Einstein's Equation and Twistor Theory — Recent Developments

0:00 Peter Stein's equation and twister theory, recent developments. A talk by Roger Penrose at the Institute of Theoretical Physics in Santa Barbara on May the 13th, 1999. So, your question, yes? So, what is the, what are the goals of the Twister program nowadays? Is it to understand the Einstein's classical theory? Is it to provide a quantum theory? Is it to provide a new quantum theory? Yes. Well, now you've asked that question, because that's in direct conflict with what the special requests from the experts were, that I shouldn't waste any time on preliminaries, and she goes straight into the new stuff. No, but it's not a preliminary. It's a question, now I have to. Yes. Well, I should explain the idea behind Twister theory, which is the same as it always was, it hasn't changed, is to find, try to find some appropriate union between Einstein's general relativity and quantum theory. But it's somewhat kind of backhanded or through the back door sort of approach, because, as you would have heard from my lecture particularly on a couple of days ago, my view is that quantum theory will also have to bend, that one's looking for a union which is more even-handed between the two and not simply imposing quantum theory as it stands now on general relativity. So the idea is to look for things which might link the two subjects rather than directly try to quantize space-time structure. and so it sort of proceeds in a kind of not very direct way but I would say the basic motivation is from complex structure because quantum mechanics is fundamentally based on the complex number field

2:30 and one is looking for features of general relativity which also seem to reflect complex number structure just trying to make connections between these things It's not kind of a directed program in the same sense that most quantum gravity programs are, but its goals are basically the same thing, to try and find a union between subjects, but in a way which allows maybe not to take quantum theory exactly as it is, but to look for features in quantum theory which might persist and other features which you might expect to change, most particularly looking for things which are manifold structures rather than vector space structures, because quantum theory has this linear background, and one is maybe looking for things which are complex manifolds rather than necessarily linear spaces. But that's all very vague, in response to your question. Is it non-computable enough? Well, that should say that not only is it not obviously non-computable, it's not obviously non-quantum mechanics in the sense of incorporating state reduction, so that it's a long way from any of those goals. And the only thing I want to talk about today is much more how it fits in with general relativity, recent ideas for making Einstein's vacuum equations, not even that at the moment, vacuum fit in with the Twister program, and recent advances over the past, well, couple of years have certainly changed one's perspective on this problem somewhat. I suppose I do need, I was going to make a concession to those who are not experts by just saying something about the motivation, even though my colleagues who have requested me not to do this will be upset by this, but I won't take very long of it, I'll just take a few minutes. Thank you. Here one has space-time, light rays are to be regarded as the first approximation is points in twister space, and the light rays through a point represent a point in space-time, would be a sphere, and space-time I should say is four-dimensional, three space and one time, so the light cone, the directions along the light cone form two-sphere, and naturally has a conformal structure, and I'm identifying that with a Riemann sphere, a

5:00 complex one-dimensional manifold, and reflecting the fact that the Lorentz group, or the non-reflective Lorentz group, is isomorphic with the group of polymorphic transformations of the simplest complex manifold, namely the sphere, the Riemann sphere. And specifically, one makes a correspondence The key relationship in this correspondence is what we call incidence. Here we have space-time points in Mikoski space, just flat space for the moment. These R's arranged in that emission matrix. The room is small. Oh, yes. Okay, I'll do it here, yes. That may be giving away someone. This emission matrix represents the space-time coordinates, and we have four complex parameters being the components of the twister. matrix relation holds. For this thing to be a Hermitian matrix, in other words this to be a real point, one requires the equation here, that's V3 DC from here, which is the equation of this space here. I should say that these four coordinates are projective coordinates, that is the ratios of these you're interested in, which give you the points in this three-dimensional space, which is what's called projective twister space, complex projective free space, and points in this space which lie on this subspace here, given by that equation, represent light rays, possibly at infinity, points over here represent these complex projective lines over here, which are the Riemann spheres I was talking about at the top there, because a complex projective line is topologically a sphere, and that's basically the correspondence. The incidence relation tells you that this point lies on that line, or this point lies on this line, and by holding either the blue things fixed and letting the orange ones vary, in which case we get the line here, or holding the orange ones fixed and let the blue things vary, in which case you get the light ray over here, which is that relationship. So that's the basic Twister theory correspondence, in a nutshell. I want to rewrite the incidence equation in terms of two spinners.

7:30 That is, the Twister formalism is expressed in terms of two-component spinners very neatly, so I have to assume that people are familiar with the notation to some degree these are two-dimensional indices, the primed ones, complex conjugates of the others. This R represents the emission matrix I had before, and that's the incidence relationship. So if you hold the blue ones fixed, you get the space-time representation of a twister, which I'll come to in a second. If you do it the other way around, you get the twister representation of a point. These two pairs of components of the twister, omega and pi, are the two spinner parts of the twister. They play rather different roles, as I'll indicate in a moment. The equation I had a moment ago, which was this thing at the bottom of the previous transparency, finds its form, z z bar equals 0, where z bar is the complex conjugate of that. conjugates to the primed and unprimed indices interchange, and so I'm going to change into two of them also in my representation, so that this thing is a dual thing to that, that's a dual thing to that, and this is just omega pi bar plus pi omega bar. The things with lower indices, as you notice, that the complex conjugate is a downstairs index, therefore a dual object to the original twister, and its representation therefore in the projective 3 space is as So for twister, in twister space, projective twister space, this represents as a point. The dual twister is represented as a plane. And the condition for this thing to vanish is that that point lies on the plane. It's quite usual to talk about complexified space-time. That allows this R to be complex, in which case the incidence doesn't give you, as it did before, just a light ray, here I'm fixing the twister when it's on the space PN and letting the points intimate with it vary and you get the light ray in space-time. If we go to the complex space then you don't have, you don't necessarily need the condition of what I call null twister

