Timeless Relational Physics & the Arrow of Time

0:00 Thank you very much for the invitation to speak. I'd like to just sort of start off by saying my overall idea of what the universe is like, where our notion of time comes from out of a complete timeless universe and where the arrow of time comes from. That's a conjecture, that idea. Then I'm going to say why I think it's a reasonable conjecture and provide some argument for the structure of non-quandemic physics or why one should take it so quite seriously. Let me start with what I think is actually the single most significant empirical fact about the universe which is that I think it's full of, it's absolutely full of things that I call science actually. And the two best examples I know are the Earth, its entire geological record, and the human brain. if you if you take one would say that my brain could be frozen as it is now with my neurons and all this connection perhaps not now but in a few decades somebody examining my brain could read out the entire story of my life, everything about my life, out of 10 billion or so neurons with a trillion or so connections between them, somehow encodes my entire life and within that from my own internal experience that sort of within my memory I continually get confirmation of these things that they're mutually consistent I remember what I think of as earlier encounters with Bert's car with his beard and just as he looks And when we meet, and he starts talking to me about it, it's evident that he carries those same records. So we could now put Bernard's brain together and my brain together,

2:30 and some other scientists could examine the two together, and they would find mutually consistent records. And the thing that I think is most striking about that is that all of this evidence for a past, recorded in structures in the present, entirely static structures as far as I'm concerned. Once my brain is frozen and a record has been taken of all the various connections, the essential bits that the neurologist would need, nothing more changes. It's there forever. And yet this static thing contains incredibly eloquent evidence for evolution, formation of records, some sort of processes taking place in time that laid out records and lead to a consistent story. And equally remarkable is the story of geology. And that is, of course, actually where people started to get the idea that the Earth had an immensely long history, much longer than the 6,000 years of the future, etc., and all that. And it all comes from mutual consistency of records. You can drill down through the ice in Greenland determine a record of what the temperature, find the temperature of the earth for, I don't know, whatever it is, 10, 20,000 years. You can then go to some desert in the middle of China and look at the sand there and analyze the sand and you get the same record of temperature dates from there. An absolutely incredibly mutually consistent thing. And the truth is everywhere you can get your hands on anything in the universe, actually it tells a story and done with and read with sufficient skill these stories are all this is the story of cosmology of evolution and everything it's a fantastic story and I think this is the single most significant fact and the point that I especially want to emphasize is that these things are static the rocks from which Lyle and Darwin deduced the antiquity of the earth and things like that, they're essentially unchanged from the time those insights were the evidence is still there. So, I propose a theory of the universe in which there is no time at all. There are just probabilities for different static configurations of the universe, and my suggestion

5:00 is that actually the law, the timeless law, which is just like Schrodinger's time-independent wave equation. The law that determines how the quantum probability is distributed in the space of all possible static configurations of the universe has a built-in tendency which makes it have a high value, the probability have a high value, on precisely these configurations that I call time capsules. And my conjecture is that timeless quantum cosmology works in exactly the opposite way to classical physics. The problem with classical physics coupled with statistics is that everything argues that we should live in an utterly uninteresting universe. You always finish up confining yourself that it's statistically hugely unlikely that we should be in such an ordered state, organized state as we are now. is that the passage from time-dependent classical dynamics to time independent or timeless quantum cosmology reverses that and that there is there is precisely a built-in thing about the way the world works which enhances the probability of remarkable structures very rich structures above all time Now, let me now talk about, I'm mostly going to talk about the Newtonian three-body problem, which is still an extremely very difficult problem. In Newtonian physics, let me talk about the variables used in Newtonian physics. Now, a triangle here represents the relative configuration of three particles. I just imagine I have three particles in Newtonian physics, so I've got three particles in Euclidean properties. So they will always form a triangle. And so just imagine, please, three mass points of the vertices of my triangle. Now, in Newtonian physics, the three-body problem is described by no less than ten variables. There are three variables which describe the separations of the particles. There are three There are three that describe the position of the centre of mass. There are three that describe its orientation in space. You can point the normal anywhere on the celestial sphere and then you rotate about the normal.

7:30 And then there's the time, the absolute time. So that's no less than ten variables describing the instantaneous state of the universe. Now if you take a relational point of view, a la Leibniz or Marx, You would question whether those ten are really necessary. In fact, objectively, if this three-particle system of the whole universe, you'd be inclined to think that actually only the three sides of the triangle is what counts. Because you can't see anything else. You can't see absolute time play. All you're aware of is, so to speak, the relative positions of the objects of the universe. So, on this view, instead of describing the three-body problem by ten variables, you also describe it by just three, the three sides of the triangle. You could have time, I mean, you can see that the three sides of the triangle changes. Yes, you've got a clock. If you've got a clock, I've got a clock, we don't have a clock sitting out there on the wall. I've actually brought my watch with me because you haven't got one here, and that's quite appropriate. I want to actually build up the rich structure of the world with the absolute minimum amount of structure that I need. And certainly I haven't got as far as I would like to go yet. There's one more step that you could take, which is to say that the size of the triangle has no meaning. And that really only the change should come. One does speak of the equilateral triangle, one does say an equilateral triangle, you say the equilateral triangle. So that's sort of a very nice way of doing it. Now I think this process could be taken further, but it leads me to think about what I call dynamics of pure shape, where the only thing that really counts is the shape of the instantaneous structure of the universe. Do you understand how you can throw the mentor away like that? Well, we'll build up momentum, I'll show you how momentum can be, because there are no individual momentum of particles, there's only changes of the relative separations. I mean, if I have two triangles, I have two triangles, these are nearly the same, they're not quite the same. Things have changed there, those don't move, I can't bring them to exact congruence, they're not exactly congruence. So things have changed. So one thing is moving relative to another.

