Change without change & how to observe it

0:00 My main conclusions are that Marx's principle can be formulated cleanly. I will go so far as saying as I think there's only one real way in which Marx's principle can be formulated. And it can be implemented by a universal method which is called best matching. And I think there is very strong experimental evidence that that is the way the world works. It is through approaches for web-catching, or at least that is how the relativity of motion is overcome in nature, and you can divide motion in a relative situation. More controversial is that I think increasingly we're gaining evidence in support of that remarkable statement of the axe that maybe space-time is not fundamental, that really dynamics and three-dimensional geometry may be more important than space-time. And then there's an interesting thing, is that we're finding, I think, an extremely interesting connection between the notion of time and whether the dynamics of the universe is scale invariant or not. There seems to be a remarkable connection there. So that's sort of what I would say is coming out of this, and I'm going to try and persuade you of that. the material from the time. Now, Chris Isham was talking about configuration spaces. Chris knows that the beginning and end of my life are configuration spaces. I think they're very fundamental. Now, the thing about configuration spaces is that you can sort of peel away some of the structure from them. If you're talking about the Newtonian n-body problem for n-particles in three-dimensional quadridden space, that configuration space has three n-dimensions. but those three n dimensions include the overall position of the system and the overall orientation of the system in space so those are six variables that are included in your 3n which if that system were the entire universe don't seem to make sense this was already clearly stated by Leibniz what on earth can it mean to say that the entire universe is six foot further to the right than the other one if the only thing that is real between the particles. So there's a very clear 3n-dimensional configuration space in which Newtonian dynamics is formulated is crying out to you to consider the possibility of doing dynamics in a smaller configuration space in which you

2:30 quotient it out 3n-6 dimensional relative configuration space. So that's a perfectly obvious one. You can go one step further which is also extremely obvious shape space where you say that the overall scale of the thing is exactly the same. We talk about the equilateral triangle. We don't, when we think of the notion of the equilateral triangle, think of its size. We regard that as immaterial. So it's really saying that if the world was an equilateral triangle, its size, if the whole universe was an equilateral triangle, its size doesn't make any sense. It's just the fact that it's an equilateral triangle. So that's what shape space is about. It's about the shapes of particle configurations. 3N minus 7 dimensional space. And all the interesting work that's happened to me in the last four years has come about from trying to extend the ideas of Machian dynamics from relative configuration space to shape space. Now there are very close parallels to this trio of configuration spaces when you go to geometrical dynamics. You either say space is described by Euclidean geometry, or you say space is actually defined by Riemannian geometry. So you can go to Riemannian geometry and then you find that there's something very similar to this thing here. You can start off with the space green, which is all those Riemannian metrics. The Riemannian metric is a 3 by 3 matrix, that's 6 degrees of freedom, on some say spatially compact 3-manicouples. That's a Riemannian metric. Now, you can quotient out monodipheomorphisms, we heard plenty about dipheomorphisms this morning. and that leads you down to super space. That's all those three metrics that describe the same three geometries. And it's the set of all three geometries that is super space. And that's very closely analogous, really, from going from here to here. Not exactly, but I mean, it's quite analogous. And super space is John Wheeler's coining there. And exactly as you can go from relative configuration space to shape space, you can go one step further And scale, you can change the scale here. There's one thing, the three geometry is essentially described by three numbers at each space point. Three numbers just describe the coordinate system you're using to describe the metric, the geometry.

5:00 That, if you're quotient out by the three difiomorphisms, you're left with three numbers per space point. You can go one further and leave the angle part of the geometry untouched, so to speak. change the scale at each point. So that's one more degree of freedom at each space point that you can take out. And that takes you down to what is called conformal super space. And that's closely analogous there. Now what I'm wanting to say is that Chris was talking about the sort of great generality of things, his notion of momentum being arrows between configurations and things. Now as far as I can see there is a huge generality of things. There's the same sort of problems cropping up in different contexts. There are different specific examples, but underlying it all is something that's very, very general. And I would say at the end of the day, it boils down to the fact that in language we use nouns and verbs. Nouns are the configuration spaces, and verbs are, so to speak, going from one noun to another. These are the arrows in causal sets and things. Well, no, sorry, from one causal set to another. So I think there is a huge generality, and I'm certainly convinced that the only sensible view of a self-contained universe is that everything is relative. There is a sort of relativity of everything and the way you implement that there's a universal method of dealing with the relativity of the universe and the fact that so to speak the essential things in the universe are nouns and verbs. So this is sort of what I want to try and bring out and say that this is very specific. And I think that there are just two really very basic problems when you're talking about motion. And this is really what is behind Newton-Scolium in the Principia. What are the fundamental issues in the relativity of time, relativity of motion? Now, it is an absolute fact that you cannot tell the time by just looking at time. You have to actually look at some object moving. You look at how the sun is moving across the sky. There is just no question that time is extracted from motion. And then you always get, and this is really just like Leibniz's statement about moving the universe six foot to the right can't make any difference to the universe. There is the statement that if all the motions in the universe were run through at twice the speed, absolutely everything,

7:30 all our bodily motions and all the clock motions would be multiplied by the same factor and you could not observe anything different. question mark about time and whether time should be any part of dynamics and I would say emphatically it shouldn't certainly not if there is any real sense in which the universe can be thought of as a completely self-contained dynamical system, some self referential relative system which in some senses is closed think of Einstein's praise of the closed universe which satisfies Marx's principle cause and effect can be closed. So if that's the case, I don't think time should play any role in these things. All you have is actually curves in configuration space. Newtonian physics, in Newtonian physics, you have curves in configuration space and you have a notion that they are traversed at a certain speed. And that speed comes into the physics. It shows up in the physics and you can identify the point where it is. And I'll show you precisely where it does show up in the physics in a moment. But the first requirement, So this is the first sort of problem. And I would say the answer to the relativity of time is that if you're going to do dynamics, if you're going to make action principles, you're going to define an action, it should not depend upon the choice of the labelling that you take along the curve and the configuration space. Your action should be re-parameterisation in variance. There is no distinguished absolute time. And that seems to me a very clear idea, very straightforward, forward, and I would say this idea explains a huge amount of classical physics as we know it today, that idea. That's one thing, that there should be re-parameterisation in variance. And I'm inclined to think that this is actually more important than the question of the definition of simultaneity, whether we've got a definition of simultaneity or not. I think re-parameterisation in variance is more important. Now the other aspect is now the relativity of motion. If I have one configuration of n particles now, and then another one which is intrinsically different, in other words it's a different point in my relative configuration space, how do I know where particle A has gone? I know that the configuration, the intrinsic configuration has changed, but to know where an individual particle has gone and define its velocity

10:00 and calculate its contribution to the action, I really have to know what its displacement has been. But that displacement will depend upon how I orient and place one of the configurations relative to the other. I can move my two hands relative to each other without doing anything to the structure of my hands and yet the difference between my index fingers is hugely different where I put my hands. so this is a problem which must be overcome Newton saw that problem much more clearly than any of his contemporaries in the 17th century and that is why he introduced absolute space he said there is something like this room in which we know perfectly well how far apart my index fingers are in this room when I hold them relative to the walls of the room like that but if you haven't got that invisible structure behind it how do you do it and what it is saying the relative position, the relativity of motion is saying is that the action should not depend upon any embedding of the relative configuration in some enveloping space. It should just be determined by the relative configuration itself. Now this is where you introduce the idea of the group orbit. If I hold my hand here, that is one position of my hand. Without changing the shape of my fingers, without change the orientation of my hand with three degrees of freedom and I can change the centre of mass or the centroid of my hand in three directions. And that is and all of all of those configurations you identify as the group orbit under the translations and rotations. So you have the concept of going from the big configuration space down to a smaller space which in the case of the N-body problem has six degrees of freedom less And all of those configurations of my hand that can be carried into exact congruence by using translations and rotations in some way are identified as the same configuration. So those are the fundamental things, the relative configurations. They are the orbits under the symmetry group of the geometry that you assume. In the case of Euclidean geometry, those will be translations and rotations. In the case of Riemannian geometry, it would be the three-dimensional diffeomorphisms that play that same role. But basically what you're saying is that your dynamics should be about finding dynamical orbits.

12:30 These are the curves in the configuration space of your dynamical system. These should be defined on the orbits, on the group orbits of the system, and not in any way use the embedding space. so that those are the two problems of relativity of time and relativity of motion and there are two obvious answers to how you get round that and what I want to say what I'm really trying to say is I think, well first of all there are two, each of those problems can be solved in a very specific way there's a very surprising I think there are indications that these go a very long way to explaining very fundamental structure. In particular, I won't say that it's any way near definitive, there's a lot more work to be done before it would be definitive, but I would say the resolution of these problems, the existence of which was clearly seen by Newton, I think they're indications that it is telling us it is explaining where the Einstein field equations come from, where the special relativistic, the universal light cone of special relativity comes from, and where gauge theory comes from. together in one package, the solution of these two problems together in the context of Riemannian geometry with one more key assumption which is to come suggests to me that all of those things can be understood as a package which come out of the resolution of the problem of absolute and relative motion. Now let me just talk about re-parameterisation variance. This is to deal with time. If you have an action which is of the following form. I'm just going to consider quite generally dynamical variables which are described by some variables QI. So these are my dynamical variables positions. Now you can write down a reparameterisation invariant action which does not depend upon any external time, very simply by just making sure that the action is always homogeneous of degree 1 in the velocities. So you take first of all any linear combination of the QI by certain which can be functions of the cubes. That's one term. Then you can take a quadratic term and take its square root. And you can go on up like this, up to any powers you like. And that is a function that will be defined

