Time, Change & Spatialization / Time in Gödel's Universe

0:00 Is it meaningful to ask if we can represent time without turning it... Is it too loud, maybe? No, it's okay. So can we represent time without turning it into space? There is a question that I will address. Of course, as you know, this problem, this question, was already asked by many so-called process philosophers. I'm thinking about Henri Madsen, Whitehead. And after Whitehead and Berkson, people like Tepec or even Abner Chironi. It is that the question that Berkson said, for example, Berkson said that he would not have relativity, a book called Duration and Simultimate. And on this book, he tries to show that, well, classical interpretation of special relativity is not really good work. You could get back to an absolute time. Well, first of all, the book was written in the 20s. And at that time, important physicists made errors about the true understanding of time and space and relativity. But the general idea, Bergson asked, is, I mean, did relativity just invented a new way of specializing time. And of course, this question was also asked by Whitehead and . Recently, as you all know, Smallin wrote a very controversial book called The Trouble with Physics. And just let me read you a few parts of that, a few lines. So this is what Smallin says. When you look back at the history of physics, one thing sticks out. When the right theory is finally proposed, Whatever else one has said about string theory or low quantum gravity and other approaches, they have not delivered on that front. The standard excuse has been that the experiment of this scale is possible to put forth. I believe that there is something basic we are all missing. Some wrong assumptions we are all making. If this is so, then we need to isolate the wrong assumptions and replace it with a new idea. What could be that wrong assumption? My guess is that it involves two things. mechanics, and the nature of time. We have already discussed the first. I find it hopeful that new ideas about quantum mechanics have been proposed recently. More and more, I have the feeling that quantum theory and general relativity

2:30 are both deeply wrong about the nature of time. It is not enough to combine them. There is a deeper problem, perhaps, going back to the origin of physics. Around the beginning of the 17th century, Picard and Galileo both made the most wonderful discovery. You could draw a graph with one axis being spaced and the other being timed. A motion to true space then becomes a curve on the graph. In this way, time is represented as if it were another dimension of space. Motion is frozen and a whole history of constant motion and change is presented to us as something static and unchanging. If I had to guess, this is the scene of the graph. We have to find a way to unfreeze time, to represent time without turning it into space. I have no idea how to do this. I can conceive of mathematics that doesn't represent the world as frozen in eternity. It's terribly hard to represent time, and that's why there is a good chance that this representation is the missing piece. So, more precisely, we can be talking about this, of course, but we'll see that we can have relational accounts of time, but we also can have substantial accounts of time. And the question that we will examine is the following. Can we have, and I will do that after Rovelli maybe, can we have change without time? And isn't the idea of having change without time the first answer to the question of specialization? So just very quickly, things that are very well-known, but it's just to understand at the level of time. Of course, as you know, physics and philosophy have struggled between two different conceptions of space. For people like Newton, in some way, from space as an entity, and of course, that means that motion is absolute. When we say motion is absolute, we mean motion is related to space. And by space, we understand space as something as a different nature than other physical entities. And in that framework, well, that's why geometry works so well with Newtonian mechanics. The geometry is not all, and of course, it's independent of the physical entity. If space is a set of relation, it means that space is just a name for naming that set of relation. And of course, if space is a set of relation,

5:00 There's no absolute motion. And, of course, geometry evolves and is developed at this time on the physical entities. So, of course, there has been, in the first half of the 20th century, much debate about the nature of Minkowski's geometry and many interpretations of Minkowski's geometry between being a Minkowski. Chapek addressed this question, but also Vaughan, and many, many others. Of course, people argue, and someone like Vibe that, well, of course, if you constitute geometry, that is the geometry of special relativity, we have a change in the world. Why we have a change in the world? Because the motion of the point in time is represented as a stationary curve in full dimension in space time. And of course, many people have criticized relativity for this, saying that, well, what is this theory when we are being in full dimension and that it's becoming just happens in three dimensions? And as I told you, virus, things were that the world simply is, it does not be harmful. Of course, people at Chapec have argued that the idea of background independence that we get in general relativity is the key understanding for seeing, not as much as specialization of time, as Chapec said, the diamondization of space. And I think that's how we see also Hamiltonian reformulation of general relativity. What can be the question of change, of a relation of time? Well, on one side, as we have seen with space, is time an entity or time just a convenient name? Not a fundamental structure, but something that emerged from something more fundamental. Can we have not change this picture, but change without time? Of course, there's very famous, as there's famous ancestors of relational space and space time, there's ancestors of relational theory of time. In ,, Lucretius was famous for writing and said that time does not exist by itself. It gets the name from the object. It is not possible that anybody measures time by itself. It may only be measured by looking at the motion of the object. So what I argue, and Leibniz was also very famous, and he fought against Newton, not only on the nature of space,

