fieldston 2, uh, Holicity 2. This object here is the conformally invariant Thorne operator, which is to be found in Spinners in Space-Time Volume 1, which is used in connection with the initial value problem. So if you were thinking about flat space and you had some cone, or it needed to be a cone, that may be not a clean vertex, and you want to specify data on this cone for a massless field of helicity 2, the null data, which is the thing which you, when you go inside and you want to integrate it out, You'll find this is the quantity. I actually, perhaps I should show you that transference here. That's this one. Yes, that's the definition of the conformal invariant thorn, using spin coefficients and so on. Ted will tell you what all those mean if you've forgotten. And this is what you find in the generalized Kierkopf-Dantemar formula for a massless field, it doesn't have to be spin 2, it can be any spin, on the cone, which needn't actually be a cone if you have a messy looking excuse for a vertex down there, and then the quantity that you integrate is simply the same quantity that I'm talking about. So it's in some sense a natural object to be talking about. Now, for what I do is I take this out to scry, you see, what I just did before. But this is just to explain that it is quite a natural-looking object. What are you going to do with that equation? Well, what I do is I... well, yes, I've done this, actually. Yes, what I do is what I did. But I haven't said it. Yes. You take the solutions of that equation, rather than the solutions of this with a zero.

15:00 And that, those are the points of my Twister space. So, that's given us the space here. The different points, you see, if I start off my equation, maybe I take a starting value for pi and propagate it by this equation. then the difference, I can do this in different ways, start off, propagate, start off, and propagate. And each of the solutions of that equation is a point on a fiber. So it doesn't look as if I've done much, if you like, but I have. Each point on the fiber here, which is the non-projective twister corresponding to that projective twister, is a solution of that equation. And I can scale it up and down by choosing different starting values. Pi, though it is grossly inhomogeneous in pi, and that's really why it's doing something. Sorry, Lee. So you, for each starting pie on the scry and on the null point of that point, you make a one-fameter family of pie, and that doesn't have to do it in the null point of that point? Well, no, it's not. You see, you don't really have a natural starting point, which is why it's doing something in a sense. Because, uh... I can do it on a scarier, or I can do it on a finite cone. Finite cone is what we need to think about. But it's a little more complicated because you have to bring the full thing and so on. But equations are all right. So it's just a solution of the equation. On the cone. On the cone, yes. On the cone. Yeah, yeah. And does it have to... Okay. Let's say, analogy does it on the light cone. It's not actually... Yes, that's the point I should just make here. On the light cone, it's not always analogy. It's usually not. If the current is non-shearing, it will be a non-geodeatic. Sorry? That's right. Because scry doesn't shear. Why don't you just look at scry? Well, I'm going to have to look at both, that's why. But I can look at scry first, yes. Sure. Let's put in the strength of the anti-back. And each such curve is a point in the direction? That's right. Now, you might think it's not done anything because you could just take some starting point. But this is a... it's a cohomological thing. That's the important thing.

17:30 There's no holomorphic way of taking a slice of this thing so that your starting points are kind of all uniform. You have to take one place to do your starting points and then somewhere else to do your starting points again, and you get a patching between the two, and that patching gives you the cohomology. The cohomology is basically the solution captures the... It captures the self-dual part of the radius. Yes, it does. Yes, it does. It's not so obvious. From what I've said, it's not at all obvious. But it does. Yeah. And the reason it does, the easiest way to see it is, not the way I'd do it first. So the parameterization is very important. It starts with . That's right. That's right. So what is it about the parametrizes? How is the parametrizes? Look, the parametrizes, it's just giving us the points on the curves. I mean, I don't know how else to say it. But I have different solutions. And I also take zero, that's not included. But anything else, I just propagate it according to that equation. And each, since I have one complex constant of integration, given the curve, I have one complex constant of integration, and that is the curve, is the fiber over here. So it's not just geometrical, geodesic, there's a cost on something here, because it's parametrizating. Yeah. Well, the geodesic is the point down there, you see. But the parametrizations, the different constants of integration in that equation, give us different So that's what it's doing, yeah. I mean, it takes a little bit to get one's mind around this thing, but that is what it's doing. In fact, just a brief comment, although I think it probably will be a waste of time to go into detail. It is related to something that Mike Eastwood did. I didn't know this at the time, but Lionel Mason reminded me. Mike Eastwood's paper is very confusing because there's a serious misprint in it which threw me completely when I read it. But it's in the further advances of Twister theory. And it's a way of representing massless fields in general with the prime index kind, so it's the positive elicitee of the normal conventions.