15:00 only on the curves. There will be no notion of a speed along the curve. Very straightforward, very obvious way of doing it. So that's the way that you can do it. Now, there's a very nice way which people should know much more about. And I think it's actually more fundamental represented, is Jacobi's principle. Newtonian mechanics can be represented in this form in a very interesting way and with a lot of relevance for what I want to say is, I don't know, is there a pointer just to go a laser pointed that's not a laser pointer oh that pointer I prefer that to a laser Newtonian theory can be rewritten. Jacobi's principle is about finding the orbit of a dynamical system in its configuration space without any reference to any speed at which it is traversed. So it's just like finding the Kepler ellipses without any use of the area law to find out how fast the thing is going around it. So this is the form of Jacobi's principle. It is of this form here. E is the fixed total energy, capital B is the potential energy, and the kinetic energy looks just as it does in Newton, except that this lambda is an arbitrary parameter along the curve in the configuration space and not at any absolute time. And then you have, just by standard calculations, you find the canonical momentum, you find that these canonical momentum just because of the square root structure satisfy a quadratic constraint, which is of this form. You get the Euler-Lagrange equations like this, which look very like Newton's second law, except that there's this thing up here, which in general will vary as a function of lambda, your particular label. You can show that the Euler-Lagrange equations will propagate that constraint. If that constraint holds initially, it will be propagated. And then you can choose, you can always choose this lambda, this time variable, in such a way that this thing is made into a constant. So this then becomes a constant. And then you exactly recover Newton's second law. So you've got a theory of time. In this representation, when you actually look at the explicit expression for how the distinguished time advances, you see that time, that distinguished time, is really an average of all the motions that are taking place in your isolated dynamical system

17:30 that you're considering. Time is the distillation of every motion in the universe. It's weighted with the mass and how far things go. But there is a complete, and I would say, beautiful theory of what time is. The metric of time emerges from a geodesic principle. And that is actually how astronomers, that's the astronomers, if ever, is time. So that is there. Nevertheless, we haven't really banished time altogether because it's just shown up in a different way. can have different values. And for different E's, you get different motions. This corresponds to the system having different energies. This actually gives you, in the two-body problem, it gives you, for a given mass of the particle, as you change the energy, you change the semi-major axis of the orbit. The semi-major axis of the orbit determines the energy. So you'd either get the orbit of Mercury, or you'd get the orbit of Saturn, depending on the energy you put into a particle of unit mass. So, and in some sense it's that energy, of, when you think of that time is flowing independently of what's happening, you have this notion that I can punch at a certain speed and that speed is what the time is doing. Now, the freedom to change the speed at which I'm punching is still somehow showing up in a different way in this energy here and it's taking you from the orbit of Mercury to the orbit of Saturn. And so it's, the notion of time is transformed. Time has been banished but it's still lurking there in the Now, so that's the theory of time, and at this stage we haven't managed to get rid of the problem of time. It's still sort of, it's been transformed, but it hasn't been, I can't say categorically there's no time at all, because there's something that looks like the effect of time in this freedom to choose that E there. Now I'm going to come on to the idea of best matching, which is the way to solve the problem of the relative motion. If I have a configuration space, I have to have some rule for defining curves and finding curves in there. The easiest way to do that and find the simplest curves is to define a metric on my configuration space, on my relative configuration space. If I can give us some rule which defines a distance between neighbouring points on the configuration space, I can then ask for the shortest or the extremal curve between two widely separating points in the configuration space.

20:00 That's two widely separating configurations. And I can say the curve that will actually be realised in nature is the one that minimises that action. So it all boils down to the way you define a metric on the configuration space. Now there's a rather obvious way to do that, which is based on just using the separations themselves. the RIJs, and their rates of change. And this method was rediscovered about nine or ten times in the 20th century. The third person who rediscovered it was Schrodinger. And it was rediscovered and is still being rediscovered now. It's a very obvious way to do it, but when you write down an action principle which depends directly on these things in this way, they virtually all lead to anisotropic inertia. and in fact this realisation led to the performing of what I think is still the most accurate experiment ever made in physics which is verifying that mass is isotropic. They all lead to an effective anisotropy of mass and there's a Hughes-Driever experiment and various refinements have done on it which I think have confirmed that mass is isotropic something to one part in 10 to the 28 or something. It's certainly one of the most incredibly accurate experiments physics. And so there's a very strong indication that if nature is Machian, it's not Machian in this way. And there's an alternative way, which Bertotti and I found that really it's the basis of gauge theory, I believe it's actually exactly what is happening in gauge theory, it's what we call best matching. The idea is, if I have two configurations, just imagine two relative configurations of N particles that are fixed I want to put some measure of how different they are apart. And the idea is just very simple. I just move one around relative to the other and try and bring them as close as possible to congruence. Now, there can't be any simpler idea than that. Now, you have to measure congruence. This is not saying that there's a unique prescription because you can define congruence in different ways. But the idea of moving around relative to some choice that you've made for how you're going to define congruence, but then using this freedom to move one configuration relative to the other until you've got the closest approach to congruence. You can't get an exact congruence because they're different,

22:30 the configurations are different, but you can get it as close as you possibly can and say that is the intrinsic difference between those two configurations and that is going to define my metric on configuration space. That is a universal method and above all it turns out that it leads to isotropic masses and I personally don't think that there is any other method that will do that and also you will see it's also very clear that this is extremely related to the very basic foundations of sort of symmetries in space group transformations it is so basic so closely related to the most basic foundations of sort of the mathematics of group transformations and symmetries things like this. I think it's sort of you know, the almighty had to choose this in some ways, I believe. And it seems to be done because it shows up we see that actually the inner structure the dynamical structure of Maxwell's theory, of general relativity of sort of gauge theory generally, seems to be based on this idea. That if you ask what is defining the metric on configuration space, it is best matching. And this again is I think very pleasing. I mean, Chris was talking about this platonic realm up there I think, you know, just thinking about the generalities of nouns, verbs, and the relativity of things, it's forced upon us, or it's very distinguished, let me put it that way. Now the question mark is not whether this is the basic principle underlying dynamics, these two principles of reparameterisation invariance are best matching. I think there's very strong evidence that this is the basis of modern physics. It's not conclusive, but that it's playing a very important part and is there, I think, is undeniable. The question is, how far does this principle go? Does it go to full scale invariance or not? And how far are we going there? There the answer is completely open. Now, let me just... This will be rather rushed, but let me just say there's a formal way of doing this best matching. don't try and understand these things anyway I'm missing out on all the key calculations I just want to tell you about what I think is a very general way where you get a very general picture of what is actually happening in gauge theory and in fact all the classical physics that we know certainly for bosons and possibly for thermions

25:00 Ed Anderson is working on thermions at the moment and is making some progress to understand them now just think about translations where, let's do just translations in one dimension where QI is just your position on a line. Now you make a translation. So this is a translational by the amount B where E is the generator, E is one for all the particles and this is the generator of translations. Now if you translate that and if you make this B now the key thing is that these transformations that you're making arbitrary functions of your label lambda arbitrary functions of time and I would say that the key thing that is happening in gauge theory is really that you're making these symmetry transformations which are about three dimensional structures but they're arbitrary functions of time or arbitrary functions of my label lambda. That I would say is the key thing so then the velocity changes under this thing and it's the change of the velocity that is not working out in Newtonian theory. Newtonian theory is invariant under this transformation if you make a one-off transformation at the start of the proceedings. But it's not, and it is actually also under this one for translations, but not for rotations and not for scaling. The interesting thing is all these Leibnizian arguments about it doesn't matter where you move the world, it doesn't matter if you take the whole history of the world and move it six foot to the right. matter how the first configuration of the universe is related to the second. The relative configurations tell you nothing in Newtonian terms how the what rotational state is in the system. So what you've got really in Newtonian physics and what is really the origin of all the problems of absolute relative motion is that you would think that the dynamics should be invariant under this when it's an arbitrary function of B It isn't. It's the arbitrary function of B that is not satisfied in Newtonian mechanics. This is the problem that you have to overcome. And I would say that gauge theory is about precisely just overcoming that arbitrary lambda dependence of your symmetry transformations that you have in space. Now, what you do to make a formal mathematical method is introduce an extra variable

27:30 to offset this thing you define an A of lambda which actually is going to transform in the opposite way you're going to add an extra variable to your system and you're going to transform it in the opposite way to what you do this one here so that you will then have so this is how the velocities will transform and then if you define a corrected coordinate which is your original one in your particular frame of reference you've chosen plus this A variable here this combination is trivially banally invariant under a time dependent transformation it's as trivial as that and so are the velocities so you can introduce these corrected coordinates which are absolutely trivially invariant and you can do the same with dilatations here's a dilatation a one dimensional dilatation so I'm going to introduce an extra variable some of my variables here are redundant I'm introducing precisely the same number I have redundancy, in this case it's only one, and I'm going to make this extra one that I do have the opposite transformation property here it's plus and minus here it's division and multiplication so that again these quantities if I introduce then the corrected coordinate A times Q and they transform in this compensating way, again these things are again invariant and this is an absolutely banal thing that's going on And the introduction of corrected coordinates and corrected velocities, it doubles the original degeneracy. And it's universal. You know exactly how to do it because this form of the corrected coordinate is uniquely determined by the symmetry that you're doing. Give me a symmetry and I will tell you how to deal with it. This is why I say that best matching is universal. Give me a symmetry and I will tell you how to deal with it. Absolutely. There's no question it can be done. Always can be done. and the question at issue is how far does nature take that I think there was one more thing I want to say this is where it's good to have you in the audience let me say

