Subsequent discussion (last part) / The role of topology in quantum gravity

0:00 So again, that's a question we can learn from this difficulty which holds Einstein up for two and a half years. So it's not a trivial problem. And again, I think many people who study general relativity today don't understand that problem. But there's still a third problem, which I call the no metric, no nothing point of view. On the basis of, again let me quote Einstein, on the basis of general theory of relativity, space, as opposed to what fills space, has no separate existence. If we imagine the gravitational field, that is the functions G-I-K, to be removed, there does not remain a space of the type of special relativity, but absolutely nothing. Also not topological space, by which I understand. For the functions G-I-K describe... That's a very serious one that physicists... That what? That the metric is what makes the matter... Well, I'm going to challenge that. For the function GIK describes not only the field, but at the same time also the topological and medical structural properties of the manifolds. There is no such thing as an empty space that is a space without a field. Space-time does not claim existence on its own, but only as a structural quality of the field. It required a very severe struggle to arrive at the concept of independence and absolute space, indispensable for the development of theories, referring to Newton's, the development of the common idea of Newton. It has required no less generous exertion subsequently to overcome this concept, a process which is by no means as yet completed. And I think, again, he means that. One hadn't adopted the point of view of no metric, no nothing. And if you look at the way we... Mathematically, to formulate the general theory of relativity, we seem to have not yet reached the stage where we have overcome that. We start out by introducing a manifold, right? And then we put structures on that manifold. Is that the way you solve the field equations of general relativity? You solve in a patch, right? And then you try to maximally extend to... You don't know before you solve the equations what your manifold is. So you're really doing, you know, you're pulling a swindle when you tell students that what you do is first you solve the manifold and then you solve the equations on the manifold.

2:30 Just even the fiberbond looks a little different than that, because the fiberbond assumes you have a base manifold. So I say an outstanding problem is how to formulate the theory, you know, the rigorous mathematical formulation of the theory, in such a way that you make it clear in that formulation that the manifold, the maximal manifold, whatever criteria you use to define the maximality, is part of the problem of solving these equations. No, there's got to be more to it, because you know that there's more than one method to it. Yes, but that doesn't, no, I'm not saying one topology, one metric. But what I'm saying is no metric, no topology. You don't start out with topology and then look for the metric. You look for a local solution to the field equation on a patch, and then you see how far can I build out that patch. But you know there are all sorts of identifications you can make on a patch. Okay, but I don't start out with the topology and then say what metrics are compatible with this. That isn't the way to work, is it? I mean, I think that's the theory we've got. Well, I mean, I don't think it is, David, because you don't, you don't actually operate that way. If you might look for it, there is... Oh, I understand. Lee wants to say something online, though. Yeah, I just think that this goes back to the, the government seems to just fear that Einstein has a class that he can't resolve.

5:00 He seems to throw it back to the legacy and his mathematical legacy. You know, it's a mathematical, the legacy of the machine is somewhere in there, you know, from the mathematics. I mean, the same thing the other day. I forget how, how did the people survive the years of living with both string geometry and the culture of the human universe. Now let's do the same thing here with that. Einstein had the philosophical legacy of Leibniz and Mach that space could be relational, but he had a mathematical legacy of absolute space, and the only thing he can't resolve that. I'm not talking here about going to an ultimate theory that surpasses general relativity. I'm just talking about what we actually do in general relativity now. And it seems to me like I once investigated the Curzon metric, right? I start out with a coordinate patch and then I try to see how, you know, what topology is compatible with that patch and I try to extend it out and out and out. And that's the way it actually works. And there ought to be a mathematics... That's where it starts. It starts with a variational principle. You see, I've got this integral over this panel. I can look at all points and all different kinds of metrics and then optimizing them. If I'm interested in the spherical symmetric solution, I don't take manifold by manifold and say can I put a spherical symmetric on this one, can I put a spherical symmetric on that one. I take a patch and I say how far can I expand this out? And if I'm smart, I come to the crystal solution. And I'm saying we ought to find a way to formalize that. Maybe, I've speculated, maybe sheath cohomology will do it, but I'm not glad that yet to say. Okay, that's another good answer to that. All I want to say is that the first thing is the intersection and the objects can be sub-valuable and the actual points. You don't have enough pure topology, the topology results from the...