7:30 but also on the nature of time. Leibniz did believe in a relational account of time. So the question I want to be asking is the following. I think that what we have in relational account of time is not time the same as an exterior parameter, but as Juan said it this morning, sorry, time has built in the system. There's not any more special arts. Rovelli, he's trying to build relational physics in many ways. He has relational interpretation of quantum mechanics. He's a deep, deep relativist. And he has also, I mean, interesting ideas about time. Of course, what Rovelli says as relativists is that we have, oh, there's no background space time. There's no background space time on which things move. And we have no time along with which everything flows. And what is precisely his idea of physics without time? Well, he's talking about this very famous anecdote. Galileo was in Pisa Cathedral. I think it's not really a bill of mail. I mean, my point is not a history at this time. And this is what happens. Galileo was looking at Pisa's catedral chandelier. Pisa chandelier, I'm not using that in English. And he had the intuition that the motion of the chandelier was isochronous. Of course, he didn't have a clock. And how did he do to measure the isochronous motion of the chandelier? Well, he used his pulse. And using his pulse, he realized that the motion of the chandelier was isochronous. is that a few, I don't know, a few years after, doctors saw, so the chandelier was used and it was made up of, well, the general concept gave birth to the idea of pendulum and the isochronism of pendulum. And the funniest story is that a few years after, well, people started using pendulum, doctors started using pendulum to measure the pulse of their patient. And we might find a very similar thing in Aristotle's understanding of time in physics. What does that mean? Well, you can say, oh, there's a circle. There's a circle. There is a logical paralysis.

10:00 But what says, and I think he's very right on that side, he says, no, what it means is that we have nothing such as a t-parameter. What we have is dynamical particles not not related to parameter t, but dynamical variables related to other dynamical variables. Of course, at our level, at our scale, it's very easy to realize that we, and this is why Newton's framework works so well, that this is totally coherent with the idea of absolute time But what Revelle tries to do in his work, he tries to show that in some way, we have no good clock at the quantum level. And he tries to, I mean, with many others, he tries to interpret the little bit equation, and he tries to find an intrinsic time variable in the equation. So that's how Rovelli's, that's what's Rovelli's account of relational time. Of course, there's two ways in the problem. And if we start at the beginning with special relativity, Rovelli says, and I think that's an interesting point, Rovelli says, what we have in special relativity is, of course, partially relational time. Because we know that we have time t's in corresponding to the infinite Lorentz frame we have. But we also know that those times refer to the Rinkowski nature of the metric of the Lienkowski geometry. So Rebelli's idea is just to say that this is one way to go to a purely relational account of time, but it's not the final result. But of course, on the other side, you will have people that will say, no, we have a problem. And this is really the idea of process philosophers. Those people say, no, what we have in special relativity is not this, we have a problem, we have specialized time. That's why we have a four-dimensional changes world. And we have to build, at the root of special identity, the idea of becoming the idea of flow of time. And so people, I didn't work that much on this. I wanted to, but I didn't have them. Those things are quite new for me. John Christian, and John Christian is a student of and also a collaborator of Roger Penrose. He's doing the whole post of time. He has an experimental about the Penrose idea. I don't talk about this, I don't know much about this.

12:30 But what Christian, so John Christian is a student of Adner Shimonie, one of those process philosophers, much influenced by people like ,, and of course, this again comes from . And Adner Shimonie's idea is the idea of experimental mathematics. You just can't do mathematics like this. You need to have one at the end of your experimental thought, and say that yes or no. So what Christian is doing is trying to construct a generalization of special relativity in which that can capture the flow of time. This idea, from a very astonishing point of view, this idea of trying to find process at a more fundamental level, well, Well, I found it very recently in the work of Bezerra and Barnes-Fernandez. They have tried, again, their idea is that space, well, time is not a fundamental concept, but it has to emerge. And that we might try at the beginning to make an algebra of the process. And they're using careful algebra to understand process. Again, I didn't look at that very carefully. I was quite amazed, because I was a bit more aware of what had been done in the 19th century. People, I mean, we all know that Clifford Gratham and Hamilton made very interesting tools for mathematical contemporary physics. But what is not well known is that they have both, the three of them, they had the idea that we could understand algebra as the science of pure time. When you read, and I read that, I mean, one of the people that most influenced me, I will not be talking about him, it's Gilles Chate. Gilles Chate was a marvelous geometer and a great philosopher. And the first time I heard Gilles Chate was talking about Hamilton's lectures on Cartier-Denis. And in the preface, as you know, Hamilton was the first one to realize that i, the famous square root of minus 1, was nothing else than the couple 0, 1. And of course, he understood that after Ardant, Vessel understood that we had to see complex numbers as points in the plane.