10:00 So one of the tasks of relational physics is to show how the Newtonian concepts of position in an inertial frame of reference, a momentum, and all of these things arise out of that. And it is perfectly possible. That can be done, and that has been done. So I want to get down to the absolute bare minimum. So, the suspect, there is what I would say, suspect kinematics, let me just put it down here. So, the suspect kinematics is time, the centre of mass coordinates, the orientation and the scale. And the ideal for me, at least at this stage, is a dynamics of pure shape. And it's going to eliminate all this suspect structure, and in that sense it's going to be radical, but it's going to retain the irreducible basic concepts, the notion of a configuration space and dynamical curves. So in that sense it's going to be very conservative. My key notion is actually the thing that Euler and Lagrange worked with with Hamilton in setting up variational mechanics. So in that sense it's going to be very conservative. Now let me remind you, when Schrödinger created polymechanics, he took actually Newtonian Kinevarius, the Schrödinger wave function for a three-body system, and this is all too seldom spelled out with books on polymechanics, even a text, but the amazing thing about the Schrödinger wave function for the three-body problem, and this was why he immediately lost his argument with Heisenberg, and in fact he was aware of himself already when he wrote the same paper on something that his attempts to interpret one of the mechanics of his wave function as a probability distribution of charges. The way that controlling a wave function depends upon the same ten arguments that I listed. It depends upon the relative configuration of the system, but it also depends upon the central mass position of the complete system, its orientation, and the time. So you have this wave function which depends on all these things there. Now, I'm saying that all the mechanics of the whole universe should not depend upon that suspect, kinematics. It should only depend upon the part that is truly irreducible. And the ideal would be that it would only depend upon the shape of the three-volume problems,

12:30 that you should have a function which depends on that. let me just say, so just very rule of thumb, quantization, what is happening to the, what will happen if we eliminate the non-shaped variables, the wave function will become progressively simpler, it will not depend upon the time, so it will be a stationary, in the ordinary way you say it will be stationary, I would say it's static, so it doesn't depend upon, it's just a static wave function there. All you need to do is say the system is isolated and then static anyway, because the time is handled apart from the solution is handled out. Yes, I would say yes, but it could be a superposition of energy eigenstates and still be isolated, but in fact the suggestion is that that's not the case. there would be no translations so this would mean that the energy is essentially one value, preferably zero this would mean there's no translation so the momentum of the system is zero well you can always achieve that for an isolated system non-trivial is that there would be no dependence on the orientation that means you would have a state of zero angular momentum and finally the scale, it would bend on shape. So that your wave function of the universe would just be a function of shape, of the shape of the configuration that you're considering. And that is actually the simplest and most naive interpretation of the Wiener-DeWitt equation, the equation that DeWitt passed in. You've got a wave function without time. I've got a wave function without time, and it depends essentially, it doesn't go as far The wheel of the width equation sort of stops at this point. What would a wave without time? Every time you walk along the beach and you see wave patterns on the beach, that's a wave without time. It's a wave in space, not in time. And since it's timeless, you're saying it's an instantaneous configuration? I'll now show you how I see the overall universe if I may, This is my picture here. In my view, one of these triangles

15:00 represents what I mean by an instance of time. If we go outside, we're getting clocks, and clocks I think are really quite minor things in the universe, really. They reflect the deep property of the universe. If you were to ask yourself, how would we recognize this instance now again? Or we take a picture of us all here and say, this actually defined the instant that I'm talking about now. And in terms of my three particles, one instantaneous configuration of three particles is an instant of time. So there is one instant of time, and there is another instant of time. And when I give popular talks, I say, now watch carefully, I'll show you what two instances of time are like, and I'm going to show you what eternity is. This is eternity for the three-body problem. three axes represent the three sides of the triangle, okay? My three axes just represent the three sides of the triangle. So each of those triangles is represented by a single point there. So a single point here is a configuration of the universe and simultaneously what I call an instant of time, okay? There's no, there's not going to be any linear time or anything like that in this, but there's going to be a great even instance of time, and they're all possible configurations of the universe. And because there are three sides of the triangle, the relative configuration space of the three-body problem is a three-dimensional space, which has, I believe, an extraordinarily interesting structure to which not sufficient attention has been drawn. It's what's called a stratified manifesto. There are frontiers to the space of all possibilities. There are structures, a very structured thing. The equilateral tribe will fly on the line and run up the middle of the thing there, okay? Now, what are these sheets here that they're... collinear configurations. A collinear configuration where one particle comes down and is high between the other two is just a triangle. It doesn't violate the triangle inequality. So those are all the points that lie on the sheets represent all possible collinear configurations. And all the points out here which are mathematically possible in my plot correspond to triangles which violate the triangle inequality. They don't exist. So it seems to me a very interesting thing that the space ball and it has frontiers. Now these ribs here are the two particle collisions where you have a linear configuration where two of the particles are sitting on top of each other and one is at some separation.