20:00 I don't think I'll go through this, but I just wanted to indicate that it is related to a fairly natural construction. That's, I think, all. If people wanted to know more about it, I could tell them. but it is related to a natural construction. It's not quite the same. It's kind of taking a small part of this construction, but it's a small part which contains all the information. Just given the . Is it possible to jump ahead? Well, I guess there are a couple of things I can jump to. One is this picture. And the other thing I should jump to is all this stuff. I'm not sure which order to do it. Let's do this first. These are the forms. Now, in the standard nonlinear graviton, or the leg break thing, you have all these forms. That's what I was saying before. In this case, you still have them, but not completely. What you have is iota and phi up to proportionality in such a way that they scale oppositely, so that the tensor product of iota and phi is an invariant object. There's also this relation, I mean, in fact the forms all have homogeneity relations, which are telling you that they're lead derivative with respect to this funnily defined. So you don't actually now have the phi given. You only have it up to scale. So that you don't actually have the Euler form. It gets messed up. And which relates partly to Jim's question about can you have fields of arbitrary spins on these things. But what happens is the helicity gets kind of messed up. The Euler operator now gets made, it's not even defined globally, it's defined in local patches, and when you go from one patch to the next it's not the same. But the homogeneity relations on the forms can be written in this funny looking equation here, where this funny symbol here is something that all relativists can write down without any difficulty, a pure mathematician, you have to go through some rigmarole. Because basically what you're doing is you're taking the forms and skewing all the entities of one with one index of the other. And that's all it's doing. And that's what this thing means. And this equation

22:30 is the, is the homogeneity equation. It expresses all of them, in fact. These ones are automatic and these ones are expressed like this. So I have this relation, and I have this thing. I'm supposed to know that. What else am I looking for? Let me just say the other thing, then. I don't know about transferences. This is really getting to the point of what one does here. See, if you do this construction on Scry, I have a Twister Space. It's the twister square, I call it curly T plus, it's the twister space with respect to square plus. I could do it for a finite point and I have another twister space relative to the point P. Now if I take, let's assume everything is nice and P is close enough to I plus that I have a broad enough family of these things, and I look at twister lines which are close to generators, then if they're close to generators they will get up And then I can take the same pi, so I propagate the pi along here by the equation I just gave you. When I hit scry, I then propagate it along scry by the equation I gave you. So I have a natural way of identifying any twister line or any part of the, let's see, any twister which belongs to the twister phase relative to P of some open set. You're getting a different trister space for each point? Yes, I have a different trister space for each point, but in a certain sense they're all the same. And that's the slightly curious thing I want to try and get across. Somehow this means that... You could do that, but that's not what I'm doing, unless one of them is at infinity. If one of them is not at infinity, I don't think it's what I want. But that's, yeah, I mean, I certainly was wondering about that. But if I'm allowing one of them to be at infinity, then I can identify an open set's worth,

