A Gravitational Time Arrow

0:00 Thank you. Anyway, that looks promising. All I need to know is... Yeah, that looks perfect. And then I just do... Yeah, maybe that this works. I don't know if you want to try this. Yeah, that looks good, yeah. Maybe that I need to plug this in there to make it up. Oh, wait a minute. What did you do? Yeah, there we go. And then there's a laser pointer as well. Yeah, so when I said you've got two boards, there's the middle, and there's also that. Yeah, if I need it, actually. Show me this one over, do you think you might? No, I think this will be a little bit full, but something to write with this here, yeah. Right, so that's just, yeah, that's fine, yeah. I mean, is there a look there? Yeah, okay. Let's check all of these. These are all pretty good ones. Thank you. So Julian, this is a fresh report. Thank you.

2:30 Thank you. Thank you. Thank you. Thank you. Thank you.

5:00 Thank you. Thank you. Thank you. Thank you.

7:30 Thank you. Thank you. Thank you. Thank you. Well, welcome to this week's Philosophy of Physics seminar. It's a real pleasure to have Julian Barber as today's speaker. Julian will be well known for several of you as a pioneer of a Muckian approach to dynamics, which he's been developing for several years, for a lifetime. But over the last decade, it's taken this particularly interesting turn

10:00 Shape Dynamics, which has several people working on it now, including two of Julian's collaborators mentioned there. So it's a great pleasure. Julian, over to you. Thank you very much. Very nice to be here again. What I'll say is basically pretty simple. Virtually everything is based on Newton's law of gravity for end-point particles. But I think there are aspects of it which haven't really attracted attention and might just give us some hint how to attack the problem at the hour of time. I'm not a great expert on the hour of time. I've taken an interest in it for a long time. I've read Paul Davis's book. I attended an international conference in Spain over 20 years ago on the subject, and every now and then I'd dip into it. No, everybody's interested in the second law of thermodynamics and the hour of time, but in no way I would regard myself as a great expert on all the details. But nevertheless, I hope what I've got to say is of interest. So this is very much a tentative conjecture, but the conjecture is that there is a law of the universe which generates complexity in all generic solutions, and there's no way that you need to invoke special fine-tuned initial conditions to get that result. So that's the basic idea, and the arrow of time is identified with the direction in a solution in which complexity grows, structures form and stabilise. That gravity does this, it's It's been recognised for a long time that gravity has basically an anti-thermodynamic behaviour. But what we, I think, have found is a new way of making this precise, which we find at least interesting, and I hope you will. Please interrupt at any stage with questions and comments, if you would, because I'd be glad of any comments. so first of all I want to ask whether we are thinking about entropy and the arrow of time in an unnecessarily complicated way when we think about the entire universe we know that the history of how the second law was discovered through the behaviour of steam engines a key thing there is that you've got confined spaces where you can change the pressure and the temperature

12:30 and you haven't got any gravitational forces. However, if you're thinking... Now, another fact, of course, about the arrow of time is that there are several arrows of time, the growth of structure through gravitational effects, retarded potentials in electrodynamics, collapse of the wave function, the second law of thermodynamics in the laboratory. Now, is that right to... appropriate way to think about this in the case of the whole universe. Now, the whole universe certainly is not inside a box. So that's the first question. And secondly, I'm only going to talk about classical physics in this talk. And in classical physics, we think there is a unique history of the universe. There's no need really at the conceptual level for anything to do with coarse graining, I beg your pardon. I'm still only a few days back from crossing the Atlantic. So that's one reason why I wonder whether we're thinking about it the right way. And I'm not persuaded that phase space is necessarily the right way to think about it, because certainly in classical physics, all the information that is in phase space actually encoded in the trajectory and configuration space. So maybe at the most primitive level, we should be looking at the behavior of systems in the configuration space. So that's what I'll be doing. So let me just, though, remind you of Boltzmann's suggestion about the arrow of time, which at one stage he suggested that it might be something to do with Poincaré recurrence or certain very rare fluctuations out of equilibrium that you have. So basically the picture that he described was that, so first of all this is the line of time which doesn't have an arrow on it but here is what the universe is like or the area around us and it's most of the time in thermal equilibrium and then every now and then separated by eons there are very deep statistical fluctuations when the entropy goes down to a very low value. And then there is the comment which anticipates the anthropic principle

15:00 that any intelligent beings must be in this area where the entropy is low. And these people will say that the future is the direction in which entropy increases. This is very commonly accepted. And then the people here will say that's their past, and the people here will say that's their past. So you have a scenario with one past and two futures, and if this is in a box of some sort, then these recurrences will occur infinitely often. Now, there's all sorts of problems with that, so I don't think anybody now, so far as I know, takes that too seriously. What more people take seriously, it seems, is what is called the past hypothesis, particularly since the notion of the Big Bang took such a firm hold on people's imagination, that there was a beginning of the universe, and I find very widely the suggestion that to explain the low entropy and the fact that entropy is still increasing around us, you have to say that the universe began at the Big Bang or at some initial state, epoch, in a very special state of very low entropy, and that is referred to as the past hypothesis. Quite a lot of people have suggested that. Now, the suggestion we're going to make is that that may not be necessary at all, and that, in fact, it seems to me that it's a mistake to talk about initial conditions for the universe. Instead, one should try and identify what the law of the universe is and then just look at all the solutions that it has and see what their characteristic properties are. And that's impossible with Einstein's general theory of relativity, which is our best theory at the moment, because it's such a complicated theory. Very wonderful, but complicated. content is that's appropriate, and things like that. But there's a remarkably good model of gravity, which is the Newtonian in-body problem. And that's what I'm going to talk about, and I think bring to light certain properties that have not been recognized. Now, as Oliver said, I've been working on Machian ideas, relational ideas. And Marx said the universe is given only once, with its

17:30 relative motions alone, determinable. And if you think of the universe, it seems very strange that it should be rotating with respect to an external frame of reference, which would give it an angular momentum. And it shouldn't have any position. Leibniz said that long ago, that if you moved the universe to the left by six feet, you wouldn't notice any difference. So that must be something meaningless in that statement. but the thing that's going to be really key in this is what can it mean to say that the universe has a given size and that's going to be absolutely key to my talk so what I'm going to consider is the n-body problem with vanishing angular momentum and the condition on the energy preferably that the energy is exactly zero but most of the things that I'm going to say go through if the energy is positive it just has to be non-negative on. So let me tell you about, you may not know about the n-body problem with zero total energy and zero angular momentum, and in particular the toy model case of the three-body problem. Now this is very interesting, and the behaviour I'm going to describe when the energy and the angular momentum are exactly zero has been known for essentially for 100 years. and it is very interesting so this depicts a complete evolution it's called hyperbolic elliptic escape and there are two histories you can read two histories out of it in the normal way because of the time reversal symmetry of Newtonian dynamics of course I forgot to say that all of these mysteries of the arrow of time arise because the underlying laws seem to be time reversal symmetric So in one picture we have one particle which is coming along this way and at the other side there's a Kepler pair which is a very well formed Kepler pair out here but as it approaches the other particle here there's a period of non-trivial three-body interaction which is pretty chaotic in general and that can lead, as in the example we show here, to a swapping of partners so the blue one that has come in here then pairs up with the green one and they go off and form a Kepler pair down here and the red one goes off up there so that's one way of reading the history but being time reversal symmetric you can just reverse the arrows and you get the opposite history