10:00 relationship. You don't need that. And the representation of a twister is to say you hold the twister fixed and let the complex points vary, is what we call an alpha plane. An alpha plane in complex McCoskey space is a totally null plane. Totally null means that the metric vanishes completely on it. Any two tangent vectors are orthogonal to each other, including cells and so on. The matrix just disappears. It's what we call an alpha plane. If you did this corresponding thing for a dual twister, you get a beta plane. These are the two kinds of planes in complex Minkowski space, which are completely null. They're either self-dual or anti-self-dual. The alpha planes are the self-dual ones, if I have them the right way around, and the beta planes are the anti-self-dual ones. The alpha planes represent up to proportionality twisters. The beta planes represent up to proportionality dual twisters. Okay, so that was a very rapid description of basic twister theory. The relationship to physics, well that's just what happens when you shift an origin, I should just mention that. The pies are independent of origin, whereas the omegas change as you move the origin by this expression here. So the omegas are In fact, that's the physical description of these things, says in a certain sense that the pi's are momentum-like things and the omegas are angular momentum-like things. More precisely, you relate the twister pi and omega to momentum and angular momentum through these formulae here. If you take the spinner pi, multiple obites complex conjugate, you get something, I should say, when you have two indices, one primed and one primed, collect them together, and you form a tensor index. So this is a four-momentum, and the fact that splits the product means it's null. It also, because this is a complex conjugate times something, it is a future-pointing null vector. So the fact that P is a future-pointing null vector is incorporated in that formula there, And what I'm trying to represent is the momentum and angular momentum of a massless particle

12:30 in terms of twisters, these two conditions here being requirement for a massless particle. The angular momentum is this expression here in terms of the twister parts, the round brackets meaning symmetrization. Here we have omega times pi bar times the epsilon. This is the Levitschevice symbol, skew-symmetrical. And you add this to its complex conjugate order with an I in there, so you subtract these two parts here. This automatically gives you the relationship. Well, first of all, it's skew-symmetric automatically in the pair AA prime, BB prime. You need to change the two pairs. this follows from its expression, so that the m you get from here in intensive form is skew symmetrical. Moreover, if you construct the Pauli-Lubansky spin vector from it, you take the p and the m and utilize this object s as the Pauli-Lubansky spin vector, for a self-respecting mass is possible, it should be proportional to the momentum, s being the helicity, the factor of proportionality, which is a real number which could be positive, negative or zero, in this classical scheme here, it turns out, if you work it out, that this is actually the norm of the twister that I had before, with a factor of half. So that this quantity here represents the helicity. So that's the sort of physical interpretation of the parts of the twister. You see that pi is a momentum thing, and the omega together with pi as its angular momentum structure, as I've indicated there, you find that the origin does affect, the angular momentum transforms as it should under origin shift. Moreover, we have this quantization procedure, or if you might call it first quantization, the commutation rules for momentum and angular momentum, as you know, are a little bit complicated, but if you look at the commutation rules for twisters, naturally arise, you find that the z's are canonical conjugates of their complex conjugates. So you have the canonical commutation relations here. There's no i in this because this is

15:00 a complex conjugate and that's the right way around. And you find, working it through, that with these commutation rules, the P and M indeed satisfy the correct commutation rules from momentum and angular momentum. There is no factor ordering problem just because of the forms of these expressions. You do find that if you work out s, that now it has to be polarized in that way there. That just comes out from the formula that you have up here. That's all pretty well forced. Now, if you adopt, if you want to look at a quantum picture of a particle in a twisted description, it's going to be a massless particle because the massless relationship is automatic in this twister scheme here. You can take the z representation, twister representation if you like, or the z-bar one, you can do whichever you choose, take the Z representation, then the wave function of a particle represented in terms of twisters would be a holomorphic function because, as with ordinary quantum mechanics, you want to say you take one or other of these variables, you take the function of one of them, and the other one becomes differentiation with respect to that one. So your wave function is supposed to be just a function of one, not of both. That means it's a function of z, not a z-bar. So therefore, you have this relation, which is simply telling you that f is a holomorphic function, complex analytic function. So then you also have z-bar represents the derivative with respect to z, this formula here. And the helicity then, as an operator, becomes h cross over 2 times minus the homogeneity operator here, minus 2. because of the symmetrization that you have here, but it's basically just the z-bar, which is the Euler homogeneity operator. So what that means is that any function of a twister, which is an eigenstate, gives you an eigenstate of helicity, must be a homogeneous function, homogeneous and polymorphic function.

17:30 And then we have, I'm always trying to finish this as introductory stuff, And we have where you get the fields from the twister function. Here's your twister function with the appropriate homogeneity degree. And you find that you can write down contra-integral expressions to give you all the different massless fields. So this guy gives you another sheet here. I've written these all in two-spin annotation. In general, one has n indices symmetrical, satisfying this field equation. It's just a field, not potentials I'm talking about here. Helicity zero is just a wave equation, and these ones would be the... If you're looking for positive frequency things, these things are negative helicity, these ones are positive helicity. So you have the different helicities represented by these equations here, and just to spell out what you have... Oh yes, I just should say that the relationship between the homogeneity degree and the helicity is given by the formula at the bottom here, and if I just apply that in various different cases, you can see what you get. Scalar wave, you have helicity zero, the twister function is homogeneity to green minus two. If you take the Dirac-Val equation for the neutrino, massless neutrino, you have the neutrino with minus half helicity, one homogeneity degree function, and minus three if it's the anti-neutrino. For the Maxwell photon left-handed you get zero homogeneity for the left-handed one and minus four for the right-handed one, and the linearized Einstein graviton, talking about a master's field of spin two, one has minus two for the left-handed helicity, and therefore or plus 2 for the homogeneity degree, and minus 6 for the right-handed one. Notice this is extraordinarily lopsided. One has a complete chirality in this description. The right-handed and left-handed things are described by functions which really look quite different. So that it is a very chiral description. Of course, I could start all over again and use the dual twisters instead of the twisters, and then these things will be reversed. mind, and just use one or the other. Particularly, for example, if you wanted to describe a photon