17:30 So those are the two particle collisions and finally the state where everything is sitting on top of itself and that's the three body collision. And this is a sort of nested hierarchical structure. The strata of higher symmetry and lower dimension are nested within the bigger thing and sit there. And so the thing that really strikes you about this is that it's highly asymmetrical. I think this is very... I think that the deepest reality about dynamics is that it's defined on a configuration space. The configuration space, by their very nature, somehow to do with the way structures can exist at all, and spatial relationships become, is massively lopsided. There is a huge asymmetry in the arena of the world. Let me just mention that you can go shape space. All of the triangles on one of these cups through here have the same perimeter. So each point on here is a representative for a possible shape of the triangle. And now you have another very interesting distinguished point, which is the equilateral triangle. when everything is sitting upon each other, that's a bit like the Big Bang, where everything is sitting on top of each other, and that's the distinguished state. If you have a theory of physics in which the scale has no significance, the distinguished point is the most uniform state that you can have. Equilateral triangle is the most uniform structure that you can have, and clearly if you have n particles, there's no theorems about which maximizes the Newtonian potential, for example. The n-body configuration which maximizes the Newtonian gravitational potential is in a well-defined sense the most symmetric state you can have. Very interesting. And interestingly, the physics at such a point is going to be extremely well behaved, completely without singularity, very much unlike what happens in theorem and scale, where you have big banks and things like that. So that's my idea of the sort of the timeless configuration space of the universe. This is the model of it. Obviously, it's multidimensional, infinite dimensional if you have fields here. But my main conjecture is the following. That quantum cosmology is just about a time or a static wave function

20:00 defined on this configuration space, which satisfies something like the time-independent Schrodinger equation. That is actually, now if you think about it, I think that when Schrodinger found his time-independent Schrodinger equation, he actually found the rules that create structure, the great dream of explaining where structure comes from. The time-independent Schrodinger equation contains no time, and yet it explains the structure of everything ultimately, really, that we see. the DNA molecules and things, I think you can only understand their structure partly in terms of the history of things like that. But basically, what the Schrodinger equation does is say which configurations have the highest probability. And it's done in a completely timeless fashion. There is an explanation for it, but there's no... The structure of the water molecule has nothing whatever to do with the history of the water molecule. It's a completely timeless thing. What I'm saying is that really the whole universe is like a giant molecule in which there are just different probabilities, timeless probabilities, static probabilities for different configurations. And these different configurations are actually what we call instant stars. And this is where I really start hand-waving. My conjecture, and I'm not going to give any of them now, but there are arguments which point in that direction. is that it is somehow this hugely asymmetrical structure of the configuration space, which causes the, for a much more complicated thing of the three-poly system, because I can't represent a time-tancel with a three-poly configuration, I need at least ten particles, and I need to begin to get something which would look like carrying semantics of meaning of those. But my conjecture is that actually this huge asymmetry of the configuration space causes the timeless wave function of the universe to be concentrated on configurations which are always time capsules, so that if scientists could look at these structures, they'd say, ah, these have arisen through a process that has taken place in time in accordance with sort of laws which are some sort of strange mixture of quantum and classical laws, and starting with a very special initial condition of low entropy. And I'm just wanting to completely sweep away all that story

22:30 and say that the really fundamental thing is that there's just a very configuration and these things are collective and have a nice possibility. And therefore, special structures, particularly structures that seem to carry a memory of a past, are just built into it. If you like, the past is either this special state down here in some senses, But there's no unique past, there's no unique curve of history. This would be a classical history inside my configurations, but there's nothing like that whatever in this story. You just have these configurations which come like that. So that's the conjecture. Let me just say that there is one huge asymmetry in our existence, but the asymmetry between the past and the future. A colossal asymmetry. And I think it's very striking that when you look at the foundation of dynamics and think about it, there's a colossal asymmetry, lopsidedness in the foundations of a thing. I conjecture that actually the most fundamental concept, there are various spaces which play a very fundamental role in physics. There's the configuration space, there's the phase space, there's the space-time, and there's the Hilbert space. My conjecture that actually the deepest and most fundamental of all is the configuration space. And it's the necessary asymmetry in any realistic configuration space of the universe, which is the ultimate origin of the arrow of time and our experience of the profound difference between the past and the future. Now, in my book, The End of Time, I give some arguments supporting this thing, and worked out not by me, by other people, which have gone some way, I think, to making that picture of mine plausible. I'm going to stop now talking about the quantum side of the universe and say why I think that that's a sensible picture to take. Now, what happens so far in quantum theories, all the really significant quantum theories we have, up to getting to the Yank-Mill theories, in all cases, one has started with a classical theory to which one has then applied rules of compensation, these sort of add-on rules that they react by. and you go from a classical theory to a quantum theory. What I want to just give you some idea is how you can define a perfectly good classical dynamics of the universe