25:00 and why I should say it's a kind of big open set, it has certain globality properties, a big open set of one space, that is the canonical one at infinity, with the one defined for P. And this will apply for any P, so long as it's close enough to I-class, if you like. I'm looking at the neighborhood of I-Plus, probably, when I do this. And it takes the scales over as well, so the whole twister space is identified. Now, looked at from the point of view of the twister space defined at Scry, what it looks like is that I drew a hole in it, and I don't have a picture which is doing what I'm trying to say here, so let me just draw it on the board. I have the twist of space at infinity, which is one thing here, which has certain global properties. I have another space corresponding to space-time twist of space, which has certain global properties. And I make an identification between these, which is sort of the following place. I take a whole, I remove part of this space, and neighborhood of that, and identify that neighborhood with something corresponding down here. I've got a hole from here too. And identify a cross from one to the other. Yes, but I can't, I don't expect to do this globally. What I expect to do, see, I'm trying to find something more general than what we had before, which was that here's your twisted space, or projective one, and I've got lines in it representing points in space-time. Now I'm not doing that here. I'm saying, I'm doing something a bit like thickening that out. I cut a hole here, take that hole out, which is what I'm doing here, and stick something else in it. So I'm calling that a surgery. So that the points in space-time are not identified with with these holomorphic curves, but they're identified with certain surgeries. And it's what's going on in this picture. And, uh... So we've got what is the picture?

27:30 Okay. Oh, it's in another building. Well, that's more serious. Well, um, do I have to stop now or do I have a few minutes? Okay. Okay, so let me, perhaps I should end with this particular family of questions. I should say that the forms that one has here, they don't give you the Euler vector field, but you can see how to build your space up patching together, preserving... I've lost some transparency, which says exactly what I was trying to say here. It preserves this product, and I think it's another thing which you have to preserve as well. And the freedom in that is these minus-six functions. The minus-six function shows up in the oiler vector field. When you patch from one patch to the next, the relationship between the oiler vector field in one patch and the next patch is given by a homogeneous degree minus six function. And it's a crazy-looking thing, but it does correctly encode the information. But let me just put these basic questions here. First question is, how do you construct the twist of space from this analytically thing? Well, I've told you that. That's this strange-looking construction. How do you reconstruct the space-time from T? Well, that's what I've been sort of hinting at with this surgery. The surgeries would have to have the property, and this is, there's some conjectural, importantly conjectural aspects, that is I don't know the answer, and they could turn out very wrong. But the idea is that the family of these surgeries, if you specify it right, exactly what you want to do, should be a four parameter family. It's like saying these holomorphic curves give you a four parameter family. This is something much more nebulous, but nevertheless, it's pretty, I think I can see how to make This family of surgeries has to be a four-parameter family. And that, each surgery corresponds to a point in the space-time. So that would be the idea of how you reconstruct the M. And as I say, there are conjectural, strongly conjectural aspects of this. And then you ask, why should the Einstein vacuum equations be implicit in this construction? And I think all I want to do here is to indicate a reason for why one should believe in three.

30:00 I think the key is number two actually, that's really, you want to make sure you get it tied down so that it's the right family of surgeries which correspond to the space-time points. But the reason for believing in maybe the third one there, oh dear, I thought I could I had it a minute ago and it's encluded down underneath this pile of things. So I can show you the structure. I said there were certain things, that's all I was looking at before. The form, as I told you, that's one of them has to be part of the structure. This form here is also part of the structure. And you need to specify that and that, and the freedom when you go from one patching to the next is given by minus six functions. The freedom on the fiber is given by minus six functions. Yes, and here's the other thing I just wanted to say. Why do I believe the Einstein equation is likely to be true? And here we just, this is just a program, I suppose, more than a proof of anything. Here is the, I've just written here, this is the twisted space here, and I take the product space of that with the different points. These are the different, each different surgery, you see, will give you a point up here. and then you form this differential form and you find that at least it looks as though it's Sparling's three-form, George Sparling's three-form. And the Einstein vacuum equations are the condition that that three-form is closed. And the way this is constructed, it comes out as sort of automatically closed. This is, I think if we've got to stop, I should probably, I can't say more about it. because, mainly because one doesn't know this is a four-parameter family. But you have to make sure the form, the interpretation of the forms is what it should be, and it really is Barling's three-form. But certainly, if you look at it in flat space, it certainly is. So the question is whether the deformations make it still a three-form. But that's the way I would see the vacuum equations being encoded in the constructions that I've been given. Thank you.