7:30 I'm talking about mathematics. But then it's a bit fluid. Do you understand the point about the... It's not the points, it's the complex. This is kind of the experience that I'm seeing the... I'm trying to put an entirely different interpretation into that statement and see if anyone can help me. Suppose that you take the metric out of the manifold, the still mathematical amount of the manifold is. How do we decide that chemical, by any physical process, that a point A is in the neighborhood of a point B, or that something is an open set? It seems to me that this may be the interpretation of the statement, rather, how do we start in solving the equations, namely, when you take the metric off, what is the physical process by which we determine the consensus of the goal? How do you deal with the Lorentzian metric? Expose it to null. Well, that's fine. Now, that's fine. There's no separation of points. You still mean something physically. You know, it's a possible life propagation there. But I'm going to finish up with a sentence or two, and then you go on. It's on your time. You can't tell me anything. The moral I want to draw from this, if you accept this point of view, I think this is related to what Chris was talking about this morning, although I'm not sure if it's quite identical to anything he said. But it's something related to what he was talking about. If you take this point of view, there's something fundamentally local about general relativity in the way it starts. And that you then build out from that. We have here another fundamental tension between the basic approach of general relativity and quantum mechanics. Because quantum mechanics is somehow basically and fundamentally global. It doesn't make sense to talk about the wave function on a patch or the state of a patch.

10:00 You have to probe the whole manifold to do this. And our way of treating general relativity and mathematics is by starting out by assuming the manifold and mask that. Developing a formalism for general relativity, which showed that locality from the beginning, would at least bring out this tension more honestly, and you'd see the fact that there was another fundamental tension between the general relativistic approach and the quantum analytical approach, which I have no answer to, but at least we should face it. I mean, this way of thinking also suggests other possibilities. Maybe the patches won't fit together into a manifold. That's why I think it might be connected with the time. Maybe a manifold is only something that's going to happen in certain special cases, and you can think of it like a classical limit, and if one starts to think in these ways, if you don't build a manifold in at the beginning, then maybe you're better off and want to get rid of it at the end. Okay, now it's on your time. I have finished. All of these are very complex regions on neighborhoods, and then add them up in a way of doing a sum of the paths, but not doing it the way that I'm comfortable when I go into the community with, parameterizing them in math. Yeah, but if I have, let's say, you know, instead of a two-squared square, I have an eight-squared square or something like that, right? I can't take the amplitude from here to there if I want to do one or two squares, right? I've got to take the amplitude for every square. No, no, no, no, I know, but the point is that I can compute it through a series of local regions. Oh yeah, but you need all the patches at the beginning. When I'm saying it's the thought work of general relativity, you only need one patch. It's something basically local about general relativity. No, but if I was doing WKB, solution of eternity, I'd start with a local solution and extend it through the space. What if you automated one pair of photons in a correlated state to move far away from each other? Exactly. Exactly. And you can't start with one state locally. And from the beginning of the universe, you can't have... There are many photons that are correlated with each other, you know, the Hubble distance of time, all through the whole universe.

12:30 On to the movies! Yeah, but if I have, let's say, you know... Instead of a two-square square, I have an eight-square, something like that, right? I can't take the amplitude from here to there if I want to switch, right? I have to take the amplitude for every square. No, no, no, no, I know, but the point is that I can compute it through a series of local regions. Oh yeah, but you need all the patches at the beginning. When I'm saying it's a thought work of general relativity, you only need one patch. It's something basically local about general relativity. No, but if I were doing WKB solutions, I'd start with the local solutions and extend it through the state. What if all you need is one pair of photons in a correlated state that move far away from each other? Exactly, exactly. And so on and so forth, and so forth, and so forth. I ask you if anyone wants to make a short contribution in the discussion period tomorrow, what is it, 11 to whatever, 11 to 12.30 to let Chris Isham know about that. Okay, last chance, yeah, last chance to howl. Okay. Right. Okay, so it's...

15:00 It's very appropriate that the session on topology got split, so we're now in a non-connected session. So first we have John Friedman on the role of topology in quantum gravity. I hope I'm not in the incoherence sector. Alright, so let me start. This is for Andrew's benefit. Some people are profoundly depressed by this. I'm not. It's for Bill Unruh's benefit who told me a couple of years ago how much he disliked them, so... So let me start with a quote from one of the sages here saying that once you abandon Euclidean geometry... You think immediately of abandoning Euclidean topology as well, and Biles suggesting that in relativity one already has the possibility of topologies with, of manifolds with non-Euclidean topology. So I wanted, this will be one of these talks that people were... People were talking about before, which gives you the impression that the field of quantum topology is conducted entirely by pictures, so it may be inappropriate for our formal papers, but it's good for talks. A manifold is said to be prime if you can't decompose it into two disjoint non-Euclidean pieces by an n-1 sphere for an n-dimensional manifold.