15:00 And that seeing complex numbers as points in the plane and think that i is nothing else than the couple 0, 1, Well, there was none anymore of any reason to call them major numbers, but the complex numbers. So when you read Hamilton's briefcase on the lecture of the question, well, Hamilton said that he got through the understanding of complex numbers as couples because he was persuaded that algebra was the science of pure time. This is his work. He's talking about algebra as the science of pure time. Now, one can say that this comes from Kant. And it's falsely true, because Adelton was a, well, one of the books he read, those were critics of pure reason. Of course, there's this idea that you might see, algebra and fantascience of pure time in transcendental aesthetics. But I think it was more. And we'll find also, I don't know, there was a French edition of Grassman's work. And I think it's a very important book. I think there is much more in what Grassman wrote than what we call now, what we use now with Grassland ideas. And again, we find also with Grassland, but with Clifford, this idea that there's a relation between algebra and pure time. And as I say, as I write, Clifford emphasized that the criterion product was based on motion and dynamics rather than on a static structure. So this is what we, I mean, as I told you, this is a work in progress, and I'm going to work on this again, but I think there's interesting analogies between those two. So if we go back now to Ravelli's understanding of relational time, well, of course, if you say that you have, as he says, change without time, you must, in some way, try to explain how do you get, at one point, the flow of time, time flow, time flow. And this, again, was a very difficult thing for me, but I think Colin Ravelli wrote a a paper in 1994. And in that paper, they tried to show that the time flow is not a universal property in generally according to quantum theory. And it is related and determined by the state of the system. So the idea was, again, as I said, to reconcile the time lapse picture of quantum gravity and the evidence of the flow of time. And what I've understood is that a relation

17:30 between the thermostat of a system and there's one parameter group of the observable algebra. And this parameter group, one parameter group, is interpreted as a time flow. So I think that's a very interesting point, because to do a quick summary of what I'm going to say, this idea of specialized time, I think, is a big problem in physics. into a relational account of time. What is interesting is that if we understand time relationally, that means we have time built in the system. And that in one way or another, we have to find back time as well, to find time as a time And that's what we have with the picture of the relational time and the original time . That's it. Thank you. Thank you very much. I was fascinated by everything you said, but there's one thing I totally beg to understand. How can algebra and time, I don't even see what they have to do with each other. other words to explain that? Well, I mean, as I told you, I'm also, I mean, everything is not very clear for me. All these ideas are very new. Kant understood it this way. And I won't like very much my answer, because I think it's more profound than what Kant said. And Kant's aesthetic, transcendental aesthetics, and pure reason, he makes a difference between geometry and algebra. Well, I don't think it's going to be very provocative to say this, but I don't think it did say amazing things about geometry. I think Hegel said the most more amazing things about mathematics. Usually people don't say that because usually your answer that you know that Hegel tried to show that there was only, I don't remember, seven planets in the solar system, and like two years after, someone followed another one. So it means that he's a poor philosopher of nature. I don't think that's a way of seeing things. If you look at, for example, Hegel,

20:00 science of science, you will find an amazing account of his understanding of the concept of infinite in differential formulas. And Hegel really understood that, finally, mathematics, he says it like this, had free us from a poor concept of infinite, from a negative concept of infinite, and we have got to a positive concept of infinite. But again, to come back to this idea, Yeah, because this is related. I think there was a great period in the 19th century. It was called . And that period was around people like ,, and . And I think that many of these people had the intuition of, well, in a way, was a real world. And we could find many connections. And so this, I'm sorry, I'm not going to be very clear, but this idea of time, energy, right, And I think there is a genealogy of this idea But I'm working on it now. I'll try to get it. Thank you. On one of your slides, you showed that general relativity is space-time is identified with the gravitational field. Yeah. I think this idea is not quite appropriate. Because in general relativity, the principle of . The principle? The principle of . eliminate, and it is allowed, to eliminate the gravitational field at one point, or even along the world line, special relativity works, and special relativity has a space-time, so. Well, that's again the big problem of what we're going to understand in what is space-time in general relativity. I don't, I mean, you will agree with me to say that in special relativity, we don't really have gravitational felt on space-time, and that in some way the an identification between the gravatures of the film and spacetime. So it's not a contradiction with the .. It's in the other way. No, so much more. It's special relativity because of spacetime. Yeah. It's a spacetime. Globally, GR breaks with SR. Locally, there are- Locally, you're finding constant of spacetime, of course. It shouldn't be so consistent. You cannot separate global concept from local concept. Yeah, but the geometry of GR does not become such a geometry. So it's a geometry. You can say that geometry is associated with the gravitational cube, but not space-time itself. This is true.