20:00 which was plain polarized, you'd need to have both the left and the right-handed parts, and you'd need therefore to, I mean, you shouldn't sort of switch over and say you used twists of function for one and a dual twist of one function for the other, that wouldn't make any sense, you couldn't even add them. So you've got to make a choice, one or the other. and that choice, well, maybe nature prefers one to the other but we don't know that at the moment at the moment you just make a random choice and I tend to choose twisters rather than dual twisters just because it saves a word or two now, okay going back to the twister function the counter-integral expression In order to write down your contra-integral, you've got to balance the homogeneity degree of this function. It's minus 2s minus 2, where s is the helicity, if I remember that. And you've got a 2-homogeneity degree from the natural differential form that sits over here. And then you have to balance the rest by putting a lot of pi's there, or else a lot of d by d omega's there. Remember that in the quantization procedure, the complex conjugate of pi goes to d by d omega. You can never use the complex conjugates because that's not a holomorphic thing. You have to keep everything complex analytic, that's part of the philosophy. So you can either put pi's in, that's complex analytic, or you can put d by the omega's in, that's also complex analytic, but you shouldn't put bars. These contra-integral expressions, to understand what's really going on, what you want to do is fix a point in space-time, and that means in the integral here you're looking at twisters which are incident with that point. That's the incidence relation, remember, I started with. So you're letting the twisters vary, the omegas and pi's vary, keeping the r fixed, but this really means you represent the omega, you rewrite the omega in terms of r and pi in the integral, and you integrate out the pi dependence, and then you get a function of r. Now, in more detail, what you do is you'd have some function here, which is holomorphic in some region. It's going to have singularity somewhere. It's holomorphic in some region. The point in space-time is represented, remember, by Riemann's sphere.

22:30 I've drawn this as a sort of sausage here, so you can think of it as both a line and a sphere, which it is both, of course. here is this thing. It intersects the region of definition of the function in some region which, if you choose things right, will be an annular region on the sphere, and your contra-integral just goes around the annular region, and you find your field, and lo and behold, having done that, you find all these massive equations are automatic consequences of the twister expressions, the homogeneity degree, and so on. So it gives you a way of solutions to those field equations automatically not now because i'm talking about flat space so in flat space there is nothing uh to rule out about higher spin but that will be a an issue which comes up when we talk about curved space absolutely and of course if you have interactions if you start worrying about that, too. But is the only thing you inspire? I think so. I mean, there's an issue there which I haven't... One really needs to understand the curved space stuff properly before getting to that. Because what happens is that twister space gets curved. I'm going to go back and listen to that question again recorded very well. These contra-integral expressions, to understand what's really going on, what you want to do is fix a point in space-time and that means in the integral here you're looking at twisters which are incident with that point that's the incidence relation, remember, I started with so you're letting the twisters vary the omegas and pias vary keeping the r fixed but this really means you represent the omega you rewrite the omega in terms of r and pi in the integral, and you integrate out the pi dependence, and then you get a function of r. Now, in more detail, what you do is you have some function here,

25:00 which is holomorphic in some region. It's going to have singularity somewhere. It's holomorphic in some region. The point in space-time is represented, remember, by a Riemann sphere. I've drawn this as a sort of sausage here, so you can think of it as both a line and a sphere, which it is both, of course. here is this thing. It intersects the region of definition of the function in some region which, if you choose things right, will be an annular region on the sphere, and your contra-integral just goes around the annular region, and you find your field, and lo and behold, having done that, you find all these math equations are automatic consequences of the twister expressions, the homogeneity degree, and so on. So it gives you a word solutions to those field equations automatically not now because i'm talking about flat space so in flat space there is nothing uh to rule out about high spin but that will be a an issue which comes up when we talk about curved space absolutely and of course if you have interactions you start worrying about that too but I think so I mean there's an issue there which I haven't when one really needs to understand the curved space stuff properly before getting to that because because they what happens is the twister space gets curved and you can't write these expressions they're just straight down like that. You've got to think about what they mean. And that issue comes in. But yes, you're right. I mean, that's obviously a... On curved space. On curved space, yes. On curved space. Sure, on flat space, there's nothing wrong with these equations for all spins. They're just a general procedure. Well, there are twists of formalisms in other dimensions and in other signatures. They're not so neat. But you can certainly write down versions of contra-integrals, and these have been done. You don't have the same complex structures? The complex structures aren't so natural. I mean, you can complexify everything if you like and just talk about the complex. But you don't have in any other dimension the sort of initial motivation, which is the Riemann sphere

27:30 as being the light cone and the directions along the light cone so here is the only case where you have I think it's the only case you get a complex manifold so it's something very special and the philosophy is very different of course from those in string theory and so on and so forth where you want to have higher dimensions and you hope to do things with higher dimensions You're taking the dimensions that you've got, and you want things to come out from a different direction, not from higher dimensions. But yes, I mean, that's it. There's a natural suggestion that some people think, I wonder if you've thought about it, that a way to go to higher dimensions would be to go from the Riemann sphere to the analog, Yes, I would be very doubtful that that gives us something useful. The main reason is that with complex numbers, you have a nice function theory, and you get tremendous richness from the holomorphic function theory structure. If you talk about quaternions, then it kind of gets very stuck. I don't know of a good function theory in quaternions, not just that I don't know it, I mean, you can see that there is a good one in a certain sense, because you can write down the complex conjugate of a quaternion in a form which looks as though it's an analytic expression. So it doesn't lend itself to the kind of thing that one does in Twister theory. I mean, sure, that's a thing you can do, and there are higher versions of Twister theory But they don't seem to have the kind of richness that one is looking for here. It's very tied to the space-time dimensionality that we see directly, so that the philosophy is very different from much of current theory. That's the way it's gone, and that's the way I'm going. It's a bit like, well, it has something in common with Fourier transforms, yes. It's better if I, perhaps if I just go to the bottom of this transparency,