17:30 In two dimensions, life is very simple. You have only two prime manifolds, a cross-cap RP2 over here, which you get by... Removing from a two-sphere a disc and then identifying diametrically opposite points on the surfaces on the edge of the disc. You can take the 2-sphere and identify diametrically opposite points on the 2-sphere, if you like, and that identifies the bottom hemisphere with the top hemisphere and leaves the equator with diametrically opposite points of the equator identified. So I could have put the other half of the 2-sphere up here, and then these points would... So if you didn't do that, it wouldn't qualify because there'd be one point that would make it... Oh, you mean the 2-sphere itself doesn't qualify as RP2, right. No, as a manifold. As a manifold. Oh, I'm sorry. Right. So that's just a formal question of whether you want to call the two-square prime or not. It's like whether you want to call one prime or not. Okay. So any closed manifold is... So I'm just calling the two-square not prime. And any closed two manifolds is then... It gets called the connected sum of S2 and copies of RP2 and T2, the two torus. And you write the connected sum like this. So here's an RP2 attached by removing a disk from the RP2, removing a disk from the 2-sphere, and sewing the two boundaries together. You can always diffeo the boundary around any, to any position on the two spheres as long as, as long as it's, you take two connected manifolds then you can, you can remove the, you can shrink the boundary down as small as you like, move it along any path and then you have it someplace else. The two sphere is a base ball. The two sphere is the, is the surface of the base ball.

20:00 There are prime factors or prime summands by the connected sum. Another way you can think of constructing two manifolds is this. You can identify edges of a polygon. So if you want to get the torus, you take a square, you identify the red edge and the blue edge. Here's identifying the red edge to get the cylinder. Identify the two blue edges over here and you produce the torus. And that identification is obtained by looking at the action of this discrete subgroup of the Euclidean group consisting of translations by, say, one in the x direction and translations by a constant, which you can take to be one in the y direction, and then the identification of Euclidean two space, E2, by this group, the quotient of E2 by this group action. That is the manifold of trajectories identifies every point has a unit cell here. This unit cell gets mapped into successively into all the other unit cells so if you just want to count each identified point once you have the interior of the unit cell together with this identification of the boundary. So an equivalent way of describing this identification is to talk about Euclidean two-space. Here's another example of a unit cell that produces the double torus, the connected sum of two toruses, and if you just make more sides to the polygon, you get all of the genus and two metaphors.

22:30 Well, if they'd only left the branches here, that would be useful. In three dimensions, life is more difficult. In two dimensions, there are a countable number of different manifolds. In three dimensions, there are already a countable number of different time manifolds. You can then add together to get again the countable number of three manifolds. So typical examples, as in two dimensions, you can obtain typical examples of three manifolds by looking at the symmetric spaces, the hyperbolic free space, the Euclidean free space, and the spherical free space. So these hyperbolic free spaces just are three, and so is Euclidean free space, but you've got the two different metrics on it. And then with this metric, you can look at discrete subgroups of the group of isometry. So for example, you could look at the action of a group of boosts acting on the hyperbolic manifold and quotient out by that action of this discrete group acting freely on the isometric. Right, you need math, but it turns out to be very useful in classifying the topologies to talk about the group of isometries that you use to produce the manifold. So, for example, the spherical spaces are obtained by S3 modulo the group of isometries, and we'll see when we discuss the groups of diffeons. The topology is related to this group of isometries. One of the key features of work that's been done in classifying free manifolds is showing that in some sense topology is really, that topology is closely related to geometry, the special geometry, the natural metrics that you can put on these free manifolds. So it's very helpful to think of them in terms of these. So we'll see that when I talk about these.