22:30 Well, what do you call space-time, then? This is an IP. I don't know. You have to tell me what you call space-time. Yeah, just to sort of comment on this question, which I was also interested in. As I recall, Kant associates arithmetic with time. What he thinks is that he's primarily about counting, and counting has something to do with criminal succession. So the convention, I could be wrong about that, but that's only because of what algebra has to do with it. And in particular, I realize you just can't read this now, but I'm looking for a little bit more information. In particular, one of the people you quoted, I think, was Clifford, thought that the fraternians were, you know, it wasn't just algebraic general, but some of you were fraternians type of dynamics. So this is just to say that the connection with Kant is too illuminating to me, and maybe there's something. Well, Kant is not, I mean, I wasn't a bit criticizing Kant, because he makes this relation. Sometimes with algebra, sometimes with the mathematics, he's not very clear on exactly the distinction, first of all. I've always been puzzled with Kant because what he says about mathematics is very poor. When you say that mathematics is opening up with 7 plus 5 equals 12, it's a poor account of mathematics. And at that time, people had done already amazing things about differential calculus and integration. So that was my point. The intuition I have on that subject is that it comes from Kant, but I think it went through the tradition of natural philosophy. And I'm sorry, I realized this in reading a paper yesterday, the paper of and I wanted to talk about it. But if you, I mean, I was talking about Gilles Châtelet. Gilles Châtelet can become a philosopher of mathematics. There's a great tradition of epistemology in France, and particularly on the mathematical side, people like ,, who are absolutely not read in other countries than in France, not very much read in France, are really doing philosophy of mathematics. Much of what has been done in the 20th century of logic, and most of the people reducing mathematics to logic. And to the day, I had a famous word saying that reducing mathematics to logic is like reducing Kant and Shakespeare to grammar. And there was few philosophers in the 20th century, and among them, ,, who did really

25:00 philosophy of mathematics. And what does is to go back to the tradition of the philosophy. And his book was translated in English. It's called Figuring Space. say, figuring space, and he makes those connections between, well, natural philosophy and, first of all, complex analysis, well, first of all, the identification of complex, not only complex planes, but also complex analysis and relational electromagnetism. And this is relative, isn't it? Thank you. That was very interesting. I think maybe we'll need to close off the discussion. Thank you very much. Thanks. This discussion is going to continue. Well, I'm happy to welcome once again to the front of the room, Professor Wolfgang Pindlein from the University of Texas at Dallas. Wolfgang has actually been up here before, pleased when he's able to come and visit us. He's going to talk to us today about the Viral Universe. You need the overhead? You bet. I may want to leave that. We don't need to know that. We're going to need this too. Yeah, I will, sorry. I thought I got it turned off, but I didn't get it. Thank you.

27:30 I'd like to talk about Gerdel's universe this afternoon. Most of you know that Gerdel, of course, was a famous mathematician, but he made one extraordinary excursion into technology. He was at the Institute for a Great Study in Grifton. He was a colleague of Einstein. He was a friend of Einstein. In fact, Einstein had a 70th birthday, somebody made a test trip for that birthday, and Gödel was obliged to write an article, and he had three years to cook up that article, so he lost the time to think, and he thought this was an interesting gift to Einstein. Somebody offered Einstein's own theory, something way off Gödel's main interest, but Gödel quickly got into relativity and invented this obvious universe for his birthday, and for his night, which has been extremely proud of this, which were very pleasing to Einstein and to love a lot of people doing. It has to do a lot with time, as we should see. So, here is a picture of Gödel's universe. Gödel invented this, here's a picture of Gödel's universe. I thought I had a pen, but I can use this one here. Goethe's universe is completely homogeneous. It fills, well, like all universes. It fills everything there is. It's made of layers. These look, in the diagram, these layers look like planes, but they are, in fact, all these planes. All these planes, all those electrons in these planes here, two-dimensional space with constant curvature, and they're stacked, however, in a Euclidean way. So sideways, the geometry is that of a two-dimensional space with constant curvature, and orthogonal to it is just Euclidean. This is just a spatial diagram of space. These red dots are supposed to be the galaxies. They're more or less uniformly distributed throughout all space during this universe actually is, of course, it's more homogenized, as one always has a symbology of all the galaxies,

30:00 and then you redistribute that sub-homogeneity to create itself a homogeneous universe, and the assumption is that that homogeneous dust will, the dynamics of that will be exactly the same as the original dynamics of this lumpy universe. So here are the galaxies. They all float freely. It's a static universe. that the actual universe was actually expanding, but he built this model, nevertheless, on a static universe, so he knew it wouldn't fit the actual universe we live in, but it was to start with just an exercise in relativity. So it's a static universe, nothing ever changes, same today, tomorrow, the day after, and it looks very harmless, really, for looking at it like that, but it has two extraordinary properties. The first property is this, that if you take a gyro compass, some box that contains gyroscope in it. If you had such a gyroconverse and put it in an airplane and the airplane threw all sorts of loops through space then, and you pointed the gyroconverse towards the sun, then in spite of all the loops, the gyroconverse would always point towards the sun. It's just put gyroscope in there. If you put one of these gyros, in fact, if you put lots of these gyroconverses, one on every galaxy, there's a gyroconverse sitting on one of those galaxies. There's no reason. So in looking at this stationary space, the first surprise is that you put these gyrocontuses in various places and they all rotate. They rotate in unison. So the local inertial frame of every point in the period of two minutes actually rotates, which is an amazing property. Why should it rotate? There's nothing to indicate that there's anything that causes that rotation. Otherwise, it's completely homogeneous. but the driver is rotating everywhere. Not only that, that's probably the amazing property number one. Amazing property number two is that if you start at a point and you can go around a large circle, because it doesn't matter if you can go in helios also, but maybe, suppose you stay in one of these planes and trace out a large circle, you need a big powerful rocket ship because you have to go very fast. Not at the speed of light, close to the speed of light, and describe a very large circle, and the amazing thing is you start at a certain event here, or whatever it is, say the beginning of, let's say the end of World War II. So if you make your trip, you can spend, by your own clock, you may have spent 60 years, and you come back, and after your trip,