30:00 For the examples that seem to be realised in nature, this seems to be working. This scheme seems to work for particle mechanics, for general relativity and for gauge theory, and we think it may work for spinners, for thermions. Quite our part will go, no, I should stand corrected on that. Sorry, this is my intuition taking me too far beyond mathematical precision. Certainly, there's lots of examples where it is absolutely obvious what to do. And it seems to suffice for all of known classical physics at the moment. So thanks for that. Now, oh yes, I know what the example was, I wanted to say. Just think about, now exactly the same thing is happening in Maxwellian electrodynamics, except that you've got many more variables. You start with the three-dimensional vector potential, of which only two are so to speak physical degrees of freedom there's one redundancy in those three things there and then you add a fourth variable which is the scalar potential in the four dimensional representation of electrodynamics and that is doubling this degeneracy. You start off with a three dimensional object of which one of the degrees of freedom must there's one redundancy in it and then you add on another one. So if you want an example for what my A variable my auxiliary variable is there is the scalar potential in the dynamical Hamiltonian representation of electrical dynamics. And this is why I'm saying that I think this is really what's key going on. Now, the best matching variational dynamics is, this is the universal method, you just identify your corrected coordinates, and then you start off by taking any Lagrange function you like, which just depends upon the corrected variables, and then it's bound to have this transformation property under arbitrary time-dependent transformations. You can't fail to do it. It's absolutely inevitable. And it's quite banal the way it does it. So before you had the liberalism allowed to refuse the b's, the b's, the b's, the b's, the b's, the b's, the b's, the b's, the b's, the b's, the b's, the b's, the b's, and in addition, this is the relativity of motion. You get the relativity of time, it must also be a re-parameterisation of everything. And to what I'm... And for what I'm talking about today, bosonic physics, bosonic type physics is going to be quadratic.

32:30 It's going to be homogeneously quadratic. It's going to be a square. It's going to have squares of the velocities and then you're going to take the square root of the hell. So it's going to be homogeneously to pretty well. That's going to be the case there. Now, so, but these q's, you see, these q's are functions of these variables and the corrected velocities. These have this thing here. And you have canonical momenta which are, first of all, you divide by the correct... You take the derivative with respect to the corrected velocity and then you take the derivative of the corrected velocity with respect to the physical thing. And the same for the auxiliary variable, you do the same thing there. and so these are the canonical momenta and this gives you because of this sort of because these are always combined together in a different way in the velocity here, you get a primary constraint which relates the canonical momenta of the PI to the canonical momenta of this auxiliary variable that I've introduced. Now what is best matching about? Best matching I've got these transformation parameters which are determining how much I'm translating and rotating one configuration relative to the other. This is an arbitrary function B. And to get to the best matched position I can do two I can be doing two things. I can be moving both of my configurations together along like that. Both together with the same transformation. But I can also be moving this one differentially with respect to the other one. are two freedoms, I've got to explore, I've got to have a device which explores all possible positions where first of all I'm relying all possible positions going that way where they're both going together and then another freedom which is describing how this one is moving differentially with respect to that other one. Those are two freedoms and that is so there's going to be standard Euler-Lagrange equations but the best matching conditions are that however I move them all together that must not change the Lagrange function, this is this condition here, that the variational derivative of the Lagrange function with respect to my auxiliary variable must be zero, and at the same time, with respect to the differential, however I move this one differentially with respect to that one, and that means the velocities of the auxiliary variable, that must make no difference. So

35:00 I get two exactly precise things that the Lagrange function must be invariant under this must give zero, which says that the canonical momentum of my auxiliary variable must be zero, and in addition this condition must hold. Now, this condition, if this thing is zero, this thing says that this combination is zero and that is a constraint. That's a constraint on the initial data. The initial canonical momenta and the initial positions must satisfy a constraint which is linear in the canonical momenta. And that's an inescapable thing of this The banal part is doing this thing here. This is non-trivial because I'm now doing this variation and requiring that both of these conditions hold. And when you look through this, you see that this condition leads to a constraint linear in the canonical momentum of your physical variables. But remember, some of these are still redundancy in those. But it's constraining that. And that linear constraint on the canonical momentum is precisely telling you that you've got that amount of redundancy it out of it by using this constraint. And then you have this condition, and when you look at it, you see that this second condition, which is the one of moving it when you move both together like that, that gives you a condition which ensures that the constraint which comes from moving one differentially with respect to the other, is propagated. So you have a beautiful self-consistent system where one condition leads to a constraint, and the other condition, both of which come from this universal idea of best matching, This one ensures that that one propagates. And this, I believe, is actually what really is deeply going on in gauge theory. And that the idea that you start with some global gauge group and gauge it and get compensating fields and all this sort of stuff, I don't think it's really getting to the heart of this thing. I'm sure Chris knows much more about the theory of gauge theory. I'd be interested to hear your comments on my claim there. Chris, I haven't looked at all the literature of books on gauge theory to know how much that book will stand up. So let me just tell you what happens then in particle dynamics. So let me just take as an example in particle dynamics something that looks very like Jacobi's principle. I'm going to have some general function, which is a function of the coordinates and my auxiliary variable. This is like a potential, and then I'm going to have a kinetic energy, and essentially that's the usual kinetic energy.

37:30 So then I have Euler-Lagrange equations like that. And then for the various, for the seven Euclidean symmetries, I just apply my best matching rules and I get certain conditions. If I get the linear constraint that the total momentum must be equal to zero, and I get the condition that my potential must be translation and invariant, and that ensures that this condition is maintained. if I apply the rotations I get that the total angular momentum of the system must be exactly zero and that the P must be rotationally invariant and that gives me, ensures the propagation of the vanishing of the angular momentum so that comes out and that was the result that Bertotti and I got 35 years ago both of these conditions are satisfied if you actually take the potential to be a difference happens in Newtonian dynamics. Now this means that in fact the A, which is hidden, which is in my scheme, has disappeared in this Lagrange function because I've actually got differences of the corrected coordinates. And in the differences of the corrected coordinates, they are equal to the difference of the coordinates. So that my auxiliary variable itself doesn't show up, only its velocity shows up. And I would say that in all of gauge theory that we have, The vector potential, the scalar potential in Maxwellian electrodynamics dimensionally is a velocity. That is the velocity of my auxiliary variable. But Maxwell electrodynamics, as it's normally presented, doesn't actually contain the auxiliary variable itself because it's disappeared by a trick like this. And so far as I know, all the known gauge theories are of this form. But what we've found is that when you do scaling variance, actually the auxiliary variable does appear with dramatic consequences. say, I hope I can get through to that before the end. So that's what happens with translations and rotations. If you then go to scale invariants, you get an extremely interesting result, and it's completely inescapable. It says that this quantity here, the scalar product of the x's into the p's, must be zero. Now, I don't know if anybody knows that this has a definite name in physics, I'd be glad to hear of it, but I've had to coin an expression for it, and I've called it the expansive momentum. It has the same dimensions as the

40:00 angular momentum. It's got the dimensions of action. And it is not conserved in general Newtonian physics. There's nothing in general Newtonian physics which causes it to be conserved. But in a scale in varying theory it must be conserved. And if it's to be conserved, now the very interesting thing happens is the potential must be homogeneous of degree minus 2. This would mean so a potential not like the Newtonian one which is 1 upon R, this would be say a 1 upon R squared potential which would lead to 1 upon R cubed forces you would think, say for example but it is inescapable that if you're going to get a scale invariant theory which propagates this constraint it's got to, the potential's got to be homogeneous of degree minus two. And you can see why this has got to be, the kinetic energy is length square. And so this is length to the two here, so this is why the potential has got to be length to the minus two. It's absolutely inescapable. The Lascale invariant theory has got to have a potential of length to the minus two in it. And now, this is where the first interesting connection between time and scale of variance shows up. I explained to you that there was nothing in ordinary Newtonian mechanics which would fix that E. And in some senses, the fact that there's a time in Newtonian dynamics appears in that capital E, which appears in Jacobi's principle. But if you have E minus V, E is homogeneous of degree zero in the length scale, whereas the Newtonian V to the one upon R potential is homogeneous of length to the minus one. now if the total potential E minus V is my total P if my potential has got to be homogeneous of degree minus 2 I cannot break it up into a constant E which has no dimensions in length plus something which has dimensions in length there is no scope at all for introducing anything like E but E when you translate into configuration space and curves in configuration space is so to speak the is something like an absolute time tucked away in Newtonian physics. So scaling variance kills time in political dynamics. There is no residual trace of anything that you could call time.