20:00 this one now comes in and meets that Kepler pair coming that way these arrows should be reversed there's a swap and it goes off the other way and that's puzzling enough I'd like to say it gets even more puzzling if you put this into personal terms and suppose that these are three young people as a pair dancing here, a boy and a girl, coming in and meet a third person, a boy coming in that way, and he pinches the girl and goes off waltzing with her down that way, and that guy goes off rather lonely that way. So that's a sad story for the red one. However, reverse the arrows, and it's exactly the opposite. That's the one who gets the girl and goes off. So I think that brings home really how mysterious the time symmetry is now i'm going to suggest a completely different way of looking at this story which is the one which i think might be more realistic let us let us say that we're going to say that time flows in the direction in which structure forms and becomes stabilized so in this case structure forms if you go from this point along in that here you get a Kepler pair which stabilizes into its orbital elements with ever better accuracy while this particle goes off in the opposite direction. And if you use this Kepler pair, so to speak, as a clock and as a measuring rod, you can see that this one is going off and essentially obeying Newton's law of inertia when it gets out there. So that's one history which starts at that region of three-body interaction. But there's a second one which which goes off and down this way. The Kepler forms down this way, and there's the third particle going off that way. And if you say that the only thing that is observable or meaningful is the instantaneous shape of the triangle, then you should be looking at the shape of the triangle to see what is happening. And you'll see here that basically in the central region you can have typically all shapes of the triangle, scalene, equilateral, isosceles and so forth but as you get out into the asymptotic regions the triangle becomes more and more needle-like there's two particles going around relatively very close to each other and the third one is a long way away and so this is actually a little bit reminiscent of that Poincaré recurrence story

22:30 where you have one past which is here where it's that region of non-trivial interaction where you essentially only have the two-body interaction and inertial motion. And that is the case, well, with one exception, which I'll come to, which is very interesting, for essentially all the solutions. The one exception, well, I will mention it already, is when all three particles collide together. This is called total collision. So in this case, they don't all collide together. is possible, these are what are called zero-measure solutions, but they all collide at once in a total collision at the common centre of mass. And then actually exactly the same thing, very similar behaviour is realised, except that they all come out of a point and then they go off and then they form this thing here of the shape wanders around and you essentially get a kekva pair forming and a third particle going off in the opposite direction. So in this case So instead of two branches, you have just one branch. So that's the situation there. So let's go on and have a look at the next thing and get some idea of why this happened. So now these are the only real equations I'm going to confront you with. I'm going to remind you of two key quantities, really, that are the beginning and end of the n-body problem. First of all is the center of mass moment of inertia. So these are here the coordinates relative to the center of mass. And we're dividing by the mass just to get something that's completely dimensionless with respect to the masses. So this is length squared. And Leibniz discovered that there's an identical relationship. This thing is exactly equal to this quantity, where you take all pairs of particles and the separations between them, you square them and you add them up. And that's the total mass there. So that's one quantity, the center of mass moment of inertia. And if you take the square root of that, it's a characteristic length that is associated with the system. It's the root mean square length. And I'll come back to that in a minute.

25:00 So the other quantity, which is, of course, very important, is the Newton's gravitational constant. And there it is, and I'm sure you're all completely familiar with that. Now, both of these are dimensionful, and really, if you're going to talk about how shapes change, you should really deal in quantities which are dimensionless, and that's going to be an absolutely key thing. I think perhaps the most interesting things that we've unearthed, if we have unearthed it, if they've not been recognised before, is the importance of saying everything must be expressed in terms of dimensionless things. And a shape is dimensionless by definition. It's a triangle. The shape of a triangle is defined by two angles. So it's clearly dimensionless. Now, you can do very interesting things by multiplying the square root of this thing by the Newton gravitational potential. So you multiply those two together, and what you get is something which is very interesting. First of all, it's a pure number because we'd already taken the masses out, but now we've got a length in the numerator and a length in the denominator. So it's dimensionless. good. So it's a function on the possible shapes on shape space, the possible shapes of the system. Yeah? That is an equal sign. That's an identity. That means an identity. It's not a definition of it. Right. now this quantity let's forget all about gravity and suppose I challenged you to find a quantity which is scale invariant and characterises in a mass-weighted way democratically so that each particle comes in in proportion to its mass that characterises the extent to which a set of particles a collection of particles is either uniform or clustered And it's got to be dimensionless. So you've got to divide one length by another length. And it turns out that in this case, there are just two fundamental lengths that come into consideration. They are the centre of mass moment of inertia,

27:30 the square root of it, rather. That's the root mean square length. And this thing is the inverse of what is called the mean harmonic length. And those are the two fundamental independent lengths that you can form. so if you take the ratio of them which essentially means multiplying this by the square root of that by that you get this quantity here which we call the shape complexity or perhaps it could be better called the shape clustering and that's a good quantity to do it because this thing, particularly if n is more than 4 or 5 changes relatively little if two particles even collide and are sitting on top of each other. There's very little change in that because it's the sum of all these squares here, the square root of the sum of all these squares. So the length associated with this collection is a slowly varying function of the configuration, the shape. On the other hand, the Newton potential here blows up if two particles collide. ultraviolet instability in Newtonian dynamics. So that is very sensitive to clustering. So that's quite interesting in its own right. But what I think is really remarkable, and as far as I know it's our observation that this is the case, if you were saying an island universe of n particles, the only thing that is meaningful to ask about its evolution is how its shape changes and you want some way to characterize that the first thing that you come to is this complexity that we've put here but now let's think about it as a potential the Newton potential does two things in Newtonian N-body theory it generates forces that change the shape of the system which is what we are prepared to talk about these are things that in some senses we say yeah that's meaningful, that's observable but it also generates forces tend to change the size of the system. They pull all the particles to the common centre of mass. And if we say that size is a gauge degree of freedom, it's not a physical degree of freedom, it's just some auxiliary thing that we've introduced for convenience of calculations, then we should take that out. So that the Newtonian n-body problem,