30:00 because it's, the point about it, what it really is, is it's a bit more like a radon transform, but it's a transform to cohomology. That's the crucial thing, because these functions, the twister function here, has singularities. You never get a non-zero answer for your contra-integral otherwise. In any case, the homogeneity degrees force you into singularities very quickly. But that's not, you see, you don't think of these things as functions. You think of these things as things, the final overlaps between open sets. So what you would do is... It's not a Fourier transform in the ordinary set. Well, it's a cohomological thing. Oh, it's the same general family of transforms in the sense that it's transform, yes. But it's not a Fourier transform. And the way to do it, you can do it using root theory. So you can realize representation. Yeah. So it certainly is related to group theory things, and there is a big theory which includes all this, which implies the other groups, bigger groups, and so on and so forth. Is it a particular case of that? Yes, it's a particular case of a general type of prescription of that nature, yes. But there are sort of special features which occur here, which one makes use of a lot. And one of these special features is that the cohomology we're talking about is first cohomology. That means that you cover your space in some way. Well, I've sort of indicated it here. The region of, there's some region of space-time you're interested in. You then, that corresponds to some region of twisted space you're interested in. It tends to be the top half. The reason for that has to do with positive frequency. remember this earlier Transfancy here, where the twister space is divided into two halves, and the top half has to do with positive frequency and the bottom half with negative frequency. It's like the Riemann sphere, cut into two, and half the functions go one way and the other half the other way, giving you splitting into positive to negative frequencies. So this aspect of quantum theory, quantum field theory, is very much taken over into the twister scheme. but what it means here is that you're looking at the top half of twisters

32:30 they say, if you're looking at a wave function you cover it with a lot of open sets maybe two, you might get away with two which is what I've basically done here two open sets, on the overlap between those two open sets is where the twister function is defined, it's not a thing on the whole space, it's a sort of transition function thing and this is important because when you generalize curved space or to non-linear interactions, Yang-Mills fields and things, this becomes an actual building of a bundle or building of a manifold. So it's important that it's first cohomology. So when people ask me about higher dimensional twister theory and so on and so forth, you do get such theories, but the cohomology goes up. For example, if you looked at six dimensions i uh i don't think about which it is exactly but but you find that you're looking at second cohomology and so on you have twisters you can have higher dimensional twister spaces um but the the specific dimension and the fact you get first cohomology here is very crucial crucially related to the space-time structure being four-dimensional so that it's all kind of knit together in that way if we talk about other dimensional spacetimes it's one of the reasons I feel uncomfortable with a higher dimensional scheme because I can't fit these ideas into them I'm not saying that it can't be done because maybe there are other ways of looking at this but I don't see how to do it now one's particularly interested in gravity here that's what I want to be talking about, and the linearized gravity, I've just written representing the curvature in terms of two spinners, one has an unprimed spinner representing the anti-self-dealed path and the primed one representing the self-dealed path, these being totally symmetrical objects. The Bianchi identities in the linearized case become the same equations that we had before, and you can represent these things by twisted contour Just as before, the two homogeneity degrees being plus two for the anti-self-dual part of the left-handed graviton, minus six for the right-handed graviton. Is the answer that the homogeneity degrees always have the minus one?

35:00 Well, that's the two in the formula. When I go back to the... Oh, sure. That's just this expression. See, that's the minus 2 there, and you've got one, and you've got the other one. You add them up, you get minus 4. These two cancel out because you're looking at the opposite. Yes, sure, absolutely. So, but anyway, at first it might not worry you so much. Okay, two numbers, and one's positive, and the other's negative. But when you start to look at non-linear things, you find these different homogeneity degrees play crucially different roles. And now I want to describe, I'm afraid it's still old history, but at least it's non-linear, general relativity. Well, there are numerous applications of twisters which I don't want to go into. I'm talking about the general program here. and in I think 1976 I produced a construction for anti-self-dual or left-handed gravitons what I call non-linear gravitons because these are solutions of the Einstein equations which are complex but which satisfy the Einstein equations exactly the vacuum equations and they're anti-self-dual so I regard these as non-linear left-handed gravitons and you can obtain these things from a twister construction by deforming twister space remember twister space for flat space-time was a complex projective three-space and what you're going to try and do is deform that well, there's a theorem which tells you you can't deform the whole thing which is just as well for reasons and you can see why it would lead to a contradiction but what you can do is deform the tubular neighbourhood of a line Why are you doing that? Well, you see, this line in the flat case represented a point in space-time. And what one's looking at, look there, the line represents a point in space-time. So you're looking at a neighborhood of a point here. So just looking at local geometry for the moment, some neighborhood of a point, finite neighborhood, that will correspond over here as you let the point wiggle around in that neighborhood, this line wiggles around and sweeps out a tubular neighborhood. curved little piece of the space-time here, I would try to form that tubular neighborhood of a line. And indeed you can do that. Basically what I've indicated

37:30 here is breaking it in two along the line, sliding one piece across relative to the other one, pasting them back down again, everything being holomorphic, you've got to make sure these are complex analytic functions that are used in this gluing. and now this line which now is broken you have to find another one which isn't broken in other words a holomorphic curve which belongs to the same topological class as the one you started with it has to be a a two-sphere it has to be a holomorphic Riemann sphere there are other ones which wind around more than once you look at one which is actually belongs to the same topological class as that one You will find by theorems due to Kodara and company that as long as this deformation is not too outrageous, you will find a four-parameter family of these curves, a complex four-parameter family. That's a rather remarkable fact, that just the condition that it launched the right topological class and is a closed thing, is a compact polymorphic curve, means that the space that represents these things over here is a four-dimensional space. mean by these things being null-separated, that's just going back to what one said earlier of space-time, that these are null-separated if these lines meet. That's a consequence of the flat-twister theory, and I just say the same thing here. And then you find, rather surprisingly, first of all, that you get the space here has a natural conformal structure, which is quadratic, and that you can then work out the Weill curvature, which is conformally invariant, and you find that the Weill curvature is anti-self-dual, and this construction is reversible. If you start with an anti-self-dual Weill curvature space, at least a small enough neighborhood, you might have global problems if you went too far away. You can reverse this construction and you find a curved twisted space. Now there are no Einstein-Field equations But the Einstein-Field equations can be imposed very easily in the presence of the Vauer curvature being anti-self-dual. And in twisted space, it just comes out as this,