42:30 All of the dynamics can be completely understood in terms of Euclidean geometry and the three fundamental symmetries on that, coupled with the idea of best matching. And these constraints only apply to the whole system. There's nothing wrong with having an island system like a globular cluster where you've got three particles, three triple star systems within a globular cluster which have got all values of the energy, all values of the angular momentum and all values of the expanse of the momentum. Only the total universe can satisfy these things there. But I think this is a, for me, this is a very pleasing result and it suggests to me that there's a profound, a profound connection between what we're going to have as our notion of time and how far we take the symmetry properties of space seriously. that this is not, and also this is suggesting that all of dynamics can be understood in terms of spatial structure that you don't need to do anything with time there doesn't need to be any four dimensional space time kinematics at all in the scheme there. Let me just mention that there's a very interesting property here, if you have, if this is the case, now the centre of, the moment of inertia plays a very interesting role in this theory, the moment of inertia in this way, and it can be rewritten in terms of the relative separations like that. Now, the time derivative of the moment of inertia is this quantity, but this is precisely my expansive momentum, which must be zero if my system is to be a scale invariant. So, in a scale invariant theory, the moment of inertia of the system is concerned. This is an absolute fluke in Newtonian mechanics. It only happens in the rarest cases in Newtonian mechanics, but in a scale-invariant theory, it must happen. And this leads to a very interesting possibility of almost exactly recovering Newtonian physics in a scale-invariant context, although you would think it would be impossible. Start off with standard Newtonian potentials. Your energy, which is the potential of homogeneous degree zero, and the Newtonian gravitational one and the electrostatic one, genus of degree minus one. Now you can make scale invariant potentials out of these in a very simple way just by dividing by appropriate powers of the moment of inertia. You divide the energy by the moment of inertia and you divide these two

45:00 by the square root of the moment of inertia. Now you're not going to... And that moment of inertia generates forces, but it's dynamically conserved by the theory. It has to be. It's really quite a surprising trick that you get there. leads so I'm not going to take you through the calculations when you work this out you actually recover the Newtonian forces that you know and love exactly with an addition just one extra thing which is just like Einstein's cosmological constant force this I down here this was the I is given by this expression here this is a harmonic oscillator type force which comes out of this thing here so what you get is in addition to the absolutely Newtonian forces, which are constant absolutely indistinguishable from Newtonian forces, you get forces which increase with distance linearly like harmonic oscillator forces and at short distances are absolutely minute. You can do calculations in realistic models and you see that it's absolutely tiny, but that it will have a big impact on cosmology. And in fact what happens, it makes, it is a way of making a dynamically stable universe. Einstein wanted to have a stable universe or a universe of constant size when he introduces cosmological constant but it's dynamically unstable this device actually makes one that is completely stable if a particle somehow escapes to some considerable distance to a cluster of things which is your universe there's a harmonic oscillator force which pulls it back in again so the system is stable it's quite confined so that's rather interesting so I've got I think about seven minutes have I Harvey I started at five past according to that I think I might just now let me just say briefly this is going to be quicker but exactly the same sort of trick can be done with Riemannian geometry there's that famous saying that the history of western philosophy is a series of footnotes to Plato I would say that the whole of theoretical physics is a series I mean the big footnote was then Riemannian geometry that came after that but I think that everything is really contained in the geometrical notions and that Euclid showed us the way and not a huge amount has happened since Euclid please know about geometry and find their way around the world

47:30 and that's really all that theoretical physicists are doing in this game I'm going to first of all talk about super space, just briefly review of the paper that came out last year called Relativity Without Relativity by myself, Brennan Foster and Neil and Wokum. Basically you can consider now this is where the first really intriguing thing comes into this. If you were to, now you've now got degrees of freedom which are defined at each space point. We're talking about fields and metrics. So at each space point we've got the variables. So these are the space point. Now if you looked for the natural generalisation of Jacobi's principle you would have something where you have a product of square roots, you have two integrals over space and I've missed out, sorry, that's the square root here should be outside, not inside, so the square root should be outside there as it is here you would have two, you would have a quadratic thing here which contains the velocity, which is quadratic in the velocities, you would integrate it over space, and there should be a d3x here, I apologise for that And then you would take the square root. And that's what we call the global square root. It's not terribly interesting. We haven't evaluated everything which happens to that. But there's an alternative way of achieving reparametrization invariance which seems to encapsulate the whole of the relativity of simultaneity which is responsible for the origin of the relativity of simultaneity which is to take what we call a local square root where we take something which is dependent on G and its spatial derivatives, that's like a potential term, and we take a quadratic kinetic energy, but we take the square roots and then we integrate over space. And I can't go into all the details of that, I'll have to ask you to read the paper Relativity Without Relativity. But that is really what is actually putting in relativity of simultaneity into general relativity. That local square root there, when you look at it in this dynamical form in the relative configuration space. Now, just as we had these constraints before, there's a constraint because you've got these square roots here. This leads to a quadratic constraint. And now you have two scalars. These are the canonical momenta going with the Gij. You have something like that. That's one scalar. And then there's another one where this is the trace. This is the definition.

50:00 Alpha is a free coefficient. And the trace is defined in this way. it's the contraction of the canonical momentum with their sort of coordinates, the Gij G is the determinant of the thing here, and this trace is exactly analogous to the expansive momentum this is the expansive momentum whose vanishing ensures that the moment of inertia in particle dynamics is concerned this condition that the P must be equal to, sorry if this condition, this P actually determines the change in the volume of the three dimensional volume of the universe and if p is zero, which will be in a scalar invariance theory, then the volume of the universe cannot change. So this is exactly analogous. So we're getting a thing significantly different from general relativity when we get a scalar invariance. Now let me just tell you what the main results that we found are. So that's the quadratic constraint. That comes from the relativity of time with the local square root. That gives much more dramatic consequences. way of achieving reparameterisation invariance, and it's actually an extremely sensitive way. The local square root is a far more sensitive way of testing, so to speak, the potential of the configuration space than the global square root. If you wanted to touch nature as finely as you could, you would choose the local square root. in addition, the relativity of motion, you get a linear best matching constraint. This, for those in the know, is just the momentum constraint of general relativity, the ADN form of this. So again, you have two constraints. Now, the interesting thing is that when you get into Riemannian geometry, this is much more sophisticated, these two constraints are much harder to propagate than you would in particle mechanics. In particle mechanics, there's really very little problem in making your constraints propagate. But in this case, we found that it's not at all easy to make these constraints propagate. So let me just summarise what the results are. The results are the following, that if you try and do propagating using only these three-dimensional notions and the idea of relativity and time and relativity and motion, you will only get these constraints, in particular the quadratic constraint, to propagate properly If you have a very special choice of the P,

52:30 and these three cases give you Euclidean general relativity, that's where you have four pluses in the space-time metric, the other one gives you Lorentzian general relativity, and then there's something called strong gravity, which I think Chris Eichel was the inventor of 20 years ago or more. So, in fact, you get out of this a completely new derivation of general relativity, into which no notions of space-time and no notions of four-dimensional general covariance have gone in at all. But you get more than that. If you try to couple then matter fields like scalar fields and vector fields, three-vector fields and scalar fields to these fields, you find that there will be a universal light cone. Once you've got general relativity, in the Lorentzian form, there's a light cone. Try and couple a scalar field to it and you'll find that the scalar field in general propagate at any speed. But in fact the propagation of the constraints forces that field to propagate along the same light cone, to have the same light cone. And when you do this for the vector field, not only do you get the same light cone but you actually get the entire structure of gauge theory comes out of it. The entire structure of gauge theory is forced upon it this way. So this I think is a completely new way of arriving at gauge theory in four dimensional gauge theory fully generally covariant four dimensional gauge theory into which no notions of four dimensionality have gone at all. Just going back to the basic issues that Newton raised in there. So vector field gauge view. Now I see I've got about two minutes to go if there's going to be any questions. Let me just say what happens if you go to scale invariance. Now this theory is general relativity, just let me put that part of the thing out. We wanted to have a unique curve in the configuration space. Now in general relativity in superspace, we don't get unique curves. The relativity of time in general relativity shows up in the fact that many different curves, these are the continuous curves here, these curves that are the be-all and end-all of my approach in the configuration space, when you do it in superspace in this way here, you find that there's not just one unique curve for which the action is stationary, but a whole family of them, which trace different curves. This is one is not extremal, but all these continuous ones are extremal. And in the space-time picture, these

55:00 correspond to different possible foliations of space-time. And this is the relativity of simultaneity in space-time. And we haven't got away from it. This is just like not being able to get away from the energy in particle mechanics until we went to scaling invariance. Now I'm just going to make the statement, if you go to curve in the configuration space and you get a unique definition of simultaneity. So, once again, scale invariance does something drastic to the notion of time. If the universe should be scale invariant in the way that we've said, that we're thinking about here, it would actually introduce an absolute definition. It will be a definition of absolute simultaneity, a dynamical one. And in fact it seems to be sitting in general relativity. General relativity is incredibly close to being completely scale invariant, but not quite. General relativity there is a dynamical representation in this language, in which general relativity is invariant under all changes of scale locally, provided this is for a spatially compact universe, provided the total volume of the universe is not changed by your conformed transformations. So it's unbelievably close to being totally that we would want, except that there's just one tiny residual thing you can't, when you do your scale transformations you can't change the total volume of the universe the fact of introducing an absolute definition of time remains, and this is sitting in I haven't got time to talk about it, this is sitting in the work that Jimmy York did, and in fact 30 years ago, John Wheeler on the basis of work that York did conjectured that funnily enough, general relativity might undermine itself so to speak, by introducing a dynamic defined definition of simultaneity. And this, I think, is a necessary concomitant of scale invariance. But this is where the story doesn't go so cleanly through. General relativity does not seem to have this absolutely complete scale invariance. It's unbelievably close to it. But the bit that it fails by is exactly what makes the Big Bang possible and the expansion of the universe possible. So we're right at the heart of all the big mysteries of the universe seem to me to be related to scale invariance. I would say that if we could find a scheme in which invariance under scaling

57:30 could be fully implemented by this best-matching technique, it must destroy the Big Bang and it must completely change the way we think about the universe. I have to admit that looks unlikely at the moment. So that's the end of my 40-year story and my 60 minutes exactly. I'm sorry. I haven't left any time for questions. Sorry, Simon. Thank you. You know we're introducing your cosmological constant like quantity but it appears to depend on actually measurable things. Does that mean in principle you can calculate your... Absolutely, yes. This is the nice thing about it. There's an absolutely explicit expression for the cosmological. And we do actually get a force that is exactly like Einstein's cosmological constant force in our theory. get the coefficient as well. You get the coefficient as well. It's determined by... But there is an element... Why I think... Why I'm beginning to think it really can't be right is that it is very fundamentally action at a distance that does it. You have to introduce the total volume of the universe into the action principle to get this for a closed universe. And this may well be the reason why that's not the way nature's gone. But it's very nice in that it gives one a beautiful explicit expression for it. Is that a question why Big Bang ruled out even if he managed to go to a more scale of air in the situation? Can you help me say that Hazel moved back towards initial singularity that's represented not by the universe against more but by things like the interstellar distance shrinking relative to the radius of the atom and the average temperature of the universe going I would say that everything is relative. the question is when you put it in the question, the way you tell whether you've got some, where you're failing in this thing is there's a very good dynamical criterion which people have heard me talk about before this criterion that I've got from Pankara a theory will be truly relative if the initial data for the theory consists of specification of a point in the configuration space which you think is the real physical one and a direction in that configuration space and you shouldn't need any higher derivatives an initial point, an initial direction should determine the future uniquely and when you look at general relativity it fails that criterion and so would I say