25:00 in which all that suspect kinematics, which I talked about at the start, is completely absolute. You can do everything with the relative configurations and nothing else. Now, in the last four years, I've been collaborating closely with Nilo Morku, the only Gaelic-speaking relativist of the world, and various students, one of whom is at Vernon's department in London. And I think we've got, we've made a lot of progress in making this Leibnizian, Markian relational view of classical physics look very much more cogent than it was at the time that I wrote my book. But we've got some, I think, some very exciting results. It's not yet by any means decisive. There are open issues about scaling variants, which I don't think we're going to be able to resolve. But certainly some things have got very much more, looking very much better shape from my point of view. And I just want to tell you a little bit about what's sort of been going on there. If anybody's interested, we've recently put four very large papers on the web, all published in classical and polygraphy in the last year. two last year or two this year, which sets it out. And as I say, I think this goes a long way of vindicating our view, but at least making it more cogent. The basic idea is really to take seriously the idea of three-dimensional structures, three-dimensional geometrical structures defined by three-dimensional geometries of some sort. I think that Western philosophy is a series of footnotes for Plato, and I don't think that I'd buy that one. What if I hit him? I would say that the history of physics is a series of footnotes for Euclid, actually. that you have actually this fabulous axiomatization of Euclidean space, and then it had one really significant development when you've got rid of the fifth axiom, the axiom of parallels, and you've got Riemannian space, three-dimensional Riemannian space. And Clifford, you may remember, who translated Riemann's works in English in 1866, conjectured that there wasn't any matter really in the world.

27:30 It was just three-dimensional remaining geometry, which was evolving in time. Now, it was just two years later that Muck started advocating purely relational physics in very important terms. If Clifford and Mark had got together and had said, let's make a Machian theory, a relational theory of the evolution of Riemannian, three jockies, and had set about it in a way which I think is quite plausible, and had added to that requirement the idea that there should be no instantaneous action of distance, there shouldn't be any growth of instantaneous action of distance, They could actually, at that stage, have found general relativity, show that there was a universal light code and got gauge theory all out of these market ideas. I just want to give you some hints to show you how that can be done and why that's plausible. So really the idea is, but there is still a great issue in my mind, and this is the issue about quantitation of general relativity when you attempt to do it through a canonical approach, which is more fundamental? Are the notions of dynamic configuration space and phase space the most fundamental concept, or is space-time the most fundamental one? And this issue is not yet resolved, but I do think that what we've managed to do, and it's very largely some lovely insights of Nilo Mokun, I think has shifted it further in the direction of saying that it's the dynamical context, space-based, configuration context. So let me just try and give you some indication of what that is. I think I started at about 10 past, did I? Shall I go for about quarter of an hour, 20 minutes, and then have sort of 5 to 10 minutes questions or something like that? I think we should finish your talk with the questions. Yeah, yeah, that's what it is. Yes, with the R. Yes, I like it. Although I don't believe in time, I try to be confident. I always tease my children by saying I'll be at home at 1507, you know, things like that. Right, so the Newtonian configuration space of the n particle problem has three n dimensions. The relative configuration space, which I've shown you there, has 3n minus 6.

30:00 And then the shape space, which is just the space of possible shapes, has 3n minus 7 coordinates. And you can do exactly analogous things to the idea of three-dimensional Riemannian geometries. There is a concept there as green. Imagine a compact three-dimensional manifold on which you define a Riemannian metric. That space is called green, and that is analogous to the Newtonian configuration space. A three-dimensional metric contains information about the spatial coordinates you're using and the geometrical relationships. Precisely, you've got a three-by-three symmetric tensor. So the six numbers at each spacepoint, essentially three of them are giving you information about the coordinates and three are giving you information about the geometry. So you can do what the mathematician calls quotient. You can quotient that space ream by the three-dimensional dipheomorphisms and you get the space of three-dimensional Riemannian geometries, that's an interdimensional space, you can go one step further and you can define something called conformal superspace, which is analogous to my shape space and has only two degrees of freedom of space. Now, if you take my point of view and you look at Newtonian structure, there are two fundamental problems that have to be overcome to set up a theory of dynamics which only uses these relational quantities and really could be said to go back just to Euclid and Riemann and is not using anything else. I'm going to assume that there's a principle of least action and that you contemplate continuous curves in the configuration space. So here's a continuous curve in the configuration space. Now there are two problems you face coming from Newtonian dynamics when you want to make a relational dynamic. of them is that in newtonian theory you have a curve like that and moreover you have a spot of life moving along it at a certain speed which is so to speak the speed at which the universe is evolving with respect to the absolute time and that freedom to have an absolute time in your theory is crucial because this is what is so to speak the speed a a momentum but the momentum

32:30 is essentially the momentum of a single particle is a direction in space plus a speed in that direction and the same goes for the whole universe is like that the history of the universe is going in a certain direction in its configuration space and it's going at a certain speed and that certain speed gives all of the magnitudes Now, the first requirement we must have is that the history of the universe is not something, it's not a path that is traversed at a certain speed, it is just the path, it is nothing else, it is just the path. That is the relational notion of time that undermines this thing. So we have to have an action principle which does not make any use of an external time. And that is very easily done. It's well known how that's done. It's done with what's called a reparameterization invariant action. Instead of defining things with respect to an absolute time, you've just introduced some monotonically increasing parameter lambda along your curve. So any parameter which increases monotonically along their lambda, you then take all your derivatives with respect to that lambda, you write down your action principle, but you insist that that action principle is not changed if you go to a completely different lambda. And that is called reparameterization invariance. And that means that the essential initial condition is not an initial position and an initial direction with a spin, it's just an initial position with an initial direction. That is what's crucial. So that's the first requirement, and that's going to be achieved by having a re-parameterization invariant thing. Now the other thing is that your action, let me just remind you of the notion of a group orbit. If I have a three-particle configuration like this, I can act on my triangle with the Euclidean generators of translations and rotations and get to a different configuration, which is a different point in the Newtonian configuration space, but is an identical point in here. And all of those configurations which can be brought to exact congruence in the Newtonian scheme by translations and rotations form the orbit of that configuration under the translations and rotations, and they are mapped to a single point down here.