40:00 that you have a projection of your twisted space down to a complex projective line. There's a slight extra thing you need to say, but this is basically it. and this gives you a Ritchie-flat complex space-time which is anti-self-dual, in fact the general one. So this, you can actually construct the twister space constructively and this gives you the general solution of the anti-self-dual Ritchie-flat anti-self-dual Einstein equations. The twister space is constructive. What's not so constructive is actually finding these holomorphic curves. found a Twister space, that's the easy part. The hard part is to find where these curves are, and that's not necessarily an easy thing to do. But in Twister theory, you don't care about that, you work with a Twister space anyway, so you don't necessarily need it to be easy to find the space-time from it. So you began here, of course, with McCarthy space. So the global projected space-time. And then you looked at the Stigular Nemo. So for part of your space-time, you obtained a solution that's going to labor at this point. I would do, yes, but since this is analytic, it's an analytic continuation. It's got all the information there. So you've certainly not... The only reason I'm doing a look for it is you might run into non-Hasdorff problems. But is that the only problem? Yes. Yes, I think so. The manifold, the metric might be non-Hasdorff, otherwise it wouldn't be going to get it. It's really the other way around, I was worrying about. Yeah, the metric might be non-Hasdorff, But I suppose I was thinking of your question the other way around. Let's say if you had a space here which was anti-self-dual and you tried to construct the Twister space for it, you might find the Twister space ended up as non-Hasdorff. And you certainly do get things like that, but non-Hasdorff-ness is not regarded as a bad thing in this subject, provided it's mild. In fact, it's a very useful thing, you find. You can have things which branch and so on, that sort of thing. I have at the back of my mind that maybe that's a good thing for state rejection, because you have things evolving and then it does one thing or the other but that's just a romantic thought I don't have any I don't have any direct association I'm forgetting, is there a reality condition so you can look at say I mean you wouldn't get a Lorenzian yes, you're not likely to satisfy it here that's the trouble because these

42:30 yes, in fact there is a whole body of theorists and people on the other side of the corridor from where I used to be in the maths institute and people who are interested in geometrical questions and they're usually interested then in positive definite spaces and then yes you can have anti-self-dual or self-dual and that's just a condition on that space. Do people know enough about those deformations to do math with? Oh yeah subject. Well, Atiyah and Hitchin and Singer wrote a paper where they did my construction over again in the positive definite case. And they look at it a different way, in some sense. You look at the twisted space as a sort of bundle over the space, over the space-time and the polymorphic conditions come out as integrability conditions on the almost complex structures and things. But it's the same thing, but written in a different signature. So there's subject when you do it that way but but yeah it does carry over okay reality condition areas yeah then the reality condition you can impose on how you yes there's a sort of hematicity condition on this and there are these uh things that they would hypercalomanifolds and where people treat hypercalomanifolds and things it's a big subject Do the solutions have singularity? Quite likely, yes. Sorry? How does it square with the animal? Yeah, yeah. It's just like a pole in a central singularity in a holomorphic function. Let's continue. Is there a control on what deformations you would have to do in order to get a global of the needs of the solution? Well, certainly there are constructions for global ones. i mean they're more like like sort of particular examples of people found uh and people like nigel hitchin who certainly played with these things and there are things like the the yao um well and it might relate to things in string theory because you you have the these complex manifolds that come up and and uh again many of these things have have twister related constructions for them So we're looking at the, well, what was it? It was the, yes. The deformation you're doing is just the definition of the complex structure. Yes, that's right, yeah. And I'm doing it, you can do it two ways.

45:00 You can either take the J, which defines the complex structure here, and deform that, or you can do it in the check way, which is what I just like better because I can see it, what's happening better. The equivalent. You can do the sort of double-type approach where you vary your complex structure and then, yeah, you just go through that. Is this designed by fact you have sort of like an inter-dimensional space? Yes, that's correct. You can characterize and you take it to general and all the solutions and characterize how many there are. Yes, well, it's an infinite, as you said, an infinite dimensional space. But you'd expect that because you're talking about solutions of the Einstein equation. on some region which has no compactness specified on it. Yes, sorry. Yes, yes, you can take a linearized one and exponentiating. Yes, well, this is, well, yes, well, yeah, I mean, it comes back to, just go back to the picture I have here, you see, where you have a lot of open sets overlapping. These are just painted on space here, when you do the contra-integral, but you think of them, I mean, the way I like to think of it is the paint dries, you see, and it deforms the things a bit. that really what it's doing is the paint is doing something active and what it does is it tells you how you're brewing these things together or it slides one over the other and that's really the way I'm thinking of it so, well I don't know, I can show you that one if you like or another one which is more or less saying the same thing this is just showing you where the if you start from the anti-self-dual Ritchie-Flatt case which is why it all works out so neatly. And you have these different parts of the curvature disappearing, different parts of this commutator on the pi. And that integrability condition on the pi is really what you're telling about this projection. In fact, I said there was a projection down to the pi space, and that translates into that geometrical construction.