1:00:00 any theory which is not truly scale invariant in general relativity in York's formulation, it's amazing you specify two degrees of freedom per spacepoint which are the pure shape degrees of freedom and then one solitary extra thing which is the rate of expansion of the universe and that's one solitary extra dynamical degree of freedom put general relativity in that form. And I would say that a truly scale invariant theory should not have that odd defect provided. So then I would say that I think that scale invariance will have dramatic consequences if it's really implemented in nature. As far as I'm concerned. What I suppose I mean is if that were implemented completely and we had something that, say, removed the total volume from your one could still imagine I mean that doesn't seem to ruin out the possibility of there being a Big Bang because we can still describe the descriptive story about the Big Bang is still told there will still be a way of describing it to some extent for quite several months I was seriously trying to find some way of describing this thing the Big Bang state is not where everything is sitting on top of each other, it's actually the most uniform state of the universe The story of the Big Bang is it actually starts from the most uniform state that you can imagine. Where, by the way, all the dynamics is incredibly well behaved, everything is very well behaved. There is no nasty singularities in scale and varying theory. All of the nasty Big Bang singularities just go out of the door there. All of the sort of, all of the shapes could change in exactly the same way. And if you think about, I mean there seems to be not much doubt about what really has happened in the universe. there's been a colossal change from a very uniform state to a highly inhomogeneous non-uniform state, a colossal change now the only way I can think of saving scaling variance within cosmology as we know it is this absolutely colossal change which is completely ignored in the Friedman-Roberts and Walker solutions is somehow rather generating a red shift and the effects and I think it's not impossible if you consider the magnitude something there, but it would need to be a completely new mechanism for generating a redshift. The redshift relies totally and utterly on that one solitary extra variable when you represent general relativity

1:02:30 dynamically that way. So we've got to find an alternative to that and frankly so far we've failed and I'm beginning to lose hope that we will find a way. It's not just GR, is it? I mean, I thought there's all sorts of massive stuff going around as well and that's massive, not that's a form of reality. actually we found a way of making, we can set our theory, by the way these papers have all come out in the last year there's two papers literally coming out now on the scale invariance, one by me on the particle mechanics and one on conformal gravity as we call it, which is by four of us including Ed who's here, and Ed actually found a way to put in matter and make it scale invariant in the theory so there is a scale invariant theory with matter is it empirically correct? It looks as if it's empirically wrong I have to say which has got lots of mass parameters everywhere, but it's not scale invariant. Don't you get some enormous coverage of the orientation field or something? We're in a very early stage in this. I mean, don't get me wrong, I'm not... I mean, I may have been carried away six months ago and thinking we were just about to get an alternative cosmology. I don't think that's the case at the moment. There are all sorts of issues like that. All I will say is that there is a theory, it's uniquely defined, and it's a unique counterpart to general relativity, which is scale invariant. relativity without that one solitary extra volume degree of freedom there. It's an extremely natural idea that you should explore that theory. We found it, it's there, I think in some sense it's unique. It's going to give GR for the solar system and binary pulsars as far as I can see with any accuracy you like, but the cosmology is going to be completely different. That's my view. A very brief question. Can you give us a short statement of the principle of general covariance in your scheme? that it is entirely instead of being a four dimensional general covariance where you have diffeomorphisms in a four dimensional thing you have re-parameterisation variance in a configuration space and you only rely on three dimensional diffeomorphisms now Ed has done some nice work on this on papers of Kukash's hyperspace thing if you start off by assuming four dimensional space time in general covariance there are three types of kinematics involved in that there's three dimensional diffeomorphisms There's something that he calls tilt kinematics and then there's non-derivative coupling which is relying on the extrinsic curvature. And the evidence that we've got, this is a bit provisional, but Ed's work is suggesting that all of known physics that we have at the moment can be got by using only the three-dimensional covariance part of it

1:05:00 plus the reparameterisation event. You don't need those tilted variants to get the observed known fields of nature. You don't need to use a large amount of the kinematics space-time, the general covariate way of looking at things seems to be redundant. You don't need it to get the results. I would say that's the sort of one of the central messages that we're getting out of our work. It is suggesting that to start with assuming general covariance is much more than you need. Okay, no, I think I'll scroll it a day there and we'll take us before more time. I must just make a comment that Julian talks about time very, very often and yet he's always running out of it. Thank you. Thank you. Thank you. It gives me great pleasure to welcome Richard Healy from the University of Arizona who is visiting London fortunately for us and able to come, despite a trip I think just last year to Vancouver. Yes sir, I got back from my country yesterday, so as I fall asleep in the middle of my own talk, you're not mine. And the topic is change without a chip. Thank you. A couple of comments on this before I start out. So, of course, based on work with I and my colleague, there's a draft version of this, which is available in the Pittsburgh archives at the moment, which is currently ongoing, some very specific instruction,

1:07:30 part of which I hope will be prompted by what you say today. And that should be complete. So that's the title, and this is, in terms of physics, extremely conventional and non-revisionary. As to whether the philosophical view of change that we're pushing, which we think is the one that fits best with that physics is itself entirely conventional and non-revisionary or somewhat surprising I will read to you to decide some people like this and some people don't first topic then is what is change oh and I should say I sort of was open on this paper for a group of L.C. I apologise to them for the things that they will hear said again today but I think they'll hear quite a number of things different from what they had before. I'll be emphasizing different points about a different audience, so I've got a total waste of time here to stay in the religion of John and some breath of fresh air. Okay, so let's start out by thinking what is change. What I'm going to try to convince you is that, contrary to some of the things that John Erman said in a recent comparative paper, there is change in the general altivistic world in fact in any general altivistic world there is change, but to make that case I have to also convince you that change in a general altivistic or even a special altivistic world has to be understood as a brain dependent notion once you've bought that if you do, I hope you will agree with me, that contrary to John Erman, there's change all around us in every altivistic world argument that he gives in this paper against that and see how to respond to it from this perspective. So what is change? Well, let's start up real basically. Something changes because different properties are different types. But this is near Cambridge change, unless the properties are real, i.e. intrinsic properties. I don't change when my brother as a child, for example. Okay. Now, philosophers have given all kinds of metaphysical accounts of change and argued with each other about what the best

1:10:00 approach is, and I'm going to simply adopt David Lewis's for endurance account of change as opposed to an endurance account for those people who know this stuff. And the idea is that something changes if known in his times-wise and different times have different intrinsic properties. This account has the advantage from my point of view is applied directly to fields and spatial geometries, changes in which need to involve no enduring objects. Okay, so if that's what change is and if we have a Newtonian space-time then we have an absolute time that feels as a privileged balance of time-slice with respect to which change may be evaluated. But in the case of any kind of relativistic space-time, all we've got is a spatio-temporal metric. We don't have a temporal metric. We need, somehow or other, to extract from that enough of a temporal structure to be able to carve up space-time, or at least some local portion of it, into time slices. Well, if we're in Minkowski's space-time, then we might decide that there's something special about inertial frames, that we could get, for each inertial frame, a particular privileged slicing into parallel hyperplanes, and we'd relativize change to such a family relative to a particular inertial frame. But what's so special by inertial frames? In special relativity, one could evaluate change not with respect to an inertial frame, but with respect to at least a local accelerated frame, or a rotating frame, or whatever you want to call it. And in no case would one be, in some sense, going against the temporal structure that builds into Negroski space-time, because all the structure you've got is the space-time metric. Many of you may have read a little article by Bell, special relativity, which is reprinted in his speakable and speakable and quantum mechanics. And he considers a nice little example which could be used to make this point. It's a case in which you have two rockets which are initially at rest, a fixed distance apart, and then they both head off on an accelerated trajectory in such a way that there's a program that is inbuilt into each rocket engine, and they institute the same program. And he asks the question, if their sort of off one a few, well, at some distance ahead of the other, and a light ray is connected with two, and they

1:12:30 head off. An accelerator can get the same program in each case, does the thread break or not? Apparently he put this question to a bunch of theorists, and soon, there was a lot of discussion, and the majority of you was no, it wouldn't. And Bell points out, although smart theories were wrong, it would. Why? Well, his story about why it would break, it goes right back to pre-Einstein day, the thread is actually Lorentz contracted as the acceleration proceeds. And the result of this Lorentz contraction is that at some point it gets contracted so much that it breaks. So in that story, privileging the rest frame, or indeed any inertial frame, what you say about the case is that the length of the thread in that frame is actually changing, or the length between the rockets is changing. And so that's what's causing the thing to break. But you could, in fact, if you were in one of those rockets, you probably would adopt a frame or adapt it to your own situation. Suppose, for example, I'm in one of the rockets and I stick this rod out. we start off, it just about touches the other rocket, and then we get going, and after a while, my rod is no longer touching the rocket. Well, I would like to say that the length of my rod hasn't changed, because in my frame, there's a rod, I'm going to get, so why did the rod no longer touch the other rocket? Well, because the other rocket has gotten further away. So, in the frame that we started off with, the inertial or rest frame of the rocket, the distance between the two rockets has not changed but the length of the two rockets is contracted in a frame an accelerated frame appropriate to one of the rockets the natural thing to say is no, the length of a rod which earlier connected the two has stayed the same but the rocket has gotten further away it's a very different story that's being told my point is that whether there's change in something like the length of that rod or the distance between the two rockets It has to be relativized to a frame. If you relativize it to any inertial frame, you'll get one answer. If you relativize it to an accelerated frame, you'll get a different answer. And the point of the example was that there's nothing special about inertial frames. All frames have their own appropriate notion of change