25:00 So we've got these these these three particular ones giving you a countable number of hyperbolic spaces for the free sphere you have all the finite subgroups of SO4 so in particular all the covering groups of the finite subgroups of SO3 the point groups so the crystallographic well all the point groups but including all the crystal all the crystallographic point groups. And then for a flat space, you have the Euclidean group with the finite set of flat spaces obtained by quotienting out the finite set of Euclidean groups that act freely over time. That's finite, the other two account. That's right. Okay, so again, as in Friedemann, well, you can also... When you do this, what happens? You again get the unit cell under the action of the group, and then you have the faces of that unit cell identified by the group action. So again, you produce these manifolds by taking a polyhedron and identifying opposite faces. Or if you think of the similar manifold that you get by removing a point from it. And opening the point out to infinity for the corresponding open manifold. Then you get that by cutting out the polyhedron and identifying opposite faces. So here's an example. Are there any choices in identifying opposite faces? For a particular polyhedron, there are often different choices depending on the group. So for the cube you have, I know, at least three manifolds you get by different choices of identifying opposite faces of the cube. So here's an example. Here's four spherical spaces. So here's an octahedron, and you're identifying opposite faces over here with the identification shown. So these, the spherical spaces, there's a rule that, at least for these spaces of S3 cross covering groups of, well, finite subgroups of S3, covering groups of the point groups.

27:30 If you look at any one of these faces and then rotate it by the minimum amount to get to the opposite face, then you end up getting the right identification. And then you just do, of course it's symmetric, so you have to rotate in a consistent way with respect to the outward normal to do the identification. So here's T star, the covering group of the symmetry group of the tetrahedron, giving you this. The octahedral space over here. Here's the Poincare dodecahedron, which is a homology sphere, gotten by quotienting S3 by the icosahedron, the covering group of the icosahedron, and so on. This is the covering group of the six-element dihedral group, so you get a prism manifold. The actinidro group, you get this truncated. Well, yeah, I'm showing you, right, I'm showing you, there's two kinds of faces here. Well, in all of these, you identify every pair of opposite faces. Down here, the difference is because there's two classes. It's not the regular, you don't restrict yourself to regular polyhedra. I do here because they're easy to draw. But if you take arbitrary polyhedra and identify pairs of faces, you do it. Yes. So you might have to take fake. Well, these are closed three-minute ones. Is that what you mean? And to get the open ones, you take, right, you take the compliments, you remove a point, or you remove a countable number of points if you want a countable number of points. But if you're just thinking of closed and asymptotically flat with a single asymptopia, then you'd just be removing one point from each of all of those.

30:00 So you can ask which of these manifolds should be interesting to us in classical relativity? Which of them, what topologies occur as vacuum spacetimes? I meant vacuum with zero cosmological passes. You can also ask which occur as vacuum spacetimes with positive energy, which occur as spacetimes with no energy condition. Of course, with no energy condition, you know they all occur. So you can ask what occurs with positive energy. They all do. That's simple to show. So the reason I put this down is that the only difficult question is to ask which occur as vacuum spacetimes. And it turns out again they all do. So there are no restrictions on which four manifolds of the form 3-manifold-cross-r occur. And then if you ask which four manifolds occur as vacuum spacetimes... They all have the form S plus R. That is, you can't get any four manifolds with topology change satisfying the vacuum Einstein equations. And now, so let me now go through this proof of the, I'll go through an easy version, I'll quote easy because I'm just going to quote the Hart theorem. So you can remove any point. I mean, it's not as three cross R because you've taken away. You can remove any set you want to from that and then take any solution to the vacuum Einstein equation.

32:30 Remove anything you want to, you've still got a solution to the vacuum Einstein equation. So if you require that it be closed and compact or that it be... I don't know if I can. He must be, yeah, so he must be, I know he makes an assumption about geodesic completeness or about lack of singularities, but it's certainly false that the space times I'm going to talk about are geodesically complete. So let me, let me wait. They may be, but they're bomb to be. Yeah, yeah, that's, no, it's certainly true. Or are not the complete. Right, well that's, in fact, exactly what I'm going to say is that classically none of them are geodesically complete. The topology is not, is not trivial for asymptotic flatness. So, so, I'm, so I'm not sure what, yeah, so, so let me, let me pass this by. Yeah, I say asymptotic flatness. So let me pass this by and go on to talk about what, yeah. Okay, okay. So now let's, let me ask, what three minutes, so this is, so you don't...