47:30 I think probably it's better if I show you the other transparents, which are saying more of the same thing here. I'll come to that bottom in a minute you see there is, as Alpi was just indicating there is a relation to the linearized theory where you had these twister functions homogeneous to a degree plus two here's the twister space and on that overlap between this region and this region you paint your twister function as a degree plus two but then you let the paint dry and it sort of distorts things and the way we think of that 2 function is interpreted as an infinitesimal shunt between one and the other. So here we have, this is just an infinitesimal deformation of the complex manifold, but the deformation is achieved in the way I've written it down here. This is really a vector field on base, and as I was saying, you can imagine integrating that vector field, to give you a finite deformation, which gives you a finite, a genuinely non-linear solution of the Einstein equations in the anti-Selvideal case. This is a vector field here, I should say, right there. And notice that f, there are two derivatives in that, and if we start with homogeneity to degree 2, then having two derivatives, it brings you down to something of zero homogeneity, an appropriate thing to somehow the homogeneity is balanced and so the plus two, you can see the particular rule that plus two is playing and you can use the same function that you had here, do the contour integral work out the psi and indeed that is the correct linearized gravitational field corresponding to the infinitesimal deformation that you have here so that all ties together very nicely and that's in a certain sense half of the Einstein equations not a very good sense I would say because it's probably you have to exponentially the point is to try and find the other half a geometrical construction for the other half and to put the two together so that you will have a vacuum solution which incorporates both the right-handed and left-handed parts of the graviton

50:00 but notice you see we've got to incorporate a minus 6 degree and it's one of these irritating problems that has been around for over 20 years I've lost track of how many years it is maybe 25 by now perhaps not quite over 20 years and it's well it's taken 20 years and I feel now that we're making some progress and the problem really is to see how to find a role for the minus 6 functions of the space. Well, the way in which it's going to work is that we somehow split it into the right and left-handed parts, and I'll show you how we can do that, and that the left-handed part is just the same as it was before. So you have a projective three-dimensional twister space that is essentially the same construction that I was just talking about when I was making these definitions of spaces and so on. I just do the same thing as before. And then the rest of the information is contained in how the non-projective space sits over the projective space. And the minus-six functions are encoded in some kind of distortion of this bundle. Did the nominee graviton, you needed the whole tristice space, The way it's actually done is to use the whole thing. Yeah. It depends on who you are, because people, the pure mathematics, I like to talk about some bundles, they tend to talk about things done on the projective space, but it's the same thing. Yes, you're right. In fact, there'd be a little bit more... So I'll certainly put another transparency up here, because this is, basically we use these forms. In fact, they're going to be important, what I want to say. Basically what one is looking at is a one form and a three form. This is the same thing I've been talking about before. Well, I've got loads of transparency, and I wasn't quite sure what orders this presented to people in. But this is, first of all, flat space, and that's these equations here.

52:30 And there are forms which come out naturally in flat space. And in particular, when you do the contraintegral, I think I wrote pi d pi somewhere in my expression. And that was, that pi, I have all sorts of different ways of writing that. That just means pi contracted with d pi, which is the thing I've written at the top. infinity twister, which is another way of writing things, but it's basically that expression there, that's pi d pi. But that is the form iota. So iota is the expression that comes into the contraintegrals in the way one normally writes it. Sometimes one writes this not in homogeneous form, but in homogeneous form as a non-projective twister space, and the advantage is in doing that. And then you take the d of that, which is a d pi d pi, this is half the exterior derivative of iota and the I use the particular Greek letters that are indicated here because they're supposed to have a mnemonic nature otherwise I'll never remember them see tor is a two form theta is a three form and phi is a four form well I tried to think of you know something else for one form but it seems to me iota will do no I can't use only because I've already used that on the the twister, the spinner part, you see, so that's, maybe if I started all over again I could, but that would confuse me. I don't want to confuse myself anymore. These are the various forms, okay, there's a natural three-form, well the four-form is really the volume form on the twister space, and the three-form is the natural volume form on the projected space, which is a weighted thing, but it's this thing here. and those are the various forms and the fourth form is a quarter of the exterior derivative of the three-form now these all exist in flat space but in answers to your question yes, in the standard construction you take those over into the non-projective non-linear graviton four-space and it's easiest to talk about the construction comes from, and so on, in terms of that. So I agree, that's what you would do normally.

55:00 But I suppose really the answer to your question is that in the standard nonlinear gravidone construction, you do use this space here, but the bundle is sort of trivial. There's no information in it. It's just a natural bundle that you can grow out of that one if you want to. So there's no extra information in that. Now we have the general standards of which gives us the subdual solutions, and now we want to get at the subdual ones. But are you now starting with the deformed information and trying to do something on it, or are we back when we come and stay there? I'll come to a little bit more exactly what I'll do in just a second, I think. I was only just trying to define the forms for the moment, but I'll tell you what's I think just to get this can be flat space or it can be the old one in the graviton which I'll call the leg break graviton cricketers will understand why because the googly is the other is the right handed one and the leg break is the left handed one don't worry me about the notation if you don't know anything about cricket I don't know much about it but this is the leg break construction it's the old fashioned one years ago, or flat space. Flat space, I wouldn't have these explicit expressions, or I could even use those in local coordinate neighborhoods. But they're always related by these differential relations there. And I want to indicate particularly the Euler vector field, which is this homogeneity function, which came in as defining the helicities and so on, can be thought of as simply the four-form divided by the fruit of the three-form. What I mean by that is that if you have a function, A, then D of A wedge, the three-form, is equal to the Euler vector field acting on A times four-form. So if you like, that's what E is. This is divided by this, so to speak. The three-form, of course, gives you a direction field on a four-space, but you want to scale it back, so it's actually a vector field, and that's what I've done here. There are these relations here which satisfy, that is, the one-form and the two-form wedged together are zero, which is just the