1:15:00 built into them in special relativity already. The corollary of that is going to be that unless you specify a frame, the whole idea of change is not well defined. or putting it differently all changes change relative to a frame Richard does that mean intrinsic profit the notion of intrinsic is also frame dependent like the length it doesn't mean that if you think about it the notion of length that was being deployed there can be cashed out in terms of the space like distance along an appropriate hypersurface and that space like distance intrinsic quantity, it's rather that what's being relativized is the notion of length itself given a definite notion of length a frame dependent notion the length of an object with that subscript is perfectly intrinsic, nothing is going on involving purported changes which aren't really because they're not changes in intrinsic properties I haven't got very far I know you haven't, but I'm terribly worried because I know that I'm not going to answer but it's initially what you said. Oh, okay. Then please ask your question. Unless you were sure of this point because, of course, one can describe change in terms of the various special-tempered intervals between events. Of course, if that which changes is something that is very dependent, indeed it would be very dependent, but there is a very independent way to describe the change in terms of in varying special temperatures between events that which is well I have to say to that any events in because of space time a detrimental point in space time special tempo intervals between them are perfectly invariant and unchanging because you specified events well it's a far as you thought that you know about events which Well, yes and no. I mean, I'm taking it that the way we identify the relevant pattern

1:17:30 is typically by pointing to something that is reasonably called a frame. there are different ways of picking out a frame you don't have to pick out a frame by saying I'm picking out a frame you can pick out concrete events you'd like your galaxy and use that to specify the particular family of time slices to which you're evaluating change that's all that's required I think maybe you're thinking that by making things frame dependent it's somehow coordinate dependent or it depends on the description of somebody or something like that the case. By being frame dependent, something is being said to depend on how you decide to slice up the space-time, which is a perfectly invariant matter. Well, this one's in... Thank you. Okay. Anyway, the next stage is going to be... What is a frame anyway? Okay. This is now supposed to extend beyond special relativity to the general and so we want to have some way of calming up a relativistic world W into classes of simultaneous points or point events and some way of saying where in some hypothetical enduring space appropriate to that frame each of them occurs so if we have a particular general relativistic model of our world W we can represent in that model a frame by foliation, which would be the dipheomorphism from three manifold sigma across some of the integral numbers into the relevant part of the manifold. The notion of frame I'm using here is going to be in general local, but later on it will be easier to restrict attention to global frames of the kind of general domestic world that has such global frame. So basically what we get out of this is an assignment of a point of sigma and a moment in theta of each point in the relative part of the manifold. And I'm going to include Minkowski's facetime in the class of general relativistic models that I'm looking at in a way that it's quite natural.

1:20:00 OK. Now, notice this could be important, that a frame is not defined as a structure on some particular model. The same frame will be represented by diffiomorphically-related aliations and diffiomorphically-related models of a space-time, because we can use a whole class of diffiomorphically-related models to represent the same generality as it will w, if we want to. And we will, as I say, be mostly concerned with global frames on empty space-time. Why are you assuming that there's space-like slice? Because I'm trying to come up with a story as to what change is. And change requires something having different intrinsic properties at different times. And a time is going to correspond to a space-like slice. You're not going to be like that. You could, but I'm just not happy about calling that change. Now, maybe I could extend my notion of change beyond the primitive idea I started off with, which involves something having the distinct intrinsic properties at distinct times, things having these deconferencing properties are different white column bits or something like that you can push me in that direction I'm not I may push it okay so that's what a brain is then let's get to general relativity I mean the thesis which I'm going to be arguing against eventually is the thesis that if you understand the deep structure of general relativity properly you will see there is no such world. And that thesis on the surface looks quite paradoxical, given the sorts of things we naturally say about general relativity. We say that even in an empty general relativistic world, there can be all kinds of things going on. It's supposed to be a dynamic theory, and even an empty general relativistic space-time can be mighty dynamic. There's a nice little quote, which I've gotten from a paperback of Akash in a book that Jeremy edited. recently, where he tells us here's a quote even in an empty space time in general relativity, we can have all kinds of stuff going on where ripples of gravitational radiation can travel around, interfere, attract each other and amplify, they can hold themselves together in a gravitational geom part of the gravitational radiation can leak out part of it may collapse and put a black hole looks like there's a whole lot going on there's a lot of change

1:22:30 so I think you'll take it really rather a radical thesis to claim that actually there's no such thing as change in general relativity, despite that very natural way of speaking. Okay. I will say more about what's going on there in order to move on to more interesting things. So on the conventional view, I think there's clearly change and need lots of change in general relativity. However, there's an alternative way of viewing general relativity which people here are much more familiar with than I am so I hope that I can get away with characterising it in the way I do without putting too many objections from the experts namely the constrained Hamiltonian formulation and that's the formulation within which John Berman's argument that I wanted to address today is located but I think it's really a kind of serious and interesting argument a serious response. Okay, so we start off in the constrained haltonian formulation from some kind of initial action for general relativity and we try and write this all out in terms of some kind of a phase space theory with position or configuration type and momentum type variables which are kind of a figure on our phase space and we come up with something like this with a 3 metric on a space-life time slice. For now, I'm just to simplify things. Let's assume that we've got an empty space-time and it's globally hyperbolic, and so we've sort of nicely carved up into time slices in the way we would like. If at any point it seems important, we'll assume it's compact as well. It'll make life very easy. so we've got a configuration variable q, which is a remodeling tree metric on each of these time slices of any particular model of the theory. And we have a momentum variable related to the extrinsic curvature of such a time slice in that way. And we can write down a Hamiltonian or a Hamiltonian density. And if we do that, then we actually end up dealing with a constraint Hamiltonian formulation. So we have constraints, which is satisfied by the variables in question, which we can

1:25:00 call the length of constraints. Now, all of these constraints can be satisfied on a certain surface of our overall phase space, which we call x, of the overall phase space omega. if you look at the of the constraint you'll see that if the constraint is satisfied then it's going to be zero so it will be zero on that constraint surface a model that is a general answer so you've got the whole base space, we've cut it down already to the constraint surface and with the constraint surface of gauge orbits. A particular model corresponds to a particular gauge orbit, but the whole diphtherophically equivalent class of models will correspond to the same gauge orbit. That's the structure that we have going for us. Now, there's a sense in which a constraint generates motion along a curve line within the gauge orbit in X. So, we have two distinct points. Omega in omega q prime, p prime. Then, they would be connected by a curve, generated solely by the momentum constraints, if and only if there's a single three-dimensional diffeomorphism on S, which takes you from the pair QP to the pair Q' P'. Okay. Now, what's all this got to do with general relativity the way we put it earlier? I have to say, the theory was primarily defined in terms of a class of models, each of which specifies a metric on a Lorentzian nanopold and in terms of which I gave you my initial account of what a frame is going to be relative to each such model. Well, that's going to be exactly what we have to work out in some detail. The issue is going to be what do these points represent exactly, and how do they represent I'll go into detail in a minute when you see why that's important because there's an argument to the effect that since on this view of general relativity as a constraint Hamiltonian theory it's essentially a gauge theory

1:27:30 motion along the gauge of it is essentially pure gauge so that in fact any pair of points like that is in fact equivalent motion along the gauge of it from one point to another correspond to change. How could it? Because any point in the gauge, you'll get corrected as the same thing. Because it's a pure gauge transformation to move from one point to another. Okay, let's keep that argument out again more explicitly. An argument against change in general relativity. I hope that I'm not misrepresenting what Jung says in this argument in this paper. If I am, I apologize to him. Let's focus this argument. In this constrained Hamiltonian formulation, general relativity is gauged here. In such a theory, motion-long gauge is generated by a gauge symmetry. Different points in the orbit simply are the different representations of the same physical state of effect. If that's the case, then a genuine physical magnitude, which we could call the scare quotes of observable in classical general relativities, are represented by functions under constraints that is the gauge invariant of many constants on the gauge orbit. But now remember what change is supposed to be. Change is supposed to correspond to a different intrinsic property being possessed by some kind at different times. So it seems there's going to be change only if a genuine physical magnitude has different values at different times. But the dynamics of a theory, in this extreme Hamiltonian formulation, written by Hamilton's equations, these yield equations satisfied by the canonical variables Q and T, and the corresponding motion to a base-based point is confined to a single gauge-over than X. But, a genuine physical magnitude has the same value at each point in the gauge-over. It's a little change in general. That's the argument that I thought was so seductive worth arguing against. Isn't that just what you should be all further and further? Oh, you've been back 30 or 40 years ago. Why don't we discover that? I'm not going to be facetious, but I mean... That may well be true. That may well be true. I haven't seen his reference to these papers 30 or 40 years ago in which this... A horrible, apparently...