32:30 you get exactly back to where you were and exactly the end of World War II. In other words, you get back to precisely the same event that you left. So there are, in this universe, there are closed time loops. Loops that you can, that an observer can trace out, and from the observer, long time, long time, will keep on taking it in the ordinary way. Nevertheless, that observer may age by, as I say, any amount you like, and come back at exactly the same event where he and she left. So that's amazing property number two. Gödel, well, Gödel, actually produce something that, amazingly enough, Einstein had foreseen. I'd say he learned the general relativity in 1915. In the end, before he had fully invented general relativity, he published a paper called The Foundation of General Relativity. It's in German, anyway. Anyway, that's how it looks, the outside of it. And then on one page, it's a very, very long paper, sketching out how his general relativity theory and he has something like the 50th page of it, one paragraph of it, and I'll just show you what he says in that one paragraph. He worries about the existence of closed line-like lines. That is really rather amazing. In 1914, he already starts by, he says, there's one thing in my theory, that might happen, closed line-like loops, and I don't see how to prove they can't exist. And it says, in the end, this goes violently against my physical intuition. And that was in 1914, and it's really utterly amazing. And then it was in 1949, going to produce the universe where that happened. You can sort of see, roughly, why Einstein worried. In special relativity, you know very well, in space time of special relativity, you have these very light forms all over the place. And in special relativity, the light forms are all lined up all nice and parallel, and the world line of anything, an observer or a party, whatever, anything that moves has to go in and out with these light cones. Whatever goes in has not to be in the past half of the light cone and come out in the future half, and it can go all over the place. Any observer, any party can do that. That's called a time-like progression.

35:00 A party cannot have a world line that lies outside the light cone. The party has to stay within the light cone. The last thing is, the particle never comes back in special relativity. In general relativity, the whole point of the relativity is that space now is curved. These light cones are no longer lined up like soldiers. The light cones are all over the place. And so it could happen that they're twisted in such a way that nevertheless, so at one point it may look like this. They're twisted in such a way that we can actually find a line that goes through the light cones and comes back to exactly the same place. That's a space-time diagram coming back to the same place. He's turned back to say there's not only space velocity time. So this would be one of those time loops that would be back to when your grandmother was born and you could do all sorts of mischief and prevent your own birth and so on. It means all sorts of logical problems. This is a quick picture of Einstein and Gödel. They were very good friends. which, in fact, produced a very, very inventive model. Here's another nice picture of Einstein in the field. Not so long ago, a student, I had this picture outside of my office. You wouldn't believe it, but not so long ago, a student came into my office and said, is that you? Anyway, I have fortunately been very good for 1956. Unfortunately, just the year after Einstein died, I had very much hope to be here at the moment. Anyway, Gödel, there is a wonderful example of how Gödel invented his universe, not only in the actual, Gödel He published his universe in two papers, one in the Einstein Festus, which has had no equation in it at all, then a very short paper, published also in 1950, but there was a very interesting lecture that he gave, which he didn't publish, but it was published costumously, and all we find in his collective works. Cohen invented Newtonian model, or Pace R.C. model. This is a Newtonian model of a rotating universe. The Deuteronian model that Goethe invented was, you start with the background of absolute space, Newton always is absolute space. In this absolute space, you choose an absolute axis. Then you get dust, and you have that dust,

37:30 uniform dust, that rotates uniformly and rigidly about the axis. And you can so arrange it, you can satisfy the equations of Newton, the inverse square law and so on, you can so arrange it that the gravitational force that pulls all the dust towards the center, if it pulls axis to the center, is completely balanced by the centrifugal force that the particles experience as they rotate. So in other words, you have an axis, and all the particles rotate about that axis, their gravitational action rebounds by centrifugal repunction, and the whole thing stays in equilibrium. going round and round and round, the distances between these galaxies, the distances remain rigidly constant. And you may say, well, that's a funny model for a universe, because it has, first of all, the international space, which is bad. Secondly, it has a preferred axis in the space, which is what it is. It's true that in its construction, there is a preferred axis in the third center, observationally, interestingly enough, all these galaxies are precisely on the same footing, and living in that universe, you could never say that you're on the axis. Every single galaxy, life on every single galaxy, would feel exactly the same as on every other galaxy. So here's a picture of life in such a greater Newtonian universe. As I say, the theory prefers the spacing, prefers an action, and, of course, ultimately, these two particles will move past, and then it will be like, obviously. Because the native universe goes out in all directions. However, there is complete equivalence. If you sit on any one of these particles, first of all, you will not feel the gravitation. Only if you have to sit still, then while you're sitting on one of those particles, you won't be pulled up. There's no gravitation on any of these particles. Point number one. Point number two. Around every party, if you put your compass, if you put a gyro compass at that point, at every single point in that universe, the gyro compass will rotate with the same angular velocity. So, in fact, although it may not seem that totally a homogeneous universe, it is a totally homogeneous universe, as far as all observations are concerned.