57:30 Hoojie-Flip relationship I knew I'd forget it at the critical moment, which tells you that the one-form things knit together to form three surfaces. thingy-jig relation Frobenius that's what I want the iota wedge d iota iota is zero that's Frobenius and so that's this thing here but then iota also wedging theta is zero I can give you the geometrical thinking of that in a minute but let's not worry too much for the moment I'll come back to the bottom of that stuff in a moment to answer Abheye's question, what am I talking about? Well, the way to think about what I'm talking about most precisely is to think of a rather specific type of space-time, and how one would generalize this to a general space-time we'd need thinking about. But let's talk about a space-time which is asymptotically flat in the strong sense that it even has a good I+. So whereas in this picture I've indicated with a lot of spots there that there might be a source somewhere in the middle, maybe there are black holes hanging around in the middle or something, I don't want to consider those for the moment. I mean, sure, I'll want to consider those eventually, but just to keep life as simple as I can, let's consider a space-time which radiates away and has a good point at the top here. So this is fortunate that we know things exist like that because of the work of Helmut Friedrich, so at least it's not a factual subject I'm addressing here. Moreover, these things can be analytic. They better be, for at least what I'm going to start with by saying. Again, one will get rid of that condition, presumably, too. There are all sorts of things that one would like to improve on this, ways in which one could improve on this, perhaps, but for the moment, let's say analytic, anthropologically flat, so you've got a good out, sky plus, and also a good point at the top, I plus, so it's just waves floating away, disappearing and leaving nothing. I might even see a brittle wave, I suppose, That's the kind of thing we're talking about here. Now, analytics, though, you can complexify. Of course, on Scry, one could have a retarded parameter u

1:00:00 and an angular complex parameter going around there, a stereographic parameter. When you complexify, u becomes complex and zeta becomes separated from its conjugate. So I call that zeta tilde. and I write them as a box like that just to make life easier to visualize. This is a complexification of scry. I should say that there are likely to be holes in this phase too. If you wander too far away from the real slice you're going to have singularities so there will be a kind of diagonal of this and you're really talking about the neighborhood of that. I won't worry too much about that for the moment. Although it plays important roles in what you can do and what you can't do. Now, the sort of origin of all this is Ted's H-space, and I'll relate it to Ted's H-space in just a minute, but how we can define in such a situation, even without I-plus being a well-defined point, you can have a hole up there too if you want, so far that would perfectly all right. You can define your twister space as the space of twister lines, or alpha lines, I might call them. Let's say twister lines. Those are curves on complexified scry, which would have been the intersections with alpha planes. And remember, an alpha plane in flat space represented a point in projective twister space. So this is doing the best you can. You may not have alpha planes because the space may have conformal curvature, but nevertheless, you can ask the question where would this line be if there were alpha planes? Can you write down an equation for this line which would follow where at the intersection with an alpha plane and does that equation still work in the general case the answer is yes in fact in this case it's just a nausea desic what you find is that the nausea desics on complexified scry are of three kinds are the generators which are these purple ones going up like that I'm sorry Ted the colors are probably going to you know all this stuff anyway so you don't mind these vertical ones are the generators The ones which lie in beta-planes on Scribe, Scribe actually has complete beta-planes and has alpha-planes too, well, complete is a bad word here, it has beta-planes and alpha-planes on it, and on the beta-planes, the null-GD6 which lie on them are twister lines, the null-GD6 which lie on the alpha-planes are dual twister lines.

1:02:30 and you can then construct I should say it's a three parameter family of these things so you construct a complex manifold that complex manifold is a twister space a projective twister space now I should tell you what this has got to do with Ted's H space by putting H space up here you see Ted's way of looking at this was to think, starting off with a complexified scry and looking for cuts, cross-sections, near the reels, but not actually real, which were given by the vanishing of the shear of the null surface which would intersect this local infinity. I never can remember whether it's with a bar. Anyway, I think this was H-space, yes. Yes, that's right, I've even got it on the transparency, just to remind myself. If sigma equals naught, then this indeed is what Ted defined as H-space. However, for Twister theory, I would have preferred him to have done it the other way around, or he would have preferred me to do mine the other way around. That is the H-star space, where you take the complex conjugate of sigma to be... always confusing. Sigma is a complex quantity, so when you complexify, it gets separated from its complex conjugate. Sigma, then, is still a holomorphic thing. Its complex conjugate is another holomorphic thing, but it's not the complex conjugate of sigma anymore. It only was on the real slice. It's wandering off to something else, and we call that sigma tilde. So don't worry about that distinction. It's only just a question of getting the signs and there's a handedness right and so on. But these cuts, which are global, appropriately global, give you the points in H-space. Now, what's the relationship between H-space and this twister construction? Well, what it turns out to be, as you take the twister lines, as I just defined them, and these are represented by points in the projective twister space,

1:05:00 if you have a good cut, that means you have a family of these twister lines which in this space here gives you a compact curve which is a Riemann sphere so you have a closed you have a closed curve a compact curve, a Riemann sphere and remember the twister construction of points in the space-time in quotes coming from this space was to take these holomorphic curves this projection, that's another way of thinking of it. So you have the projection down to the pi-space, you lift them back up here, and those curves are the points in h-space, in fact, and if you translate that over here, see as you move that point along here, it gives you a family of these twister lines, you have the cut ruled by twister lines. That is just head 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 multi-v6 Is this connection between head-to-eight states and twister states leading to a corresponding connection between quantization that carries twister lines? I should ask Ted No, I'm just doing at this moment I just reformulated I'm trying to do very something very similar to a logic reformulating that's right we had a paper together oh that's it you're not going to be off the question in this picture you never really needed this you just said somebody uses science together with some function oh yes that's true you're right 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

1:07:30 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, 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 that we're thinking of H-spaces, is you start with an asymptotically flat space and try to extract from it a sort of space-time, a complex space-time, which has the anti-self-dual radiation of the space but where the self-dual part has been put to zero. And so 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, of splitting it into the two parts which you might not naturally see how to do otherwise so that's that's why that's where this comes in yeah there's nothing special about the self-dual no no space time and that that was the original idea and the original space-time was not uh meant to be anti-self-dual or self-dual it was just a radiating space-time yeah you could take That's correct, yes. Yes, sure, 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. 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...