1:30:00 I'm sure John is aware of the early late of June. I mean, what's the difference? If he is aware of it, he'd be interested to see what the difference is. And that's why I said that I might as well have this argument wrong by presenting it this way. this is the object to which I'm going to respond and if this is the old frozen formula that was only 40 years ago then what I'm saying is totally old to people who know that stuff it still seems to me to be worth saying only because it helps us to think carefully about how things are getting represented within this formulation of this theory and what notion of change we should be hanging on to in order to defend the claim that there's change in general let's go back to the beginning of this argument different points in the orbit simply offer different representations of the same physical state of effect how do we assess that claim? well, to assess the claim we have to know how any point in the gauge orbit represents anything so let's think a bit more carefully about how that can be I think there are several ways in which you could take a point in the gauge orbit to represent something and let's see what they are I think part of the seductiveness of this argument I believe might come from sort of confusions and ambiguities in the way that base points get to represent things if you flip back from one mode of representation to another you can convince yourself of something but you can get all these modes of representation clearly separate and explicitly you wouldn't do it So, what and how does the base base point represent? If sigma is an abstract 3-manifold, an abstract one is not now supposed to be sitting in one of these models of general relativity, just abstract 3-manifold, that we want to try to relate perhaps to one or more models of general relativity. at least we have the possibility of mapping it into Kocci surfaces of this particular model of general relativity and we have Riemannian metric Q and some symmetric tensor field about which we don't say anything else at this point defined on our abstract

1:32:30 three manifold We can pull the space remining metrics of sigma and we can define something which we'll call the phase space of A to be the total of that and define canonical variables Q and P in that way and then sigma may be naturally invented as a type of space just in this model N just in case those two relations hold Okay. Indeed, there's a natural embedding of sigma in M by any trigonorphism. That's an isometry between this metric that we've imposed externally on sigma and what we might call the actual Riemannian metric with S interets from the Lorentz metric on M. And this natural embedding also maps our symmetric tensor KB with respect to G. So that's a natural embedding. There might be unnatural ones as well that we might want to think about. Okay. So I'm going to say the following. I'm going to say that face-to-face point omega-qp of this space that we just defined initially represents S in Mg. If there is a natural embedding of sigma as a time-slice S by a sum if you want this in H. So two comments on that. I say initially represents Because so far, all we're saying is how that face-to-face represents something in a model of general relativity. We're not saying how it represents something in the world. To settle that question, we're going to have to ask the question how the model represents the world. Note also that there might be lots of natural embeddings of sigma as a time slice, S, by lots of different hippomorphisms. So to say that a face-to-face point initially represents S is not to imply that it uniquely initially represents S. There might be others in S. As I say, initial representation, the way I define it, is descriptively. So if the description is multiplyly satisfied, then that point initially represents the one-time slice of M. And just to repeat what I said, it deserves repeating. That was only one model providing only one representation of the So even if we've settled on which of these potentially multiple descriptions we're going to take to be the one by virtue of which this represents a particular slice of that model, we still haven't figured out what, as I might call it, instant of the world that that model represents, that face-to-face weight itself represents.

1:35:00 So that's sort of setting things up. We still haven't settled the question, what and how does a base point represent? We've gone part way to providing a number of different answers to that question. So let's try and go a bit further and see what those different answers would be. Okay, first mode of representation. In the first mode of representation, we essentially forget about the distinction between our particular model of general relativity and the world take the model as a stand-in of the world. That's the way I put it. What we do then is we take that particular base-space point to represent an instant of general relativistic world W in the following way. We fasten on our particular model that we take to be the stand-in for the world from the entire class of models that we could have chosen, and we let every particular sign-size of that model that that face-space point initially represents by some dipheomorphic embedding, H. Now, we take that face-space point to represent that instant. Now, this is some other face-space point. What does that represent? Well, it may be that that other face-space point initially represents a unique time-slice S prime for that. If it does, then we take that face-space point to represent in W whatever is represented by that time slice of M. So, essentially, we're looking around on the gauge orbit at a particular point connected to our original point on the gauge orbit and asking the question what does it represent? And to answer that question we just sort of look at the description satisfied by that point. What in fact is the metric and the extrinsic curvature labeled by that one privileged model to see if we can find anything with that metric and that extrinsic curvature.

1:37:30 If there's more than one, we just pick one. If there's none, then that point of view is nothing at all in the world. So, as I said at the bottom, that mode of representation effectively takes our single model as a standing for the world and picks up the instance or instance of that world represented by points of the constraint surface by a mixture of ostention and description. we're using the description that's provided by the actual values of these Q's and P's to help us to figure out what a face-to-face point associated with those values represents and when description runs out, if it does, we just pick one by ostension now notice something interesting there might be other models which fail to represent that W even if they are diphyomorphic And it may be that we can find the point Q prime, P prime, which simply doesn't initially represent anything in N. And on this view, it doesn't represent anything in the real world, even though Q prime and P prime are related to QP by a single three-dimensional on the surface that's not quite accurate I guess we have to go back to sigma that's not the only way to go we might think that there's something unfortunately about privileging a particular model to represent the world because there's a whole of models out there and what about all the others represent that world. How would face-to-face points represent in that way? So the second mode of representation I want to look at is representation by equivalent models. We'll start off the same way. A face-to-face point omega to P initially represents anything times slice of S in the new symmetric Q and extrinsic Q corresponding to P in any globally hyperbolic model of general relativity here. And similarly, we just let this other face-to-face point do its own thing and reach out to the whole class of models and see if it represents anything initially in any of those models. Okay. Now, any two models of which that face-to-face point initially represents a time slice S

1:40:00 are going to be related by dipomorphism, that's the identity on S. similarly for any other point and its selected time slice in any of those models so how finally then well not quite finally we'll take those two time slices in any such gipiomorphity related models to represent a particular instance of which these models often equivalent representation so we've got the base space point, the whole class of models face-to-face points reach out to the models, hook into certain time slices of those models, and then we say what it is the face-to-face points represent by saying what the slices represent, and we take the slices in such a difficulty-related model to represent the instant in the world on which these models offer equivalent representation. Finally, then, what do the face-to-face points themselves represent? This point, for example, represents I in that case, because represents S and S, but this represents R. Okay. Now, if you have two models that are not difficult to be related, then these two points by this mode of representation are going to lie on different age orbits. They're going to represent instances in different worlds. So there's no notion that we're moving from there to there, one could be talking about or representing change in a given world, but that's okay, because those were not degree-mortically related anyway on the other hand, if the models are degree-mortically related, then these two points may be taken to represent instances in the same world in this way okay, now suppose we just look at the two points and suppose they're connected by a path in the gauge orbit and that path is generated solely by momentum constraint then they will initially represent slices as s-primed in models m-g and m-d-star-g where s-primed has a used metric d-star of q and x-primed it's over to d-star of k. And on this way of thinking, insofar as those two models operate with a representation of the world, we then may take those two points to represent the same instance in w, even though the q-primed p-primed are

1:42:30 difference in that way. So, notice the difference between the two modes of representation. The first mode of representation, we could have points like this, related by a single three-dimensional diphthomorphism, essentially, and if one of them represented something in the world, the other one wouldn't. On this mode of representation, if one of them represented something in the world, then so does the other. The same thing. when you say any two models of which omega pq initially represented time slice s related by a diphenomorphism that is the identity on s presumably that's only true for models without symmetries which format distinct space-type surfaces onto each other initial representation was not a relation that takes out a unique slice, so if there's lots of them Well, I guess, yeah, so I think that will be true in that case, right? If it initially represents a time slice x, and initial representation doesn't pick out a unique time slice, then it will initially represent lots of other time slices in that same model, and then we can, you know, for each of them, we can pick something such that Is that not going to work out given the multiple representations that we can get? It doesn't actually matter for the argument that I'm giving, but it would be nice to be saying something true rather than the plot. Perhaps we should come back to this later if you still think it matters. I don't think it matters, and I still think it's true, but we could argue about it and miss the name of the pavement. The third mode of representation, I think, is actually significantly different and in many ways more interesting. And so far noticed that most of these modes of representation I've talked about didn't any place talk about frames or anything like that, and moreover, none of them gave us any

1:45:00 clue as to what time any of these face-to-face points was to be referred to. If we really say something about what times are involved here, which at the end of the they would then be able to do it, or to even raise the question as to whether that motion along a gauge or a bit involves change, then we're going to have to pin a time onto some of these face-to-face points. So we can write omega of P of T and Q of T. That's what a frame will do for us. And the third mode of representation takes it to frame simply impose representation. They impose it perhaps completely without regard to the descriptive content is packed into the particular values of the three-metric and extrinsic curvature that are coordinates of a space-to-face point. So now I'm going to back up to global frames. And think of dipomorphism that represents a global frame on a particular model, which in turn is one way of representing a generalistic world, W. now if that's a global frame this thing is or can be made into a four dimensional differential manifold which has time slices like that on it and we can impose a Riemannian metric in a symmetric tensor field on those time slices in that way and now we've got a parameter t appearing because of the fact imposed a frame which picks out or gives us the labels for these different surfaces in that differential manifold and because this represents a global frame that's actually going to be a time slice not just going to be any old surface and as such it will inherit the Riemannian metric and extrinsic curvature from itself. So there's a potential clash here. The frame has just plonked us down on the time slice of the manifold, but that time slice of that manifold has its own Riemannian metric in extrinsic curvature, which is inherited from the Lorentz metric for the

1:47:30 manifold, independent of the PMQ that we started with, and tried to impose upon it, as it were. So indeed in general, if f is an arbitrary frame the humanitarian metric in extrinsic curvature that f of sigma t inherits from g will differ through these things. So we have failure of initial representation. But if we wanted to we should be bloody minded we could take that face-to-face point to represent the instant represented by that anyway. We could do that that. We are sort of ignoring the intrinsic geometry of the time slice in the model, and just imposing a representation relation, such that it's omega of Q and P that represents that is that, not omega of some other space point, which would initially represent this time slice. Okay. Now we've got the world, and we've talked about how this frame poses a mode of representation with respect to this model. The frame we're using to impose the representation looks different in different models, because a frame is not defined just on a single model. A frame is something which, for any model, picks out the relevant foliation of that model, and as we change from one model to another by the three-mortism, so the representation of the frame also changes from model to model by the same different model. So if our world is represented by this other model instead, in the same frame, we'll be represented by a dipiomorphic object, dipiomorphism, if you like. And then that takes the same instance that we represented in this other model by the appropriately transformed object in the manifold. So here are two points to keep in mind here. that the mode of representation that we're using here differs in two important respects from the first two. The way it embeds sigma in the manifold is unnatural, insofar as it fails to be an isometry. We might have chosen a frame for which it actually is an isometry, but that's sort of,