30:00 treated in the way that Mach would require, Lo and behold, the potential that drives it is precisely this quantity which measures clustering. So the potential is just built in at the most fundamental of a Newton imagine to generate clusters, generate structures. And we'll see how that happens. So that's that part of the story. So let me show you this. It's very interesting. Now, this is what we call the shape sphere. Now, the only case in which you can represent shape space, that's the space of all possible shapes of the system, is for the three-body problem, because two angles define the shape of a triangle, so that you can plot the triangles. you can represent them as points on the surface of a sphere of two dimensions. So here is the equator down here, and all points in the northern hemisphere have a mirror image. They represent a triangle, so each point on the sphere represents the shape of a triangle, and the points in the northern hemisphere have mirror images in the southern hemisphere. So the points with the same longitude but opposite latitude are mirror images of each other. And then we've plotted here with contour lines and colour shading the values of the shape potential or the complexity. So let me tell you about this because it's really very interesting. At the North Pole, we have the equilateral triangle and at the South Pole, its mirror image. And Lagrange showed already 150 years ago, 140 more like, that the absolute maximum of that shape potential is achieved at the equilateral triangle, whatever the values of the masses. By the way, this is plotted for equal masses, which I think is realistic for field theory. So that's where there's an absolute maximum. And then there are saddle points at what are called the Euler configurations. These are collinear configurations. So all points on the equator are collinear configurations. And here are collinear configurations. They're

32:30 called the Euler configurations, three of them, where one of the three particles is between the other two in such a position that this shape potential passes through a saddle. And these are called central configurations and are the subject of a huge amount of interest in n-body theory. One of the reasons, well, one aspect of them is that if you could imagine holding the system at rest in the shape corresponding to the stationary points of the shape potential and let the particles go from rest, under Newtonian gravity till they all collide at the common center of mass. And that's called a total collision. And in n-body theory, it's asserted that you cannot continue the solution past that point of total collision. Now, we are actually going to question that, and I'll come back to that. I hope I won't forget to come back to it. So then here, where we have infinitely deep wells wells of the shape potential we have where two particles are getting relatively very much closer together than they are to the third one because this would be what is happening in a Kepler pair but it's also purely kinematic and the shape potential gets infinitely negative here not in the way that the Newton potential does. The Newton potential gets infinitely negative if two particles get very close to each other or coincide. That rests on your notion that size has some meaning. Of course, size has meaning in this room. I'm six foot tall and the ceiling is whatever it is, ten foot. So there it's perfectly meaningful, but it's always a ratio. I'm six-tenths of the height to the ceiling. But if you're considering the whole universe, it can't mean anything to say how big it is. It's only ratios that count, and so the shape potential becomes infinitely negative because you've got essentially one short side and two very long ones, and it's that ratio that is going to infinity. It's not because the two particles are colliding, it's because the ratio is going to infinity. So that's the shaped sphere. Now you can't, as I say, central configurations are of great interest. As the number of particles increases, the number of central configurations goes up, rockets up for 16 equal mass particles.

35:00 Somebody who was doing numerical calculations for us readily found over 70,000 on a computer. And so there's lots of them. But as I say, they are of great interest in n-body theory. Next slide. Now comes the thing which is very important as well. And this is the effect of what is called dynamical similarity and the Lagrange-Jacobe relation. Now a potential, and I've got the example of the Newton potential, the Newton potential, well let me take a general potential, any general potential which is a function of the, in this case say the separations, but it's mathematically much more general than just a physical potential. if you multiply all the position vectors by a common constant if that changes the value of the potential by that constant raised to the power k then you say that that potential is homogeneous of degree k and the Newton potential is therefore homogeneous of degree k minus 1 and in the case when then you get what's called dynamical similarity if you find some solution of your equations you multiply all of the distances by this constant alpha and you multiply the times by this quantity alpha to that power then you get another solution and the well-known example of this is Kepler's third law where the period goes is the three halves power of the semi-major axis K is minus 1, this makes this three halves here. But that means that if you have a system with such a potential, then every, there's a complete one parameter family of solutions of the Newtonian theory, which if you believe that size is real and has physical meaning, are all different. But if you project them down to shape space and just see the succession of shapes through which they pass, is the succession of shapes, you don't have any time parameter or any scale, then they

37:30 all collapse down to just one solution. So they should be identified. So this really is a strong indication that you want to be very careful about time and size. They're intimately related and they're both suspect. Now I come to what's called the Lagrange-Jacobe relation. It's also called the virial relation and it's behind the virial theorem. So first And I'll talk about the time derivative in Newtonian theory of the center of mass moment of inertia, which we had before. This is equal to something which we call the dilatational momentum. In n-body theory, strangely, it hasn't been given a name. We call it the dilatational momentum because it has exactly the same dimensions as angular momentum. And whereas angular momentum is a measure of how much motion is in rotation, the dilatational momentum is a measure of how much is in just overall expansion in Newtonian terms. So that's what the moment of inertia is in its time derivative, is that quantity there. Now, it's a very easy application. It's a one-step application of Newton's second law to get what is called the Lagrange-Jacobe relation, the second derivative of the moment of inertia is equal to four times the total energy minus two into this k plus two and then times the potential where k is the degree of homogeneity. So now let's think about Newtonian theory. So if in Newtonian theory k is minus one so this bracket becomes just one. So in Newtonian theory that's the second derivative of the moment of inertia. Now you look at this thing and you recall that the Newton potential is negative definite. So this quantity is positive definite and if the energy is either 0 or greater than 0 the moment of inertia is positive and that means that it's the time derivative of the dilatational momentum is greater than zero, and D is monotonic. Now, this result was very important. Let me see what the next slide is. It's sometimes since I looked at this. Yes. All right. I'm just going to show you one or two key things. So first of all, plot it as a function of Newtonian time, but that arrow shouldn't really be there because it could be just as well point that way. But it follows from that result that

40:00 if the energy is zero or positive, that the moment of inertia is U-shaped upwards. It must go up to infinity. And that was the first qualitative result obtained in dynamics. Very important one. Lagrange realized because of this result, that if the energy is non-negative in the n-body system, at least one particle must escape to infinity, because that is the necessary condition for the moment of inertia eventually to come all the way up to infinity. So that showed that the n-body problem was unstable in that case, and it has lots of implications in n-body theory. And it also stimulated a lot of interest in the much harder problem of what happens if the energy is negative. And that led on to the famous study of the stability of the solar system, the long-term stability, and Poincaré's discovery of chaos. So then it's the time derivative of the moment of inertia. If you start over here, it's infinitely negative. This is the d, the dilatational momentum. That's the time derivative. It comes down, it passes through zero at that point and goes back up on this side here. But it could equally, if you reverse the Newtonian time direction, you just get the opposite slope for the dilatational momentum. But now what is interesting, this is the result of a numerical calculation And what we've also plotted here is the only thing which is really observable, which is the shape potential calculated using the successive shapes of the triangle. All you need is the successive shapes of the triangle. And what you get is a curve like this, which clearly shows a minimum here. And then there are fluctuations as you go up either side. They're not the same on the two sides. and the reason you get these fluctuations it's because of the formation of the Kepler pair the Kepler pairs are formed in general with eccentricity which in this case is relatively large and then you so the particles at some stages are closer to each other than they are at others and that gives rise to these regular oscillations there but meanwhile the third particle is getting ever further away and that's causing the moment of inertia to grow the steady growth of these fluctuations and