35:00 In a smooth way, get topology change. And now let's ask, suppose we look at three manifolds of this form S cross R, four manifolds of the form S cross R. What three manifolds occur? Okay, as vacuum spacetimes. Now this was looked at over the last couple of years by Don Witt. The claim is that they all do, and it relies on, in the compact case, it's a simple consequence of a theorem by Kasdan and Warner that says if you're given any... Any function f on a closed 3-manifold for which the function is somewhere negative, then you can find a metric g whose scalar curvature is that function. That is, you can find a metric to reproduce any scalar curvature you choose as long as that scalar curvature is not everywhere positive. If it's everywhere positive, you're down to these spherical spaces. If it's not everywhere positive, then you can, for any space, you can find a metric that reproduces that scalar curvature. Alright, so that's the key to it, and Don noticed that if you use that, you can immediately show that you satisfy the vacuum Einstein equations. You set the extrinsic curvature of the hypersurface equal to GAB, and now what do you got? Well, since KAB is proportional to GAB, the momentum constraint is automatically satisfied. You've just picked the r equal to minus 6, so that when you take this KAB equals GAB, KAB KAB plus K squared is 3 plus 9. This is the proof. R is minus six, and that's equal to zero. So those are the constraints, and that's all there is to it.

37:30 Okay. So now let's go to the asymptotically flat space. There the proof is not so trivial, so I won't give it here. I'll just mention what's involved. Again, Don finds that every three manifold... So now you remove a point from an arbitrary free manifold and ask whether that occurs as the space-like hypersurface of an asymptotically flat spacetime. Well, the claim is it does, and the proof here requires you to patch the manifold. So think of some three torus down here with opposite faces identified for the blue lines. And you're going to patch that to Schwarzschild and the construction of the proof at the horizon. So this is like Brill's example where he constructed a free torus to show that you could get an asymptotically flat free torus with positive energy. He took dust. This is similar, but now you have to satisfy the vacuum constraint equations after you've done the fashion. You can't just reduce the dust in order to get it. So this is not trivial to show that you retain a solution of the vacuum Einstein equations, but it's true. Let me quote Gannon's theorem over here, saying that if you have, that classically, if you have any space-time that satisfies weak energy condition and has a non-simply connected Cauchy surface, so all of these three manifolds, if the topology is not Euclidean, and the three manifolds has only this one boundary, then... Then the Cauchy surface is always non-simply connected, the free manifold is. So then it follows that M is geodesically incomplete. So this is all of it. This is all of it. Any asymptotically flat manifold satisfying the energy condition with non-Euclidean topology, all classical topological structures collapse and presumably leave flat holes.

40:00 But it's not, it's not regular near infinity. It's not asymptotic as well. I think it's stated. I'm not, I doubt it, but I don't know. Are you going to talk about maximal prices? It's not valid by common sense. Okay, go ahead. There's a, there's a very interesting conclusion that was mentioned with another result. You can argue that these singularities must be very interesting, they cannot be the standard four-field type singularities. And the argument goes, there was a student at Yale, Robert Bartnick, who proved that asymptotic and class-based types, which had singularities of the four-field type, always have electrical flights. It looks like this is a space that can be turned into a zero. It's a world-wide action like a volume. It's a dynamometer of a minimum surface in an ordinary human space. It can be performed at a time, number, place, and time. The anthropometry classifies time to be an ordinary type of singularity. It's like this. The data can show that on any surface like these would not fit in the topology. You can't have a mathematical science because a mathematical science would require a positive scale of curvature and one can prove that there are no such methods. Right. And that's the Gromov and Rossi result and shown now that there is. Right. So none of these space times that you construct by evolving these couplers and not using the data have mathematical biases. Therefore, they must violate the molecular theorem and the only way they can do it is by having very strange curricularities. Yeah, that's right. So the statement about maximal slices was actually what Don emphasized in the article where he showed the existence of the asymptotic graph.

42:30 No, no, you can still have a mentor on it, but it's still going to be a black hole. You have to find out what happens when you get right beside the black hole, and you're going to be nervous. You're going to have to have questions, and you're going to have to, you know, try to figure it out. You know, when you find it. When you, yeah. When you find it. If you don't. If you don't. If you don't. If you don't. If you don't. If you don't. If you don't. If you don't. If you don't. If you don't. If you don't. If you don't. If you don't. If you don't. Okay. So there's an open question, I think, also in this, in this, that's relevant to this, well, at least to what I'm going to say next. And that is that after you, after you collapse, you know there's a horizon. You might ask, in the classical geometry, can you have non-Euclidean topology that does not have a horizon around it? That may, I don't know the answer to that. I'm not sure that it's known. It's... So certainly for the hyperbolic or the flat manifolds with the point removed, you can't have them without a horizon. I don't know whether you can classically have spherical spaces, whether classically you can detect the fact, you satisfy positive energy, whether classically you can detect the fact that you have non-Euclidean topology. Classically, topology is essentially uninteresting. You may not be able to detect it at all. If you can detect a non-trivial topology, it's only around briefly and then it collapses. It suggests why it is that we don't see large-scale non-Euclidean topological structures left over from the early universe.