40:00 And, of course, if you study anywhere, there's a Coriolis force pushing you sideways, and because the Coriolis force everywhere pushes you sideways, if you actually, if you lived in that universe, if you lived in one of those galaxies, and you shut up a particle in any direction, that, say you shut it up in one of those deep things, that part of it would actually trace out the circle because it's constantly pushed sideways by the gloria on this course. But the push sideways reserves your forward velocity so the forward velocity never changes and of course the sideways iteration is constant so you trace out the circle. So that you least the orbits in that Newtonian, go to Newtonian universe, all the orbits are circular Unless you also push the original part of the graph with a slightly, I call it vertically in the C dimension. You push it up, clearly in the C plane, it's a circle with a slight C component into the helix. Clearly, that's how the periodic force would shape these three orbits. The interesting thing is that in the actual retinistic, full retinistic go to the universe, the two VCs are exactly the same. They're also circles in z-place, and they're heliases otherwise. So there's a very close analogy between the Newtonian model and the final Gerdel universe. Final Gerdel universe, I think I don't have time to go into the details. Gerdel had a very complicated way of constructing it, but there are also simpler ways of constructing it, and this is a way, and I won't have time to go into the detail. So we start with a standard metric for, let's assume we have a stationary universe. A stationary universe means one of us changed the day, the same as the day afterwards, but rotation is allowed. If no rotation, then it's static, but if you allow rotation, then it's for a stationary universe. It has a set of lattice, and given the lattice metric, which is this, and given the gravitational field, which is this, and given the rotation, which is this, essentially, you can write down, in special activity, the metric of such a stationary space. Then you can use the symmetries of this space for one of the homogeneous and simply by demanding homogeneity you can restrict the metric further. Those of you who have done more generativity know virtually well in Schwarzschild. For example, you get the Schwarzschild space-time

42:30 would use this symmetry a great deal to narrow down the metric and only really narrow it down as much as possible. Then you've got a clear equation and then finish the job. Exactly the same in the variables. You use the symmetries that you want to apply and then solve the field equation. And incidentally, in Schwarzschild, a miracle happened. A miracle that, in fact, Einstein apparently had not believed. The miracle is that you put in the symmetries and you apply the field equation and you have a simple solution. Einstein, I think, had no faith in that and he always did. His original work was done with approximations. he had found it in the finished problem, but he was so complete, I imagine that it didn't exist, so then he got me in a dishwasher and found it. Much the same happens here. The metric looks quite complicated, the equations look quite complicated, when you solve them, you find there's a simple solution, which is very lucky for Goethe because that's how he found this beautiful university. So you have the metric, you use all the symmetries to narrow it down, finally you narrow it down to this form, and you see here those who have a little experience, is the metric of a two-dimensional space of constant curvature, which at least specifies the lattice there. And then there's no gravitational field, so there's just a simple constant out here, and there's a rotation, which gives us a W there. And then we apply the Einstein field equations, which have, well, how much time do I have to do it? So the Einstein field equations On the left-hand side, you have something representative of the geometry, which is the so-called Einstein tensor. And on the right-hand side, you have something representative of the sources, of the content of the universe, which is simply the energy tensor. And then there's also a cosmological term. The Eisen's integrations without the lambda term, those were the original integrations, and Eisen added the lambda term. For Eisen's original universe, Eisen needed a positive line. And Götl, strangely enough, and that's quite a bit of a mystery, Götl needed this land determinant. He couldn't have satisfied the difference unless he had a land determinant. And it's hard to understand just what role the land determinant plays in Götl's universe. It's very easy. I say religion for the land determinant in a static universe, which gradually tried to