42:30 you can prove there are exact results from n-body theory which shows that you get this that the fluctuations grow between monotonically rising bounds when you get sufficiently far from the point where d is equal to zero. So these are exact results I'll just mention here what happens with the zero-measure solutions, which are these ones that are, well, you couldn't manufacture them because it requires incredible fine-tuning, but this curve would just be, you would only have half of that curve there, and the moment of inertia curve would come out of zero, and it would rise up in a concave way, and you'd get half of the curve for d equals zero. Interestingly, d is zero when the moment of inertia, if the energy is zero. So I'm now going to assume that g is 0, because that's a particularly interesting case there. In that case, you can prove that d is equal to 0 when they all collide together. So you get this common pattern. So in this case here, I think you really can say that there's one past here and there are two futures if you look at the way structure is formed. I'm just going to show you some mathematics which brings this home a bit more. You can introduce coordinates on shape space. One measures the extent to which the system is isosceles and away from that. So you've introduced spherical coordinates on that. Don't really worry about these things, the behaviour of this system really as if you were ants living on the shape sphere so that all you're aware of is how the shapes change and then you say how can I describe that well in fact you can describe it as a Hamiltonian system and there is the Hamiltonian it's not a standard Hamiltonian in the first place it's a Hamiltonian constraint a certain quantity is equal to zero this is the statement representation, the energy of the system is exactly zero. And then you can, because that d, and this I think is really very significant, this d is monotonic, and it can serve as a time variable. And I think at this stage I need to put one or two things on the board there.

45:00 So, first of all, time is invisible, and so is size, and time we believe is monotonic. It seems to go forward all the time. So what does it really mean to say that we have time as an independent variable in Newtonian dynamics? It means the following. So, if I give you initial conditions rA for the positions of the particles, and I give you initial velocities, I will get a certain solution. But I can multiply all these velocities by... I can leave the initial positions unchanged, but I can multiply all the velocities by a constant. and then I get a different solution and that's a different solution in shape space and I would say this is really fundamentally what that absolute time in Newtonian theory means it's the ability to generate a one parameter family of solutions and what is happening in this case that you're changing the total kinetic energy at the initial instant and the total energy of the system at the start. Now, we're insisting that the energy is zero. Now, the energy zero and angular momentum zero are particularly interesting, that case there, because that is scale invariant. If you have a system with non-zero energy and you choose particular units, inches or centimetres, the value of the energy and the angular momentum will change. But if the energy is either zero or infinite, infinite. If it's 0, you can't change 0. 0 is 0 in all units that you can choose. So this is what the n-body people call the scale invariant case. And in that case, you get down to the situation that if you go back to the shape sphere, a point and a tangent vector define a solution uniquely. Now, a tangent

47:30 vector, or it's a velocity on shape space, but what exactly is that velocity? And the really interesting thing is the following. You can, in the Newtonian n-body problem, if you approach it in a marking way, you can kill the energy and you can kill the angular momentum. But you cannot kill the energy that is in overall expansion or fact it's due to the fact that the Newton potential does not commute with the Hamiltonian that's the technical reason so in Newtonian theory there is always and that means that the moment of inertia will always change, the moment of inertia cannot stay constant, it must change that means that at any instant there's a ratio of kinetic energy the change of shape divided by the energy in change of dilatational kinetic energy. And this is a one-parameter freedom, so that on shape space, when you plot the, when you generate Newtonian solutions and then you project them down to shape space, you will find that if you choose at any point So that's an initial shape of the triangle and an initial direction. There will be a one-parameter family of solutions which deviate from there and peel off. And that's entirely due to this fact that in Newtonian theory you can't kill that. And this is very interesting because that dilatational momentum is monotonic and it's invisible if we say only shapes count. So this is actually having exactly the same effect as the Newtonian absolute time does in the standard way that you think about it. So our suggestion is that really, there isn't any independent time, but if you buy this idea of the Newton potential, and there is a very close analogue in general relativity, then really the time variable is to do with this fact that in Newton at the moment of inertia can change, and in a dynamically closed universe in Einstein theory, the volume of the universe can change. And what you ultimately get, the dimensionless quantity, is you choose in the Newtonian theory a certain value of the dilatational momentum here,

50:00 and then you define a time parameter tau as the ratio of d over d0. And that's a monotonic quantity. So it has the two key properties of time in Newtonian theory. It is monotonic and it's invisible. and then you actually get a really interesting fact I can just go into this a little bit there is the question of if we accept that this is our theory of the universe our toy model of the universe what is the minimal data you would say that you understand the theory if you know how to generate all the solutions with the minimal data. They must be dimensionless, and it's the minimal data. And there's a very interesting way to do this, which I think has considerable implications for quantum gravity to do with the search for observables, and we can perhaps discuss that in questions afterwards. If you go to any point that is not at d equals zero, then there will be some proportion of the kinetic energy the Newtonian kinetic energy in overall expansion and a certain proportion in change of shape but if you go to this magic point here this is precisely the point where the moment of inertia is not changing there's no overall expansion it's all pure change of shape so now what you can do is So I'm going to specify all the solutions in the following way. I'm going to specify, I'm going to say, I'm going to go to that point at which the solution is going to pass through d equals 0. So my initial data is d equals 0, but I'm not giving a number for that, and that's scale invariant data. And then I specify a shape on shape space, this is the shape of the system at that point, in which it's moving. And that data characterises, parametrizes every single solution of the Newtonian n-body problem with zero energy and zero angular momentum. And I'm not aware that this is sort of... people in the n-body... I talked to n-body specialists in Paris and

52:30 And, I mean, essentially they've been doing shape dynamics all the time, but they always start from the Newtonian picture with the Newtonian second law. And although these things are immediately obvious to them, it isn't the way they've been thinking about it. They don't think about the problem about the hour of time. They're doing wonderful mathematics, finding exact solutions, finding when you can get integrable results and all sorts of things like that. So I think this is quite an interesting thing. And as I say, I think this has great significance potentially in quantum gravity. So that's really, I think, quite promising from this point of view. Where have I lost the pointer? Where have I put it down? How smart are you, Ollie? Thank you. Right, let's go on, because I'm getting near, I think, to the end of the thing. Ah, yes. So let me just show you this thing. So this is when we take D as that time parameter, and we get a time-dependent Hamiltonian. So D divided by D naught is, or D squared, this can be your time parameter. So what you've got here is a Hamiltonian, which is time-dependent. And those of you who've been following what I've tried to do for many years, about 15 years ago I got very keen on the idea of getting rid of size altogether from dynamics, because I thought shapes are the only thing dynamics of pure shape so I hit upon the idea of a theory which is described by geodesics on shape space where a point in a direction you would only need a point in a direction wherever you were to define a solution then you would get a geodesic solution and this is actually the Hamiltonian you get in that case you change, instead of having the Newton potential so the Newton potential is that and you divide it by the square root of the moment of inertia that gives you a potential which is homogeneous