45:00 But it means that if you're interested in topology, you have to be interested in quantum topology. We're not going to find classically any interesting solutions here. That's right, you might find, yeah, you might be some big thing that's still out there. Why would I, why would I argue with that? I simply need a transition sentence to get the... So let's begin by looking at the quantum theory. What we'd like to, I'd like to start out looking, dividing it into two parts. Quantum kinematics, which I think I, which I think we can give a reasonable presentation of, understand reasonably well, and then quantum dynamics, of which I would certainly not say the same. The, the kin, by the kinematics, I mean the momentum constraint that the divergence of pi AB is equal to zero. You can state this equivalently by smearing this with a test vector psi that vanishes at infinity. If you integrate by parts, this is telling you that the integral of the linear derivative of the metric times pi a b is equal to zero for psi vanishing asymptotically. If psi doesn't vanish asymptotically, you get a surface term. And then if psi looks asymptotically like some killing vector of the asymptotic metric, then the volume integral is just the associated ADM momentum, and you can write it as a surface term in infinity. That's the classical statement, with which you're all familiar, and the quantum statement you're probably familiar with too. If you look at G and its conjugate momentum, pi, the commutation relation implies that peak psi, that the commutator of the operator peak psi, this momentum constraint smeared with the C number vector field psi, I mean the C number vector field psi gives you the lead derivative with respect to psi of GAB smeared with psi and...

47:30 And exactly the same thing with pi AB, so that the constraint acts as the generator of spatial diffeomorphisms. And of course, in the classical theory, it again acts on phase space as the generator of canonical transformations that correspond to diffeomorphisms. So, peak psi equals zero says that the state space is invariant. Under asymptotically trivial diffeos and asymptotically trivial diffeos that are deformable to the identity. Now, that means there are two classes of diffeos that do not leave the state space fixed. There are those that are non-trivial at spatial infinity. So if you look at a diffeo chi that behaves at spatial infinity like a rotation or a translation, an element of the symmetry group of spatial infinity in general, then For chi lambda, a family chi lambda of diffeos, starting at the identity and generated by psi, the ADM momentum is this surface integral, well, over here, we can integrate by parts, write it as the volume integral, and it's d by d lambda of psi of the diffeo acting on the metric, so the... So the state vector changes, if it has non-zero momentum or angular momentum, it changes under the spatial diffeo, and the change in the method is measured in those degrees, momentum or angular. There's a second type, analogous to the large-gauge transformations of Yang-Mills theory, of diffeos that are not in the component of the identity, but are trivial at infinity.

50:00 So, if we call dip the set of these dipios that preserve the asymptotic structure of spatial infinity, they leave the point of infinity fixed, and this is, you don't have to say it more precisely, because it doesn't matter for... For the group, for the action on the state space, whether you fix a point in a frame at infinity, which is a minimal asymptotic condition, weaker than what you want to impose, or whether you fix a point at an entire disk surrounding that point at infinity, completely insensitive to what you choose for what... Topology, and now we'll talk about it. Now I'll talk about it. So, pi zero of diff, this group of inequivalent diffios acts on the state space. And now let's look at what these groups are. For asymptotically flat spacetimes, John Witten and I spent some time looking at it in a final classification for all manifolds in form S3 slash G, where G is a finite subgroup here of SO4, only the spherical spaces. Yes, this is saying what you're going to, how strong you're, how strong you're, you want to make your asymptotic conditions at spatial infinity. Certainly the first part of what, what the symmetry, what these guys are. This is my, I mean, I don't regard my job as telling, as saying what you're supposed to pick out here for momentum or angular momentum at spatial infinity or what larger class you want.