35:00 So what my action must do, I must have an action which does not depend upon the individual points in the Newtonian configuration space, but is completely determined by the curve in the relative configuration space. So we're talking about an action which depends only on the orbits in the configuration space. If I use the Newtonian configuration space, the action must depend only on the orbit. There are curves in the Newtonian configuration space, these are group orbits, and these are curves that go through the same orbits. These are just different representations because at each instant, at each instant I've taken a different translation and rotation. I've done a different Euphlidean transformation on it. And I must be able to do this changes completely arbitrary. These must be lambda dependent. Lambda is my label along here. transformations which take me to different points along the orbit, shift along the orbit, must be completely independent. I must be able to do them at each point in a completely different way. So each transformation as I go along is independent of the ones that went before. And my action must depend only on those things. That's a very well-defined problem, and there's actually a very well-defined way of solving it. What you're really going to What I'm really going to do is to define a metric on this relative configuration space. I'm going to say that between any two neighboring points here, there's a distance which depends only on these two points. And what I'm actually going to do is consider those two triangles. What I've really got to say is find, associate some number, which I call the distance between those two particles, which is actually just determined by those two triangles and nothing else, okay? So I'm going to do that. And the answer is very simple, and I will actually conjecture that it is actually the most fundamental principle of classical physics, and it's applied to the whole of known classical physics, and it's a process called best matching. Suppose you had to tell me, give me a number which depends upon only those two triangles. Peter has got to come up with a number which depends on the end of his triangle. Bernard has got to do it in two separate groups, and they've got to come back and give us the same answer.

37:30 How are they going to do that during those two triangles? Well, there is one obvious way which is very closely related to the underlying geometrical structure we're talking about, and that is what I call best matching. Let me start off by just putting those two triangles like that in an arbitrary position with one on top of each other. This will give me little dx's here. So if particle 1 is in the 2 configuration, this will give me a dx1, this will give me a dx2, and this will give me a dx3. So if there's a short colour, you can write down the formula. I can contemplate this thing. These are my little dx, let me take these as factorial. So I weight them with the masses like this. I square them and then I take the square root for some arbitrary positioning like that that clearly is not going to solve my problem because it depends on the arbitrary position I've done but I now use the Euclidean symmetries to move one of the triangles relative to the other and bring it into a position of best matching where this thing is reduced to a minimum and that is a unique minimum and further than Peter will find the same answer if they go into different rooms So this defines a metric on this configuration space. And then I can say the history of the universe is just a continuous curve, which is a geodesic of this metric here. I'll give you your exact expression for your dynamic time that you were talking about yesterday, It's coming in the next triangle. But it's not done with time. It's done with triangle. There's no time. Everything is done with a triangle. I do nothing except with a triangle. So let me just take you through that, because that's really very interesting, the two things. Let me first of all show you the elimination of time, and then I'll take you through the... Just a little bit about that, and then I think I'll stop. There's a thing called Jacobi's principle, which far too few people know about. Jacobi's principle is the following. You can ask in Newtonian physics for the orbit of the system, irrespective of the speed at which the system goes round its orbit.

40:00 The classic example is precisely the planets. When you solve the Kepler two-body problem in Newtonian physics, you start off by finding the orbit, the elliptical orbit, and then you use the energy theorem to find out how parts of the planet can go through other systems there. And Jacobi's principle is just the formalization of this to an end-body system. Very interesting. It is a geodesic principle. These are the Newtonian displacements in an inertial frame of reference. I won't get the Newtonian integration to place at the moment. Space hatching will be doing something to these DXIs here to get rid of that problem there. The E here appears as a number, but it does turn out in Newtonian physics to be the total energy of the system. And V is the potential energy. So the square roots tell you that you've got a geodesic principle here. And so the orbits in the Newtonian system at one fixed energy are all described by geodesics with respect to this metric. this is what's called the kinetic metric and then it's multiplied by a conformal factor which gives you introduces the forces there now the key thing about this which I think is very important so that the here's a t here which is this expression there it's manifestly invariant under reparameterization when I do the x by the lambda I've got two dx by the lambdas here but I take a square root so they just cancel against that lambda out there so it's obviously I want to tell you, I want to draw particular attention to the structure of the canonical momentum. The canonical momentum are the key things in both Lagrangian and Hamiltonian mechanics. The canonical momentum, because it's your Euler-Lagrange equation, are the PI by D lambda equals DL by D lambda, which is the forces. Now, if you look at these canonical momenta, you see the relational notion of time in it. These are like direction cosines. You're not dividing by time. You're taking a small displacement, and you're dividing it by the displacements of everything. Notice that you've got your capital T.