55:00 of degree minus 2 and that actually gives you then a geodesic theory on shape space and for a while I thought this was just what I wanted but the interesting thing about this is when you plot, and Flavio McCartney did it when you plot these geodesics and you look what that shape potential does now that shape potential is a measure of how clustered the system is how much structure is formed and when you plot this, the curves of this thing and see what that shape potential does or the complexity it just goes up and down and no interesting structure forms it never forms stable structure so it's actually a very boring world And then you add just one more thing, which really is, in Newtonian terms, size itself is never meaningful, but in Newtonian terms it is meaningful to say the universe now is twice as big as it was at some earlier epoch. that's really what the expansion of the universe is about and this is really this time parameter we're introducing is really just it's more or less the Hubble time parameter and it's related to red shifts which are ratios, dimensionless numbers so we're very close to very basic facts but I find it really interesting from the point of view of dynamics of pure shape One sort of flaw in the system that you need, just one more parameter to characterise the thing, actually leads to all that's interesting in the world, the formation of structure. And I suspect that it's the same in general relativity, because there is something called the York time, which is closely analogous to our dilatational momentum. Let me just go on, because I'm getting near the end of what I want to say. Yes, if you actually, I won't go into all of that thing. I just want to say, you can introduce a new time variable here, which is this thing, which is the logarithm of that quantity. Then you redefine the momenta. After we've done this, we found that n-body people are doing the same sort of thing well just to characterize the solutions but instead of getting standard equation to motion you get extra terms which look just like friction so you get these these friction terms and this actually explains in a very simple intuitive way what is going on on shape space what those uh

57:30 oh whatever i need to go forward because the one that i want to show is forward it's here so now you get this interesting situation that the what is the this shows a motion of the system so the point d equals zero is on the back here where that cross is that's where the dilatation momentum goes through zero so there are two arms of the solution which go off in different directions from there and they eventually the yellow one comes round the back comes round here and it spirals down there and the red one does that now the way you can basically describe this is when the system all the time is subject to a certain amount of friction when represented in this way, this is what the ants on shaped space would find, that they've got a system subject to friction but when it's up here away from the potentials the effect of the friction is relatively small Here are potential wells, and when they get near the potential wells, the friction is acting on them. They just get dragged inexorably into the potential wells. And you get a very clear understanding of all the entire, all the solutions just are very easily understood. They must all come up out of one, depending on the time direction, they come up out of one of the potential wells and fall down another one, or fall back down the same one. But the key thing is there's always this magic point here corresponding to where you can specify the data. And in some senses, I think really you could argue that Newtonian dynamics is not time-symmetric at all. It's time-asymmetric if you say you start at this point here. Because if you were creatures living inside these universes and you made observations when the Kepler pairs have formed or in the n-body problem when you've got more Kepler pairs formed you could only determine the orbital elements to a certain accuracy and if you tried to calculate back to what would happen as you go through this point here you would just get a fog here and you wouldn't be able to say what would be the solution out the other side

1:00:00 so in practical terms these are two distinct solutions they are separate and in some senses this is the beginning of time for each of the halves they are effectively independent so I think it's I would say that is the more realistic the more objective way to think about the end body problem when you get in this case here of scale invariance, where it's zero energy, zero angular momentum. Let me also make the point that it's non-trivial that this happens. The Lagrange-Jacobe relation relies on... Well, first of all, the dynamical similarity gets rid of that one parameter family. So that's the first thing. But that Lagrange-Jacobe relation that you have, That depends upon the Newton potential being homogeneous of length to the minus 1. It also depends upon the fact that it's length to the minus 1. It's crucially important that the Newton potential is negative definite. There are no local minima in this shape potential. and maxima. And finally, we've imposed the requirement that the energy is zero, that's scale invariance. So these are very plausible. Well, first of all, the scale invariance is a very plausible requirement. The other ones are just given to us by nature. And they seem to come out of very profound mathematics as well, because as I say, that shape potential is just the ratio of the two most fundamental, the two fundamental lengths that you can form in a democratic, mass-weighted manner. So that's all very deep. And the same thing happens when you look at Einstein's theory. As I say, we're nowhere near able to give a definitive story there like we are for the N-body problem. But the fact that the York time is monotonic also hinges on about three things that come together to give the result. And it does seem to be equally sort of fundamental and basic. So I think I've got nearly to the end of what I want to say on this thing, and we can go on too. So this is actually the solution with three particles, and that's a thousand-body simulation that somebody did,

1:02:30 where you see these fluctuations, not surprisingly with all those particles, when you get a thousand, how the fluctuations even out, and you get a very steady growth of the shape potential or the complexity. And I think this is going to be my last slide. Yeah, this is an artist's impression, and it may be a little bit overselling the thing. What we know in the three-body problem is there's always one Kepler pair is formed. That's absolutely the case. whether when you have a lot of particles that's another matter now I've long been convinced that the way to understand the universe is to look at its shape space and something in the Newtonian case that is a potential to find on it you can't do that quite so easily in Einstein's theory but nevertheless I think it's definitely possible to say which are the most uniform shapes that the universe can have And for me, the beginning of time is not where time is, it's where the universe is most uniform. And what I find very encouraging, really, is if you take all of the typical, the generic solutions, they will always go through a shape where in Newtonian terms the system is most compressed and it has the smallest moment of inertia. But basically it will be a more uniform distribution. either side of this most uniform distribution which is somewhere up here the system will evolve and form structures in both directions so you will have two hours of time emerging from there and then you get these asymptotic rods and clocks and stored information if you say that the settling down of the orbital elements if you represent that as binary digits then they're settling down ever more stably so you're getting information dynamical information is being stored Let me also say that Einstein in his autobiographical notes near the end of his life said that he'd committed a sin when he created general relativity in that he'd introduced rods and clocks as extraneous elements which just in some magical way measured the four-dimensional metric. Now Harvey knows all about this. He's written a long book on it. Basically I'm very much in sympathy with this viewpoint. So what we're seeing here is actually that the rods and clocks

1:05:00 are forming spontaneously out of the dynamics. Quite how many, it may be a bit optimistic, this many shows. We do know in the real universe this has happened, or it is there. I mean, whether in the real universe there's a bounce or whether it's something more like a total collision, I wouldn't like to conjecture, but nevertheless that is happening. And also, by the way, this is a very nice way, when you just say everything is expressed in terms of ratios, I think you get away from these puzzles in cosmology about what does expansion mean? What is the universe expanding into? But more serious is the question that many people ask. When the universe expands, what doesn't? There's a very nice paper written on that, and it's quite a puzzling thing. And people sort of talk about systems, either classical systems or quantum systems, sort of cutting themselves off from the expansion of the universe. It's a delicate argument, and so forth. But here it just happens quite spontaneously, because what is interesting here is once these Kepler pairs have formed, the ratios of their semi-major axes and their periods remain constant relative to each other. But the universe is expanding as measured by the diameters that you get here, because they're flying apart there. So you get a completely uniform account of what seems to be this very mysterious expansion of the universe. all just changing of ratios, which must ultimately be the better way to look at it. So with that sort of rather optimistic loss on our observations, I'll stop. Thanks very much. Thank you.