52:30 And this is sensitive to particular diphios that are non-trivial at spatial infinity. Down here, to what you pick for the structure of spatial infinity, what the conserved quantities are obviously depends on what you're going to get at spatial infinity. But if you ask what's pi zero of the diphios that are trivial at spatial infinity? And that is insensitive for the structure of infinity. As long as spatial infinity is regular enough that you can at least identify a rotational subgroup of infinity. Alright, so now let me give you, let me show you some examples and then tell you what the classification looks like for some spherical spaces. I'm not going to give you the general classification, but it's pretty consistent. Here's an example. Let me use this old terminology of Misner and Finkelstein, and I'll call a prime factor of the free topology a topological geon, or just a geon. So I think it's... The term geon was started by Geon Wheeler, but he used it... He didn't use it in general for arbitrary, he used it initially for this, for the concentration of energy, and then in the Miesma-Finkelstein Diffie you've got this classification, so they take the terminology and then So now, now let's look at, let's look at a particular, a particular Diffie. So this is going to be geon exchange. We'll take two identical, in this case, Poincare, Dodecahedra. So now you're thinking of an asymptotically flat space. You've removed these two dodecahedra, and you've identified opposite faces. So out here the topology is Euclidean, and then these faces have been identified to produce your free topology.

55:00 So now you've got two identical geons and a diffeo that's not in the component of the identity. The Diffio takes this geon to this one, this guy down to here, so the Diffio can be stated by saying what happens in the Diffio to a path, a green path over here. So the green path that goes in over here, when you take this guy and move him over like that, the green path that went here is now over here. And the green path that came in here has now gone down over there, okay. And you've pulled the metric along. So if this, for example, were a geodesic over here and you put a metric on the spacetime, then this would be the geodesic of the isometric spacetime pulled along by the diffio. So this is, so these would then be two spacetimes with the same metric in effect. If you keep track at infinity of where you are, then... Then there's a difference in the homotopy class of these geodesics. It's not homotopic to the identity if you keep infinity fixed or if you take a closed manifold and put a couple of extra geons out there which act in effect as markers that prevent you from deforming this to the identity. You can think of them, maybe the way to think of it is, think of embedding the manifold in some higher dimensional space. Think of a handle embedded in free space, and now we can, and now that handle and the plane is all made out of rubber, and we can stretch it around and move it, but we can't actually change the position of the rubber handle. The set of points gets mapped into itself, so you can think of the rubber handle mounted on some wooden handle.

57:30 And you can move it around and you can take some point that's on the top of the rubber handle and stretch it out to infinity, but that's all you can do. And now, we've got, now you can think, not of a diffio, but of an operation that is the following. Cut, here's a handle with a cap that goes, put this handle out. Rotate it around by 180 degrees and stick it back in. Now that's not any continuous family of diffios. But at the end, and it's not a diphyo itself. The way I said it, it's not a diphyo. But the final manifold you get to is diphyomorphic to the original manifold. And after you've done that, and the diphyomorphism takes, it's diphyomorphic under a diphyomorphism that does this. It maps all the exterior points to where they were. It maps this point over, it was over here, to this. So as you go in, it maps a succession of points, the points that are rotated by angles between zero and pi. And finally, when you get in here, it's just an exchange of the two sides of the handle. So that's a discrete diphthia. And it takes this element, this element of pi 1, goes into a homotopically distinct element of pi 1. This curve goes into a curve that can't be homotoped back to this. Here I've got this... Isn't that one of the... No, no, it's not. No, you see, I can't... That's how I set it up. This handle... I've got this rubber sheet, and the handle is mounted on a piece of wood. I can't... I can't conform it to the identity. I can't continuously... I can't continuously move the handle around like that. There's no embedding space. You can see the embedding space there. But in reality, there's no embedding space. There's just this space of points, the handle, the rubber sheet. And I can move this space of points into itself. That's what a diffio is. That's the space of points back into itself. So I can move this point and stretch it out wherever I want, but I can't change the position of the handle in the embedding space.

1:00:00 That doesn't have any meaning. You have to worry about it. Now, do these things have the singularities we've described before? In the time evolution, the time evolution of these will be singular if you satisfy Einstein's equations. Let me talk to you, yeah, in the quantum theory, there's no, you don't have to believe in a singular quantum theory. Quantum theory can be non-singular. I can certainly look at the Diffio in the class. And then above it, and the time evolution would be exactly the same. So in the time evolution, it just looks like the Diffiord singularity in the sense that means that as you approach the singularity, the metric is always the Diffiord metric. So the whole time evolution is isometric to the original time evolution because the initial data is isometric to the original initial data. All right, to the next? Okay.