45:00 collapse the land determinant, tried to prevent the collapsing, and it remains static. So there's a role of the land determinant in the obvious. But anyway, so you start with a simplified metric, you apply the Einstein field equations and the Einstein field equations. They're actually not so bad. Some friends of mine actually did it by computer, I imagine. Given enough energy, we'll put it on them by hand also. These are the various components of the Einstein for the metric that has already incorporated all the symmetries. And they don't look so bad. This W is simply something to do with the rotation part. And that's the only unknown, apart from the K, is the curvature index, and it tells you whether it's, for example, plays out positively or negative zero curve. And then also, there is, yeah, that's the only unknown, really, in that metric, except the lambda term also. Put a lambda term in. Maybe the original hope is that we get away without a lambda term. But the field equations very quickly tell you So these here equations are trivial, they're trivial to satisfy. This is the first one that really counts. And you see immediately you have a lambda term on the right-hand side and a square on the left-hand side, a minus, minus a square on the left-hand side. So it turns you immediately, you can't, if the W describes the rotations, if you want rotation at all, you need a lambda term. In other words, from the first equation, you need to know that lambda has to be negative. You need a negative lambda. The second equation will then tell you is to be negative also. So you already know you need a negative lambda, and you need these slices to be negatively curved. And so it goes to a field equation, and you find out that the only thing that's missing is what is the density, and you find a relationship with the density. And the real mirror is that all the persons are satisfied in a very simple way, which must have been a very surprising hurdle. So, in fact, this is what we've come up with, these are the very values that we find from the field equation you find, as you see, that k is negative, we find a certain lambda, g there's a relation between lambda and rho, which is interesting, and then of course there's also, there are all these relations, in other words, the universe is fully determined, except rho for a scale factor.

47:30 Why are there close time-like norms? Let's just look at the metric. Here's the Gerdel metric in one form. So we found, we put in all the findings that we just have here, and there's also the Simstern metric. So this is the final Gerdel metric. Now you look at that, and with a little experience, you don't need to have to puzzle it right here. You can see, ah yes, there are time rules. And the time rules are rather simple. Take one of these horizontal planes, the Z constant plane. And if you also take time constant, let's time be constant, let's Z be constant, and take a circle, which makes R constant also. So the Z is constant in this plane, you hope that you can make a loop along which the cosmic time T says, your own time will increase, but the cosmic time T says constant, so if you get back, it'll say it doesn't count. And so if Z is constant, T is constant, R is constant, And it would see, then dc is 0, if t's constant dc is 0, if y's constant dc is 0, there's not much stays behind, there's not much remains. This is what remains of the metric, if those things are constant. And you see immediately that you can get ds squared. Now, ds-repositive is a path that's permitted. Here's your positive is what's called a pandemic path. It's a path that an observer can actually trace out. The sweet yellow path on which the ESO is positive means you get traveling. You may need a strong rocket to help you, but you keep traveling. And you can see from this metric here, trivially, that when cosh r is bigger than, just as I have it there, particularly from the second equation there, when cosh r is bigger than 3, that curly bracket is positive, phi is the angle around the circle. So as phi kicks in, that's the only thing that kicks in around the circle, every dozen is constant. You have a time-like curve, a curve along which z is constant. And that gives you your time loops. That's what, there are more complicated time loops. In fact, the golden universe is extraordinary property that from any event, you can travel to any other event. all paths, if you go to the universe, where there are plenty of energy and pockets of age, then you can go from any other

50:00 universe, and you can prove that quite easily. So there's these interesting times after this. I'll just show you how that works out. First of all, this is a sort of a funny diagram with the world lines. The fundamental of the fixed points of the lattice. They're basically all spirals because the lattice rotates so the world lines are certainly not straight. They're sort of spiralized. So these are the world lines of the lattice points. And a relatively, very point in the universe you can define a preferred time. There are many preferred times. So this is one world line and the preferred times are going to this world line. another word language, another preferred language, that's one of, Gurgle was very interested in the fact that it's a special activity, time is not unique with this model. For each, for each, for each lattice point, there's basically a different time. For each lattice, anyway, for each z lattice, there's basically a different time. But I can then straighten out all these lines, and I can draw a diagram like this. This is a diagram where, this is not a space-time diagram, where the light is well as they want to straighten out. These are the straight, it's just a map. And these are the key for constant planes. And this diagram explains why the little term is possible. And these diagrams show you the light cones and it's just exactly the light cones and Einstein's clear as it happens. The center light cone is straight and then as you go away from the center towards the edge, the light cone tilts more and more. This is a map, and in this map, angles get distorted. So this is a light cone, but the angles are 45 degrees. But you can see what happens. The light cone at first is straight, then it tilts, then it tilts. By a time, a certain distance from the center, the light cone tilts and cuts this time axis. That means that these are the planes deep Wisconsin. This means that you can come along go right to the light corner, keep on going, and all the time, the T corner, the cross of the T corner, the states constant, you can travel that, travel a love circle, come back exactly the same spot as you left there, and you would have come back