1:07:30 Yes, no, I wasn't actually referring to... No, no, I mean, it's perfectly... I mean, everybody says, yes, the diameters of the galaxies remain basically constant relative to the separations between the galaxies. That's the observational fact, yes. But actually, a lot of the mathematical and intuitive way people think terms of space expanding. I mean, people say space is literally expanding. You will find that very, very common. And I think this may be a way of, that perhaps needs challenging a little bit. Yeah. No, no, I mean, nothing I've said would surprise a cosmologist. It's just, I mean, what I think is new in this is that we've got these precise quantitative scale-invariant measures of complexity and the shape potential. Now, I mean, there's been a long history of people trying to find a definition for gravitational entropy. And, well, a lot of them have developed the original ideas of Roger Penrose. And virtually all of them, to some extent, try and mimic standard entropy in that they base it on four-dimensional geometry, which is including the kinetic energy of change, the dynamical part, and just the shape part. And we are saying perhaps much more fundamental is just how the shape is changing. So our definition and our arrows of time are based purely on the shape. They're not on the shape and the way the shape is changing. It's just the shape that does it. So I think that this could be interesting to go to this more primitive, more basic way of thinking about things. Harvey. Julian, I'm struggling to understand intuitively how the monotonicity of D comes about in a theory that's essentially time-signatory. So let me compare this with several other cases. You think, for example, of stiscal mechanics of Boltzmann's H theorem. So H changes monotonically, but of course you're putting in, you know, molecular chaos to start to launch out its conditions. The time and that time makes a difference. So there's nothing terribly surprising there. Another case might be in thermodynamics and certain axiomatization of thermodynamics. A recent one by Weaver Ingleson, you introduce an entropy function which you show to be monotonic.

1:10:00 of the actions that lead to that are time symmetric, but in fact, hidden in there somewhere is a certain condition of error. So again, it's not altogether surprising. Well, then of course, there's just standard statistical mechanics, if you like. The standard what? Standard statistical mechanics, where if you introduce the past hypothesis, well, again, you have the possibility of solutions which have monotonic behavior with respect to some in some function or in some physical quantity. But again, that's because you're looking at a non-generic set of solutions. But you're not looking at, how? I'm looking at every time solution. You're looking at every solution. Yeah. The fundamental theory, I mean, despite what you said at the very end when you said that you don't think the atomic mechanics is really deep down time reversal invariant, when you're defining D, I haven't seen anything in the mathematics that pulls out a time direction. It pulls the time direction out from either side of d equals 0. There's two time directions pointing in opposite directions from d equals 0. Now, by the way, I don't want to put too much emphasis on saying that Newtonian dynamics is really time asymmetric. I would say it is for ants living on shape space can't see when they can only see the part of the history in which they're living so to speak where they're making observations and they can't see through this this molecular chaos what this what this does is show that every solution has a region of molecular chaos out of which structure grows in both directions that's just an inescapable consequence of lagrange's great discovery 150 years again. By the way, the n-body people emphasize this. When you read n-body texts, I've been doing a bit, they talk about the importance of this Lyapunov function. They use it a lot. It shows, for example, that the n-body, if the energy is non-negative, the n-body problem cannot have any periodic solutions. It just follows immediately. Out of that. But the behaviour of the shape-complexity function is time-symmetric.

1:12:30 Qualitatively. It's qualitatively time-symmetric. But D is qualitatively asymmetric. D is monotonic. No, but you can reverse it. I mean, let's go back to the... Where was that? isn't it? No, forward. D could be either, you could, if you reverse the direction of the Newtonian time, D goes in the other direction. Sure, but it's the monotonicity that's... It's the monotonicity. That's one of the greatest discoveries in dynamics. Lagrange was a great was it backwards I think it must be backwards yes there we are yes it's the this relation here this relation here this is the Lagrange Jacobi relation it's essentially what Lagrange found in the 18th century this is just once you need one application of Newton's second law to change one term in the time you formally calculate the time derivative the second time derivative of the moment of inertia use Newton's second law, and then this is an absolute inescapable consequence of Newton's second law and the homogeneity of the potential, then all you need is to say the energy is non-negative and it comes up. That's the degree of homogeneity of the potential. So it's K is minus 1. So this is where the marvellous properties of the Newton potential come in. K is minus 1 for the Newton potential, so this changes that to 1, and you get it just to 2 there. is negative definite, there's lots of potentials that are not negative definite, but Newton is and that means that this quantity is positive definite so that if the energy is non-negative your moment of inertia is concave upwards and its derivative is monotonic. Just a little observation on that if you look at the definition of T there if you do T goes to minus T time-reversible, B is going to go to minus B, so D goes to minus D, and then your D by DT, both D and T, both have negatives, so the time-reversible one gives a new D D by DT being right from 0.

1:15:00 That's how you're going to get the monotonic D when you do a time-reversible, because D itself is also negative. Yes, so the time-derivative, whichever way you go in that curve, the D is increasing. if you start on one side it's infinitely negative and it goes up to being infinitely positive it's just the same so it's always improving it's horizontal there D goes through zero and that's this you see the moment of inertia just must go through zero exactly once and that's where D goes through zero and I think this is this is very deep in the dynamics dynamics and it's very remarkable that the same sort of structure is present in general relativity. It depends a bit on things, and general relativity is so much more complicated. I don't want to stick my neck out too much. Oh sorry, it's a minimum, but d goes through zero. But you can, I mean what I think is interesting is those solutions where it can, the total collision, where the moment of inertia goes to zero, but it still comes up, concave upwards, and d is gross, the time derivative of d is monotonic, is positive, and d is monotonic. But the characteristic of that point, though, when there's a future in one way and a future in the other way, is the moment of inertia, or is that a moment? Yes, now, let me go forward, because I've been in discussions with the n-body people, And I think I'm beginning to persuade them that we might get them to change what they say in n-body theory. So the standard story is that the system can reach a total collision where you can't continue the solution any further. the n-body people mean by saying you can't continue the solution any further is that if you have a two-body collision, you can continue it. There's a well-known way of doing it which is said by continuity with respect to the initial data. So if you have just say a heavy mass here, an infinite mass and a particle which comes in and hits it