1:10:00 So it's going back to the original ideas a little bit more. But, yeah. Okay. Now, the twist of 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 the way. This is how to encode the googly, as well as the self-deal information. Now, let me just, first of all, instead of going back to Jim's comment, you can produce the non-projective space-time. See, this only gives you the projective space-time 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... It's just very similar to... Yeah, I was just... Yes, that's right. Yes, that's right, yes. I'm sure it relates to a lot of things that you do and so on. 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 G-D-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 geodesic. So it is a geodesic on scribe.

1:12:30 Now, what one normally did, what I would have done, up to about a year ago, was to put zero on the right-hand side of this equation. So that the natural thing to do is to propagate the spinner this is naught 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 naught 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 can then talk about the construction using the inhomogeneous, using the non-projective space, which in fact is what made me happy too. But now, what we do is the following. We take this crazy-looking equation here. It may look a little crazy, but 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 vile curvature written in spiniform, and then, oh, which one is? This is the self-dual vile curvature written in spiniform, and then And you define this object by taking the conformal factor. You need a conformal factor, of course, to make a square finite and so on. You take a suitable conformal factor to make square finite. And the VAR 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. that's conforming there. This is 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 power 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. if you choose a general conformal factor,

1:15:00 they just scale away from each other, but this one satisfies the mass of 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, and it should, because scry, conformally, you don't care. I mean, mass of field equations don't particularly care with scry, they just go shooting through it. And that was one of the initial reasons for thinking that scry was a good idea. but anyway, here we have this scaling so, okay, this is the basically the component of the VAR curvature in the direction of the generators but you have to bring the scaling in because the VAR curvature is zero and this is the derivative in that direction then, to make everything balanced I put the pi 1 to the minus 5 See, you'll count the number of one primes, you see, you've got four of them there, five 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. What do you mean, was this sort of a second derivative of the shield? Yes, this is already the second derivative. It's already the... how many is it? First derivative, third. No, this is the second derivative of the shear. No, I don't know. Is that true? Psi 4? Psi 4. It's going to be the second derivative. Second derivative of the shear. That's right. That's right. Second derivative of the shear, yes. It's a third derivative of a metric. I'll tell you why that'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 that looks a little funny, it is in fact really quite a natural thing. I1 is just this component here, and all these things I think are... ...explanatory. Sorry? You could say it's a Sherlock Holmes principle. Well, it might be, but it does tie in with other things, you see, that was the, that was the thing, but it, yeah, making, I should make this comment, or should I make another one first?

1:17:30 This is another thing with a triple. Oh, what's the time, I've, oh, I've got my watch, and I've hidden it under my transferences. Oh, I see, okay, yeah, right, so we'll try and, uh, tidy things up by then, solve the Einstein equations. complete generality and that sort of thing I don't know. Let me tell you something else first. So I'll come back to that one. What is actually a useful thing to think of is not simply to do this at scribe, as I've done here, but to do it also on a finite light cone. In fact, most of the things I was saying will work just as well on the finite light cone, but certain slight differences. This is the finite light cone. are complexified, so it seems analytic and all that. Now I've got microns instead of iotas, this is a notation, pointing up 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-Fieldspin 2, holistically 2. This object here is the conformally invariant thorn operator, which is to be found in spinners and 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 would be maybe 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 find this is the quantity I actually, perhaps I should show you that transfer that's this one that's the definition of the thorn conforming that thorn using spin coefficients and so on Ted will tell you what all those means you've forgotten and this is what you find Keikhoff-Dantemar formula for a master's field. It doesn't have to be a spin 2. It can be any spin. On the cone, which needn't actually be a cone if you have a messy-looking excuse

1:20:00 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 of the scry, you see, and what I've just said before. This is just to explain it. It is quite a natural-looking object. What are you going to do with that equation? We really accept that you want that equation. What do you mean by that? Well, what I do is I... Well, yes, I've done this, actually. What I do is what I did. But I haven't said it. That's my problem. You take the solutions of that equation rather than the solutions of this with a zero. And that... Those are the points of my twisted 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 different, 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 the fiber. So it doesn't look as if I've done much, if you like, but I have. Each point on the fibre 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. that is grossly inhomogeneous in pi, and that's really why it's doing something. Sorry, Lee. So you, for each starting pi on Sprite or on the, you know, according to that point, you make a one-parameter family of pi, and that doesn't mean that it's a copy of the Well, no, it's not. You see, you don't really have an actual starting point, which is why it's doing something, in a sense. Because, uh... 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 the equation is all right. So it's just a solution of the equation. On the cone? On the cone, yes. On the cone. Yeah. Yeah.

1:22:30 Does it have to? Okay. So it's going to say analogy that is on the right cone? It's not actually, yes. That's the point I should just make here. On the right cone, it's not always analogy. It's usually not. It depends on the shear of the cone. If the cone is non-shearing, So why don't you just look at both, that's why, but I can look at Sky first, yes, sure. Let's put in this transfer and sit back. Yes, yes, that's right. And you might think it's not done anything, because you could just take some starting points, but this is a, it's a chromological thing. That's the important thing, is that 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 in the space of the solution captures the... It captures the self-dual part of the rain over it. 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. Right. I'm now going to switch this tape over to the other side, to side B. At this point, 1 hour 17 minutes of a total recording of Penrose's May the 13th, 1999 talk at the Institute of Theoretical Physics on Einstein's equation and twister theory recent developments have passed. That's 1 hour 17 minutes out of a total length of 1 hour 30 minutes and 38 seconds. What you'll hear on the reverse side, side B of this tape, will be first a repeat of the last 17 minutes exactly, only everything from one hour into the lecture till the end. So, the last 30 minutes, 38 seconds of the lecture, of which the first 17 are a repeat of what has gone before. So, turning over to side B now.

1:25:00 Thank you.