52:30 with precisely the same, this is a space-time diagram, so each point is not just a point, but it's an event. You would come back with precisely the same event that you left. I think that's how it's up with that. Yeah, clearly we have some questions. So I think we should call the other test and then apply machine. Well, there's a big point in the fact that there's an expert in here. That's the question I would like to ask. Göring's universe, it's not the best universe, because there's time groups. There was space before, there was loads around them. There was space, there was space. All of them had close time groups. Nobody ever noticed them. Until Gerdel found his universe, Gerdel noticed that he had Manu, and only after that, the people go back and find Manu's earlier in the universe also. So in a way, it's like he invented it, but it wasn't quite the first universe that predicted it. He told them blacks to talk about that machine and so on, in general, but he never mentioned Gilda, I think. I don't know. He never mentioned? He never mentioned that you can use content. Keep going. No, he does. Professor Putnam. Yeah, he. I also have . If there is a . I agree with you, not enough. Not enough. That would be interesting stuff to talk about. I mean, keep going. There's always time to change the tunnels, with exotic tunnels, I'm wondering what Einstein was thinking about when he was worried about this. I mean, it's fairly, from a modern perspective, it's fairly trivial, although it's based on closed time to occur, that you're compacted by topologically. Oh, yeah, that is. But what seems to be remarkable about the Newtonian version that the topology is just generic as a graph graph. And was he thinking about this in the sense that Einstein, would that be context in which he was? Well, look, yeah. You think you're thinking like .

55:00 So that's a very trivial case. In fact, in the space, it's not quite trivial, but you can undo it also. So in fact, you find the diagram and you can undo it rather than make it . But larger space, larger space, and Goethe space were the first ones where you can undo it. There are so many pictures. And was that what I just wanted to say? I said, I think we never thought of actually having him, because he really didn't obviously write enough roomers who made them. I mean, he should be one of them. So we had to get any thought about doing this. People would say, well, there's not much of a danger, because I need to be under a lot of time. But he was genuinely afraid of these bits that the, and he didn't say that, but that's, I feel that that's the only way he would have thought. He knew that he was assaulting the icon. And as soon as he talked about that, he realized, my goodness, and maybe that happens. And he was very, very worried about it. It's a beautiful paradigm. He wasn't here. He gave the eyes down. It was a good thing. It was at 1730, it was at 1730. And Iceland, Iceland was very happy with it. Well, he went overboard, I was saying, often did it, it's the greatest working in general in the history of my own, but we were just doing a great job. What about the recreational state? The market, it does, it does, pure dust, and there's no question, but there's a land in it. So you have, so the land is, No, there's no pressure at all. But there is a language there, which in itself is not very... It's strange that you need a negative language. You need a negative language. Now they accept it. Yeah. Well, no, but now we need a positive one. Now we need a positive one. But it's re-accepting positive. My self-accepting is not always fine. I'm sure it's not ready. The portion H2 by SL2Z to get a compact renal surface, it can be. Is it possible? We have H2 cross R. Yes, yes. But I have the actual SL2 requirements at human history. The portion that I get all renal, compact renal surface, except this year. Let us make it go down and we make lots of, lots of compact renal surfaces cross R,

57:30 to discover it. The metric is completely more between these in all directions. It should be okay. It should be okay, yes. We have an infinite number of these things. You have never actually talked about that. Basically it seems it is in all directions. Yes. But then there are compact limits with the strata. Yes. Which of the strata is? Or of Venus. Here they are. Here they are. One more question. What is important to you? I have a paper atop and code. It dismisses these cases with 2 plus 1. There was another proposal, 2 plus 1. I'm not familiar with that. But thank you for pointing out. It seems like, from what you were explaining, time violation is something we talk about when we deal with relativity, and I was curious as to, with Gödel's universe, it seems like when we talk about a time loop, or Imobius, whatever you want to call that, we can call him Imobius for the time he comes with it itself, but when dealing with that, it seems like he's turned the dilation on his head. You had stated that as you leave, let's say, perform a thought experiment that we're leaving, and upon our return, we've spent however much time in this crap, and we're actually at the same point at which we left. Because there's not a rate of change that's marked, at the point of departure, at the point you determine, and time is looped. I'm curious as to, I guess, what would that screw? No, you're right. It's a complicated question. Time dilation is the effect of special relativity,

1:00:00 and time dilation happens in the local regression plane. Right. Now, if you take one of those time loops, there are all sorts of things that happen. At any one point in the loop, there are three times of interest. of the observer in the trials. Here's another one. There's a cosmic time T, we're on a T constant surface. But on top of that, there is the local time of the galaxy that sits at the place we just passed. That's yet another time. That inertia time of the galaxy that sits there does not coincide to the cosmic time. And from all these three things together, you find, you could work out that the observer has the right triangulation in the right place. Locally, it all works out, but it's a complicated, you know, it's a complicated, it's a complicated calculation because we have to work integrated all the way around. Locally, everything's perfectly above all. But it's complicated because you work with these three times. Thank you. I think that perhaps we shouldn't stop now because we're running after now displace ourselves to a new location where we resume at 3 o'clock. We're actually going back to the right campus center where we had lunch. To the second floor. I'm sorry. Take 31. Ask me a question. We're going to right campus center at 2.31 where we resume at 3 o'clock. So it's about the second floor of the building where we had lunch. If they didn't get back, if they didn't rest now, why didn't they call the rest on the war coming out and help them build the context to counter-example and watch them. And when we're walking, rest now. Rest now. Because you just before, you don't come into so much.