1:17:30 and it could bounce back again so it would come back in again. However, if you give it almost exactly the same initial data but with a very small impact parameter it will come in as a hairpin and go around like a hairpin and as you make the impact parameter ever smaller the hairpin gets ever closer to a straight line so they say that you can regularise the two body collisions in that way but if you try and do that with three body collisions it's very easy to show that it won't happen because you only have to slightly change the things and the history will be quite different this picture here, here's the equilateral triangle, so you can either hit it at the... Yes, that's very important. I'll do it with a... Let me go back to where I show the shapes here. In the first place. There. Thank you, Shaila. If you... You can come up to this point here, but if you very slightly change the parameters here or the conditions out here you'll find that the curve will go off in a completely different way from one that's very close to it So this is why the n-body people say you cannot regularise the three-body collision That's the end of the thing and they say it's sort of like the Big Bang However, when we thought about this and we've been thinking about this for a while we came to the conclusion that actually in some senses you can, by this device here you can and I've had initial discussions with the two N-body specialists in Paris and they didn't say you're wrong Julian, they said yeah you may have a point you can if you find such a solution then it's uniquely characterized by the fact that it hits a central collision, that it hits a central configuration there that's where the shape potential is has its maximum And at that point, D is equal to zero. And the solution is just defined by the shape, which is that you're at the equilateral triangle, and the direction at which it leaves it. And so that's a way of characterising that. That uniquely characterises that one special solution that does it. Then there's a dual solution, which is where it just goes through and goes out in the opposite direction. And I said to the N-body people, can't you think about it in that way? And their initial reaction was, after some discussion, well, perhaps you've got a point. think about it, but if you're restricted to the shape sphere. So there is a solution.

1:20:00 So even in that case, it'll go through, so in that curve here, where the moment of inertia, here's the moment of inertia, and here's the Newton potential, it'll just come down, and then it'll go off, it'll hit the point, so to speak, bounce off. And then it will start fluctuating, and the shape potential will fluctuate, because in both directions, as Kepler pairs will form. By the way, there is one thing which I'm going to meet up with one of the n-body people in Barcelona in a couple of weeks' time. There is a very interesting thing, which I've really got the n-body people struggling over. There are solutions which come out of a Kepler pair and a distant particle, where they interact, and then they go to the equilateral triangle. But it's not where they all collide, it's where they go into what's called parabolic escape where they go off forever in the shape of an equilateral triangle and the question is if you just see the curve on shape space can you tell that apart and they were really struggling with this question whether you could tell total collision from total collapse I suspect that you can but I'm hoping to clarify this question in Barcelona in a couple of weeks time but let me say another thing which is very beautiful the fact that you've got scale invariance is that you've got this complete control over every solution. All of the typical ones just come out of the deep potential and go back down again, or go into a different one. Or, in very rare cases, they might, in ordinary end-body terms, come to a rest there and end there. But in our view, they might just go on the other side there. So this might just lead to ideas about the other side of the Big Bang or something like this if these sort of ideas go through into general relativity. But that's very, very speculative. So Julian, I have a question or a sort of cluster of questions around this d equals 0, sd, we're characterising the minimum data you need. So one of the things I'm struggling with that when you have a given configuration space if you really have a theory that's

1:22:30 constructed using those variables then the initial data should be of the form configuration plus direction and of course this is giving one extra bit of information that you're characterizing by saying at this particular point, the d equals zero point the solution, you can give a point in a direction, but it has to be at that point. So I have a few questions based around that. So one is, does the fact that a point in a direction doesn't specify a solution by itself mean that when you plot all the solutions for the theory on this sphere, you're going to have two solutions, two distinct solutions passing through the same? you'll have a one-parameter family if you go to any point and you take a point in a direction there's a one-parameter family but now this is very interesting because this is related to very basic questions in canonical quantum gravity and a few years ago I had a there are things called perennials this is a coining of Carol Kuchos a perennial is a quantity which does not change as the system evolves. It's some eternal characterisation of the solution. And if you try and find these in general relativity as it's in the way it's normally described with foliation invariance so that you can choose any definition of simultaneity. Nobody knows any perennials in the theory there at all. And Jonathan Halliwell a few years ago was grappling with this problem and dealing with this question of the interpretation of Dirac's theorem and asking about perennials. and he wanted he said that actually in some ways you can say that in an ordinary dynamical theory if you specify a point in phase space which is equivalent to a point and a tangent vector that gives you a solution and in some senses you can call those perennials because they don't change, once you've specified that you've got your solution but Jonathan made this very important point I think now with hindsight It's just one bit of redundant information, because it's telling you not only what the solution is, but it's telling you where you've started it, where you are on that solution at that point.

1:25:00 And that can't be anything to do with the solution. There's a bit of redundant information there. So the question is, is there any way in which you could specify the complete set of perennials that uniquely distinguish a solution in that way? this was Jonathan's question and it's only recently occurred to me that there is a way when you've got this magical point d equals zero because if two independent mathematicians were given a solution and told to specify in dimensionless form the perennials that uniquely define it they wouldn't agree because they would each go to different points in phase space to do it but they could if there's this magic point d equals zero there because you can say I go to that point and there you get shape invariant data You get an unambiguous specification of the data. And it's just what you want. And I'm hopeful that this is significant for observables in quantum gravity. And it could be also in general relativity because shape dynamics gives us a unique notion of simultaneity. So we may get it there. So it's, and it's not just some trivial point. Ah, you've happened to take a point which is sitting there by accident. It's not there by accident. There's very deep things in Newtonian theory. and if this goes through to general relativity there too, that creates that unique point. That point there is there for very deep reasons. So is this... So, considering an arbitrary point on the trajectory, this data will not be given by... I'm wondering to what extent this can be expressed as a function of what would be the canonical data for this theory so given a solution I can give you this perennial but the way you've introduced us to it it says go to this point on the trajectory and then work out well actually that's not quite I don't say what it is I say you go to any point you like on shape space and choose a direction and then you say I'm going to find the solution that goes through this point with this direction and at that point d is equal to zero and then our equations will generate that solution for you and then you do that at every point in shape space

1:27:30 and every direction and you generate all the solutions so it generates everything for you I see I don't think I do see it, but can you just tell me again, if I take a point on shape-space and direction, you see I have a one-parameter family solution. Yeah. Passing that point with that direction, how does that one-parameter, what is it characterising? In Newtonian terms, it is characterising the amount of kinetic energy there is in change of shape, and how much is in change of size. And you can't see that on shape-space. So this was my other question. you need, you can understand as a point and a direction, just in terms of the numbers, if the configuration space has scale. If the configuration space has scale, then... I don't want to contemplate such a thing. It's introducing dimensionful things. Wait a minute, but let me think about it. So the question is, why should we think of this as a truly scale invariant theory, Because the initial data, the solution, all the solutions can be generated through scale-invariant data, dimensionless scale-invariant data, because d equals 0 is scale-invariant, and shape and direction is also scale-invariant. by the way let me just say it's fascinating talking to these N-body people who've been terrific help to me they just don't think this way you can really see them struggling to get their heads around taking shape size out of out of dynamics perhaps it might help just to there's a well known theorem called the velocity decomposition theorem which says that if you're in the centre of mass the centre of mass kinetic energy. Somebody called Sari, his name is associated with this. So Sari's velocity decomposition says the centre of mass kinetic energy, at a given instant, can be uniquely decomposed into the shape kinetic energy, plus the rotational kinetic energy,