42:30 which is quadratic in the displacement, is down in the denominator. So I've got a quadratic thing here taking its square root, and I've got the little dx's there. Now this is exactly a direction cosine. So if I go here in my model here, from the origin to some point out here, I've got various little dx's. Now I'm not, and I go from one point to another point, that generates little dx. I'm not taking the, let me draw it here, let me just do it in two dimensions. Here's an initial point, and here I go to another point. This could be a particle in two dimensions. And I get a dx and a dy here. Now, my velocities are not dx by dt, by the time, they're dx by the hypotenuse. And those are direction cotimes. Time is really the total of all the displacements in the universe. So that's what the time measure is. This is very much more satisfactory. You're not measuring a displacement with respect to something which is completely different than independent time. You're measuring it with respect to other displacements, which are actually all of the displacements that are taking place in the universe in the process that you're considering. So this is very satisfactory, and it results, if you take direction cosines, you square them, you add them up, you get one. So there's an identity. So this is reflected in a square root identity, which has this form. So essentially, these are actually direction cosines multiplied by the square root of e minus v. So not surprisingly, when you square them and add them, you get this relationship set. So that's very important there. Now let me show you Tony's dynamic time where it comes in there. The equations of timeless physics, if you write down the order of the Grosz equations for this Jacobi system, you get this rather ugly expression, which cries out to be simplified. There is a unique choice of your lambda. I've not put any time into this. but there is a unique lambda that I can always choose to make this thing, this pesky fellow here, exactly equal to 1. Then my equations simplify massively and I get Newton's second thought. So this just shows that you don't need to introduce any absolute time for an isolated dynamic system.

45:00 So a system of n-particles in Euclidean space, you can do it all with respect to that. Now, the real beauty about this is that this is your dynamic time, I think, that is. This is the ephemeris time or Newtonian time of the astronomers, where they found the Earth possible to be a new phenomenon. This is not the energy theorem now, it's the definition of time. So here you get your increment of the distinguished time, which you can call a lapse for people who know about the lapse and the ADM-pharmism of general relativity. It's the weighted average of all the displacements in the universe. So I don't think you could have a more satisfactory and a complete theory of time than that. Time is just the weighted average of everything that changes. So if I have an infinitesimal change, a small change of everything in the universe, what is really determining speed is this quantity here or there. So that couldn't really, I think, be more satisfying. Now, I'm not... I'm running out of time as well, but I always want to say you're a little bit too long there. Let me just say that this relationship here shows that you can do without time completely in Newtonian physics. Now when you go to general relativity and Riemannian geometry, and you say this to be no action at a distance, I can't go into this, you'll have to read the papers about it. Something extremely similar to this happens, except that this relationship holds at a given point. At a given point in the standard model, you need, I think, something like a hundred different numbers to characterize the instantaneous state that a given point displays. You take all the Yangville fields that are in principle, the gravity field and all the fermions and so forth. You finish up with something like a hundred different things at each space point. So if you go from one space-time point to another one, at the same instant you get to a neighboring point or to some different one, you will get a whole, something like a hundred of these things there. And if you look at the key, when you represent general relativity as a dynamical theory,

47:30 the canonical momentum, the canonical momentum are the key things because those are the things that the Oil and the Grange equations govern the change of. The canonical momentum in such a theory are not anything to do with how these things change with respect to any time label or anything that's not time. It's just a ratio of each of those hundred things with the sum of the squares of the whole hundred together. So again, you see that the fundamental dynamical thing in general relativity is not containing anything to do with an external time. It's all completely relative. And this is very beautiful. This is to do with the Gauss-Kadazi relations. the Gauss-Terranium-Agradium this is actually very beautiful this is actually, it turns out to be Gauss-Terranium-Agradium the central and most important thing which is the Hamiltonian constraint in general relativity and ADR is actually exactly expressing this fact that there is no time locally, the variations are all, all of your 100 or so things are varying relative to each other that is what counts in general relativity it is nothing about time It is about how all your hundred variables are changing relative to each other. Let me just say, because I've only got four minutes to do that, I'm going to take some shortcuts now. In the, let me just say, if you take this idea of best matching seriously, what comes out is actually you recover, from the particle system, you recover Newtonian mechanics exactly. you get a dynamic deterioration or emergence of absolute time and inertial frames of reference. Within these things you have exactly Newton's laws, but in the N-body case, particle dynamics case, you get in addition that the total momentum is zero, well that's high exactly, the total angular momentum is zero, that is highly non-trivial, the total angular momentum is zero. the individual momentum of pi, so some pi equals zero, that's nothing new, that's the momentum is zero. The total angular momentum, which is not a trivial relation, that's zero.

50:00 And then if you do a scale invariant, something that I call the expansive momentum, which is a new thing, but I won't say totally much about it, but you can get this thing dotted into the Pi to zero. That's what you get in particle mechanics, and in addition, you get that the energy is exactly zero if you're in a scale of variance theory. When you come to general relativity, let me just say what you actually get out of these things you have space-time, which I will foldate like this, and time is going up like that, these will be two space-like hypersurfaces there. And if these hypersurfaces are embedded in a reachy-flat space-time, that's a space-time that satisfies the Einstein-Teal equations, then you get two fundamental relations, which are these Gauss-Gadatsui relations, as I told you, that give you a relationship between between the intrinsic curvature here and the extrinsic curvature, this is the internal geometry in the three-dimensional hypersurface, and how it is bent in the embedded four-dimensional space-time. And what we have now managed to show, as I say, with the great help of Nilo Merkur, is that this whole structure of space-time and the Gauss-Gonacci embedding relations, which tell you that you're in space-time, follow exactly from these three ideas that there is no time, that it is done by best matching, that the dynamical principle is based on best matching, and that there is no instantaneous action of existence. Those three criteria, in the way that I've outlined them, lead exactly to Einsteinian physics, and you get exactly the Gauss-Kadazi embedding conditions, which are everything you need. The Gauss condition is a quadratic thing. It's a quadratic constraint. It's just like that. It is a local direction cosine identity. And that is actually contained everything. The Hamiltonian constraint is really everything in general relativity. It encapsulates the whole of the dynamics. And that is really reflecting this idea that there is no time in general activity. The Codazzi condition is what is called the momentum constraint.

52:30 That arises from the best matching. And the fact that you have these constraints, there's four of them all together, one at each space point, that comes about because of the idea that there's no action in the business. So it's really quite impressive, I think. And therefore, if the classical physics that we know, matter field, you get a remarkable derivation of special relativity, you get the fact that the universal light code, you can start off by doing gravity, and then you can let gravity interact with matter fields, and you find that those matter fields must reflect a universal light code, you get a derivation of that, that's presupposed in the third thing, and moreover, if you have a vector field, you try and make a vector field interact with gravity, you find that it has to be a gauge field and the fact that it's a gauge field and the fact that there's a universe like that turned out to be just different manifestations of the same thing it's all a consequence of this idea of best managing extended very coherently and very consistently. Now this isn't the whole story but still open ends and things like that but I think it's for me it's very encouraging and it's now according to my watch it's one minute past I have some questions, if there are any. Julian, can I ask? Excuse my voice. You say you have no action at the distance. But it seems you have correlation of some kind at the distance. Because how can you talk about the whole in this way? Oh yes, you do. This is... I mean, when people ask me about action at the distance, I say, well, geometrical relationships are action at some distance par axelon. I mean, there's no question that everything is knit together and it all hangs together. What I mean by no action at the distance is a very specific requirement that you write that down. So if you're just going to do pure general relativity without matter, your dynamical variable is going to be gij, which represents the three-dimensional geometry. So this is a three-dimensional dimension thing. Now you're going to have an action principle, which looks like this. The action is going to be an integral over my lambda thing, an integral over all space.

55:00 of a Lagrange function which depends upon Gij and Gij by the lambda. Now, what I mean by a local theory is that at each space point you calculate a quantity which depends upon only these things at that given space point. then you integrate it over the home of space and then you integrate it with respect to your private label or your my lambda parameter like that now if you were to introduce what you could do and we found actually quite an interesting scale invariant generalization of general relativity looks just like general relativity but to get this theory we have to introduce it to the Lagrange function but the Lagrange function in addition depends upon the volume of the universe integral over three-dimensional space of square root of the determinant of G. So I've got a Lagrange function which has within it an integral over the whole space. And this is a clear element of action of the distance. Now this is really a rather interesting theory. We call it conformal gravity. It's virtually indistinguishable from general relativity, but there's no meaning to expansion of space. And when we were creating the theory, we were I'm rather hopeful, of course, this would be the answer, but I have to say we're now rather sceptical about that, and I think it's more just an interesting mathematical theory. It does say something about trying to make a scale invariant here. Now, let me just say why... So this is bad news when we're trying to do something scale invariant, but it's good news for us, because when you have a field theory, there's two ways in which you can take a square root. If you remember the action principle, Jacobi's principle, has E minus V to the square root, and then the kinetic energy to the square root. So that's the Lagrange function of classical dynamics in Jacobi's principle. That's our. Now, if you have a field theory, what we've got is something that's called the byline-Sharth-Wheeler action for general relativity. actually by like Sharpe Weaver, has the following form. It's integral over the lambda, integral over these three eggs,

57:30 square root of g, square root of the three-dimensional square curve, square root of capital P, which is the kinetic energy. Now, all of these things are calculated at each space point. These depend upon only these things, essentially, at each space point. So what you're doing is calculating, and this is quadratic, so you're So you're calculating here something which is quadratic in the velocity at each spacepoint. Then you're taking the square root of each spacepoint, and then you're integrating over all of space. Now, this is a local action principle. Now, the more natural generalization of Jacobi's principle would have the following form. integral d3x square root of g, square root of capital R, takes the square root of that after you integrate and multiply it by the same thing, square root integral d, square root of g. There's two ways you could do things. You could first integrate, this is like a potential energy, this is like a kinetic energy. You could first of all integrate over space and then take the square root. This is what we call global square root. Or you could do the local one where you just integrate, you take the square root and then you integrate over space. This has got an instantaneous action and a distance in it. And this one has. and the idea that there's no time and we're dealing with bosonic physics means there's got to be a square root in the action and in field theory there's two places you can take that square root if you take the square root in this way you get a theory which has got action at the distance if you take the square root in this way you recover the whole of relativity you get this whole possibility of completely changing the way you define simultaneity the light curve and you get the gauge principle all coming out of this choice of a local thing so the thing that kills our idea of having a scale invariant theory that's bad news but the same principle but a locality principle saying you've got to have no intense action existence but if you say at the level without where we're not trying to do scale invariant we just try to make a theory which is not scaling there but it's got everything else that you would want then and it comes from the nocality of the requirement of the nocality.