1:30:00 it's how much the shape is changing it's a function of the change of shape and if you by mocking arguments you can kill that in eternity you can't kill that so this always gives that one parameter uncontrollable thing in this level and that's what I wanted to get rid of dynamics of shaped space with my paper from 2003. But now I've sort of abandoned that extreme idea, and now I find that this is actually in many ways much more satisfactory. There's a lot of positive features, I think. But you're denying that the true way to understand this extra degree of freedom is as a ratio of the shape kinetic energy to the dilatational kinetic energy. I don't think there is dilatational kinetic energy. I think it's... I'll accept both ways of describing it because I would call... You see, the Newtonian represent... It's just to do... It's always in all gauge theories. It's much easier to write the equations using gauge variables. And Newtonian theory is just the same. An inertial frame of reference is essentially redundant for describing what is actually happening, but it's jolly convenient for writing down the equations And this is something very... I think this is something very deep to do with the nature of geometry. I could go into that, but perhaps over supper. I don't know. People may want to go and have a drink in the Royal Oak, but I mean, I'm happy to go on, but people might want to leave. I don't know. I'm not trying to stop, but... Yeah. Quick question. Remind me, how did we get the friction terms that showed up? That's just... By the way, this is all in a... You can see all of the details in this paper on the archive. It's 13, 10, 5, 1, 6, 7 by the three of us, GRQC. Intuitively, you can see it must come, because what we're essentially doing is we're taking the kinetic... In the Newtonian terms, we're taking the kinetic... So, in Newtonian theory, energy is conserved.

1:32:30 In this theory, we're taking the kinetic energy in expansion and using it to define a time variable. So we're pinching some of the kinetic energy. So we can't have energy conservation. So it must show up as something like friction terms. Can you put the picture up where you have the orbits, the trajectories spiralling down the... Yeah. So one thing I was not clear on is to what extent the story relied on your choice of E for zero. So presumably for E positive, you still get this kind of... No, this is actually what I wanted to say, which is so beautiful about this thing. When E is positive, you can, the trajectory, you can get escape to infinity, where all the particles go off to infinity, in principle as any scalene triangle. So the curve on shape space, as it tends to infinity, would just stop at any scalene triangle. And that's actually a violation of the principle of sufficient reason. Because on shape space, you can't find any explanation for why that thing stops at some bizarre scalene triangle. But when you impose the energy equal to zero, then you get this beautiful behaviour that there is nothing inexplicable. The shape potential explains everything that happens, together with the quadratic kinetic energy, which is the only sort of choice that you have comes out of that. So it's, by the way, I think, you know, Einstein had all this, when Einstein was creating general relativity, he said, the fact that there are distinguished frames of reference is an affront to the principle of sufficient reason. What is it? Why is one frame of reference distinguished rather than another? But Poincaré, I think, had this much more basic and fundamental, much more illuminating way of thinking about it, what you must look at is the initial value problem you must try and identify what intuitively you think are the relational the true relational degrees of freedom and then say my theory must be such that a point an initial position initial relational data and the rates of change of the initial data determine the solution uniquely and that's a much more precise, accurate thing

1:35:00 and then you avoid these things with the thing So this is exactly what we get when the energy is zero. We get this, we eliminate this, the front to the principle of sufficient reason where it can stop at any of the scale you're trying to revive. It can't do that. It must. I mean, I use the analogy that the trajectories cling to the shapes here like a wet dress to a lady's body. so you know it's got to be it's got to respect that there and it's the power of a rational law I would say law is determining all the solutions that can exist now the quantum theory I think is a wave function evolving on shaped space and if you're interested that's in this paper as well our ideas about that so your quantum theory the conjecture is that it's going to be Now, yes, I'm a real apostate, very appropriate as my name is Julian. I must put this on my website soon. I'm still getting asked to talk about the non-existence of time. But it's certainly not Newtonian time. No, I think we... No, and it's... We conjecture a wave equation, a time-dependent Schrodinger equation with a time-dependent Hamiltonian. And it's very nice, as far as we can see, You specify a wave function as an initial type, an initial wave function, and you let the thing evolve. It's time-reversible, so it goes in both directions. Very soon the wave function just settles down into the potential world, so it's exactly what you'd expect. So you seem to get a superposition of three systems with a Kepler pair and a third particle going off there. It's just what you would think. So you've got all the problems of many worlds and so forth. And the interesting thing is that there's absolutely no constant in it. There's no Planck's constant or anything like that. So the quantum effects, the expansion of the universe, everything just comes out of formation of structure. It all just comes out of an equation which hasn't got any constants in it. It's just... Can you say a few more words I think I just missed it so you said twice in passing that it's striking that

1:37:30 there is an analogue of I think the behaviour of D in this theory to the volume of the universe in Jian The parallel the moment of inertia goes with the volume of the universe and the time The derivative of each is the time, in either Newtonian or Einsteinian terms, the time derivative of that is the thing that's monotonic. So this was the York time, that York, interestingly, when York originally introduced it, he just said it has a tendency to increase. This is known for 40 years, but to mine, we couldn't find where he'd actually really gone through and proved for non-trivial examples that it is monotonic, but we have done it in this, well, Flavio's done it in this paper here. certainly this plays an important part the York work on solving the initial value problem known to quite a number of people who work in relativity and there certainly is a close parallel but quite how close is I wouldn't want to say at this stage but it gives us hope you say just one more word what exactly is parallel Now it's in terms of well let me make one other observation if you look in either Newtonian theory or general relativity you have, you can have in Newtonian theory you can have as many degrees of freedom which describe the shape of the system and there's always just one that describes the size so that's in Lagrangian terms so then the moment of inertia and the dilatational momentum are canonically conjugate pair so the dilatational momentum is the conjugate to the moment the canonical conjugate to the moment of inertia and in general relativity the York time is the canonical conjugate of the volume and there's always this is very striking you see you get all these beautiful shape degrees of freedom in France, and you just get this one invisible canonical conjugate pair

1:40:00 and the one that and one of them is monotonic thank you so I suggest we're aiming to eat earlier than normal, so if we move discussion on to an informal putting now it's time to have a drink at the right rate before we move on Let's thank the speaker. Thanks very much. Thank you. I think they have a very interesting line of criticism, arguing that you can't make sense of these things even as representation, given normal ways of thinking about what it takes for something to be representation. And, of course, all of this type of objection that they have just quantify what you think is an evolving view of the community distribution. So, I... Yes, anyway, I'd be interested in it. Who was the person who was? Christy Miller from Sydney, and David Brannes. So, Christy Miller has been writing papers about how to make sense of causation, Well, I mean, I don't know whether to say, I believe in time now or not.

1:42:30 I mean, for me, the key thing is the curve on shapes. And there's a one-parameter family that is going through any given point with any given section. And for me, that, I mean, it's all come because I've finally got...