1:00:00 Any other questions? This is best similarity. Best matching. Best matching. I thought the equation was about squaring the distances between the vertices of the refinance and then sum it. Was that right? Then you take a square root. You square one and sum it. Take a square root. Is there a particular formulation that's necessary, or could there be other formulations of best matching without best covering? Well, there are different choices, but there is a unique sense in which best matching is distinguished. is if you look at, if you have types of transformations like this, you get a transformation, this transformation of using the generators to go along the orbit in completely time-independent ways, generates a transformation of your velocities. Your velocities require correction terms, and the form of the correction terms is explicitly determined by the action of the generator there is a unique change in your velocity and it's then very natural to do a variational principle which is best matching which which just looks at all possible changes to your velocities so in some senses that there are there i'm sure there are infinitely many different ways of making relational theories But the simplest one, and the one that in some senses is most closely tied to the underlying of the geometrical structure, is uniquely defined. Well, not quite, because you can still choose different forms for your kinetic energy. But essentially, there is something which we call a corrective coordinate. There's a device for correcting this thing, which we call the corrective velocity. And you just, so you're, instead of having a velocity, you have what we call the corrected velocity. And that's, and you just make your action principle depend upon the corrected velocity and nothing else. And this is, this is a very natural, and this is exactly what is happening actually in gauge theory. The, the, the, when you put Maxwell theory in, in a, in a sort of dynamical theory, the scalar potential, the electrostatic potential, appears exactly as a correction to the velocity of the vector potential.

1:02:30 You get A, you get the vector potential, the velocity of the vector potential is corrected by the spatial derivative of the electrostatic potential. But the electrostatic potential is actually a velocity, dimensionally it is a velocity. And that's exactly this correction. So, in fact, actually this is exactly gauge principle, and I think it's a much, instead of saying you start with a global gauge symmetry, you gauge it, it's much more consistent, I think, to start with gauge theory as a theory of best matching. I was wondering if you were to change that equation, have just absolute difference between the vertices and minimize that, would that give you completely different results? And one thing which is very, very significant, the reason why Beth Potley and I, 25 years ago, introduced the idea of best matching, was that virtually all other methods you use in particle mechanics lead to anisotropic effective masses. And as far as I know, it's still the most exact, precise measurement that's ever been made, experiment ever done in physics, the Hughes-Threeder experiment, which establishes that mass, the effective mass that appears in atomic physics, is isotropic, I think, from one part in 10 to 28. Absolutely staggering accuracy. And virtually all attempts to implement much principle, except by best matching, will fall foul of this and lead to our esotropic land. So I think there's extremely strong experimental support for best matching. One last question. It's actually time, but sort of... I just wanted to make a quick remark. As an old man, I look on it and write the ending of things rather than the way forward to the beginning. And I think here is the peak of an edifice with names like Gauss and Jacobi and Hells in particular, lower down. And here we've actually got it all clear. Now, apart from the continent, which I don't want to know, in fact, it's been really wonderful. I have a question. Years ago, somebody, I can't remember whether it was Del Vittro or it might have been Del Slater, There would be a very rare book by a Frenchman who came, I think, is about Roux.

1:05:00 Yeah, I know about the Roux. You do know about the Roux, do you see? It's an attempt to be relational, I think. It is not the past times to serve the past. Actually, my recollection was that it was the other way. Rather, it might have been Roux. There is some connection with that. And does he use best matching? I can't remember. The interesting thing is this alternative theory, there's a very natural alternative theory which was rediscovered in the 20th century about 15 times, starting in 19... The first person was somebody called Hoffman in 1913, who met Einstein in 1913, that's why we know about him, thanks to that site, who did an alternative machian theory where you just put the relative separations in. And actually the most beautiful paper of all that is Schrödinger's last paper that he wrote before he created a great mechanics. He actually did a mafia theory of relative separation, and was very well aware that it led to an isotropic masses, and actually very ironically made estimates. He knew that the galaxy was the term of the solar system, and at that stage the astronomical data was not sufficiently accurate. And he said, we're just not in conflict with the observations as known as the moment. Well, it's a beautiful paper, but certainly I don't, but this theory, and then, in fact, Petrotti and I rediscovered, in fact, our paper in 1977, the film paper that Petrotti and I wrote, is almost the word Schrodinger's paper in 1925, we knew nothing about it. Ironically, Bruno was even shredding this one of his last tutors. New Year's interest is much preferable, but he's not thinking about the same. So there are alternative ideas which have been thought, but the key thing is this requirement of mass isotropy. So I think there's an extremely strong, I mean there's two very powerful underpinnings of this whole approach. One is that it's mass eisotropic, and second, that this is manifest in what happens in general relativity in James theory. Best matching is the dynamical core of this, coupled with the idea that there's no time. The time, the canonical momenta, in general relativity, not in all of the James theory, in general relativity, all the canonical momenta are direction co-science.

1:07:30 So that's timeless, and now it's time to copy it. Yeah, I'm going to ask you a question, but I'll ask you guys to come. All of you will come. Thank you.