1:50:00 by the way, depending on how we chose our frame. But, most importantly, and this is where we've made progress over the first two modes of representation this is automatically indexing each other with points of omega with times maybe x anyway so let's go back a bit what I've tried to do is first of all stress that the question what does a face-to-face point represent is really up to us to decide view of representation if you like my face-to-face points represent what I want them to represent if we take that view then the question how does motion along a gauge orbit correspond to change if at all that question is also going to need to be answered carefully because in order to answer it we're going to first of all have to point for some particular story about how any point on the gauge orbit represent anything, or any instant in the world, the way I'm setting things up. Okay. So how is motion along the gauge orbit related to change? Remember, the argument that I'm trying to respond to said, well, motion along the gauge orbit would correspond to change if there was such a thing. But the general point on the gauge orbit represents the same physical state of Earth anyway. Motion along the gauge orbit doesn't correspond to change because there's no change. You mean the group involved in the four dimensions? Which group? I'm talking about motion along the gauge orbit at the moment. On the constraint surface. The full constraint surface. Well, the gauge orbit is well let's see a single gauge orbit corresponds to a dipymorphism class of models each of which can be taken in the same world any motion along a gauge orbit

1:52:30 can be generated some combination of the two constraints, one of which was the, well, two panels of constraints, one being the momentum constraints, and the other one being the Hamiltonian constraints. So there will be some restricted motions along the gauge orbit, which will correspond to imposing just the momentum constraints over and over again, without even touching the Hamiltonian constraint. But I'm not interested in the distinction between the kinds of three-dimensional diphtymorphisms that are generated by those two constraints. motion along the gauge of it however it has been generated you're taking the four dimensional point of view very seriously if I could just make one comment all this calling it gauge is I think one must emphasize that the momentum constraint is linear in the canonical momentum and the Hamiltonian one is quadratic and what one is I mean I think this is the real crux of the dilemma the problem of time and all this business of change in general relativity, is that one is saying, really when one is saying there's no time, one is saying that the fact that there's the quadratic one doesn't really make any odds to the gauge philosophy. I think a lot of it is relying on that, to say that the fact that the one is a quadratic constraint as opposed to the linear one is not affecting this philosophy that a gauge transformation doesn't onto a real change. And I think this is a huge issue. This is where the problem of time is residing. It's understanding that and understanding whether normal gauge torque applies to the quadratic constraints as well as the linear ones. This is my view on it at the moment. As far as what I'm doing today, except for a couple of minutes at the end, there is no problem of time. This is pure classical general relativity. And if you believe me, then pure classical general relativity, if you both people here account of change, a frame-dependent account, because all changes don't depend in any relatively relativistic world, but the purpose we call here is an account of change. Now, it might follow from this account of change, and in particular from the fact that it is essentially frame-dependent, that when you look at a quantized theory, when you can't simply impose a nice frame to start off with, then talk of change is going to look very problematic. So that's where the problem is.

1:55:00 Not in a classical theory at all. well the classical theory is well defined we don't know what the common theory is yes but you can see why you have troubles as well as no problem within the classical theory due to the way I'm viewing it so anyway how is motion elongated going to be related to change well the picture that I have of change, at least in an empty general statistic space time is that it occurs as varies with time, because succeeding instants have different shapes. If we put matter in, there'd be more kinds of interesting changes going on, but even in empty spaces, we'd have that kind of change going on. Emotional R gauge orbit, as I was just discussing in answer to a question, is generated by momentum and Hamiltonian constraints acting separately altogether. So, in order to understand how a particular emotional R gauge orbit is related to change, we need to be clear on how a point on the orbit represents it in the first questions. So, suppose this, suppose we've got two points in the gauge orbit, and these represent two instants in our world via a frame, represented on a particular model like F, where Q is given by Q of T1, T1 in that representation of that frame, and P is given by P of T1, and analogously then i claim no problem there's change in w relative to that frame between that instance and that instance if and only if there's some difference in those values q and q prime and q and p prime of course trivially there is such a difference and t1 and t2 aren't the same and you've got change between t1 and t2 now notice that that's so even if it happens to be the case, that there's a three-dimensional that takes you from the pair PQ to the pair P'Q' there's still a change in that case these two points will be connected by a parametrized path and whether this is generated by the hamersonian constraints momentum constraints or both together is irrelevant to the issue of change so I think disagreeing for some of the things I've understood but perhaps I've misunderstood him what I understood him to be saying is

1:57:30 big difference between the two kinds of constraints, that momentum constraints simply generate alternative representations of the same instant. And change, on the other hand, is generated by the Hamiltonian constraints. On this picture, if this is how the points are representing instance, it doesn't matter what kind of constraints you've got, you've got change anyway, if this condition is satisfied, change relative to that frame. But that's okay because all changes are relative to a frame anyway. Okay. Now suppose we take a different mode of representation here. We've got the first mode of representation in terms with which we take our model as stand-in to the world. Everything we talk about has to be restricted to that in that case we've got a face-to-face point representing instance by initially representing a certain time slice and let's ask again, does motion along the path from qp to q'p' correspond to change? Now we're going to change in this way with this mode of representation of both face-to-face points unless the second of these two slice in our model, there's no way that could correspond to change, because the second base point doesn't work to anything at all in that world. I mean, transition between omega tp and omega q'p' is taking us right out of the world altogether, pointing us down to some other possible world. So that's not change. Now that's true, even if these two points happen to be related by an actual dipomorphism in that way. still motion from PQ to P'Q' was taken entirely out of that world and popped us down someplace else so it certainly didn't represent change on the other hand if this point initially represents some distinct time slice in the model that we're taking to stand into the world then indeed there are many frames in which there is change in that world between the first and the second instant the one that this frame represents just as long as don't agree in both of their values provided, I think this is a reasonable condition, that these two

2:00:00 surfaces at least don't intersect I mean, I'd be hard for it to say that there's change between that time slice and that time slice and that corresponds to the fact that there isn't any kind of frame that will include both of those two time slices because the frame has to be a foliation so it has to separate them so, provided these two don't intersect, there's going to be lots of digital organisms, which take us from our representative space and time into the manifold and that this surface S the one that only the QP initially represents onto that S primarily associated with surface, or is the surface that only a t prime t prime initially represent. Okay, so again, this is sort of repeating what I said earlier, some of the frames that connect these two face-to-face points in that position will, sorry, some of the parametrized paths which some of these frames use to connect those two in the gauge orbit will be generated by the Hamiltonian constraints alone, others won't be. For the ones that in fact are generated by the Hamiltonian constraints alone, Luhashi's claim that it's the Hamiltonian constraints that generate change that would be correct. The both. But there will be others for which we get changed without the part being generated solely by the Hamiltonian constraints, because the momentum constraints are going to have to keep in two. Okay. Kuhash also maintained, in my understanding, that if two points in the gauge-over-directed by the length of this draft and going from one to the other is simply changing the way you represent the same instant, that's going to be thought on this mode of representation because of what I said earlier on this mode of representation we could flip out of the model entirely flip out of this world entirely and be talking about another world and we made a simple transition from QP to Q prime to Q prime generating purely momentum constraints I think you can see that

2:02:30 the approach I'm taking here is going to be partial approach. He wants to say, yes, there's change, if I understand it correctly. But it's only the change that's generated by the homogeneous constraints. And I want to say, no, it doesn't really matter which constraint is there you're generating, provided you can find a frame with respect to which all this stuff pulls through, then you've got change. right, let's just include a question. What? Including questions. Okay. I was like, so fast at the beginning, I was optimistic. Okay, so, where do we go? Yeah. The outcome against change maintain a genuine physical fact which is represented by gauging varying functions on the constraint surface that remain constant in the gauging orbit and concluded that there can be no change in observables. Well there's no change in these things but who cares about them at least within general relativity there's nothing special about being a quote observable in general relativity. Perhaps when we get to a context that's going to be important. Moreover such changes are observable, both frame dependent all we have to do is put ourselves in a frame and observe things at the appropriate instant. And if things look different, well, you observe change. We don't have to put ourselves in that frame. We could put ourselves in a different frame as long as we know how the frame we actually do occupy is related to the frame with respect to which we're trying to evaluate change. And we can do that too. Okay. Implications to the problem of time. Well, if I'm right, there is no problem of time in general at all. But there might be in a quantized theory. Why? Because on this account of change, in a relativistic world, all change is frame dependent. What's a frame? A frame is a hippomorphism that picks out a class of space-like hyperservices, a poliation. But if you don't have any structure on your space-time to start with, which you probably will not have in a Kwanzaa's theory, the notion of a frame can't be defined. If you can't define a frame, then you don't know what you're talking about when you're talking about change, because change is change with respect to a frame. So, I think that's my take on why there is a problem when trying to formulate a quantum theory

2:05:00 there's no problem with time, that's what general is to be told, if this is right I'm sure lots of you think it's wrong, so go ahead applause applause applause applause yeah, well, Richard thanks very much the global feel about you want to kind of recover an economical 3 plus 1 kind of notion of change with frames while, as you were emphasising in the talk to a question, having this four-dimensional invariance perspective, not caring too much whether it's momentum or Hamiltonian that needs you. Well, I have two questions. the verbal question really is why think that change must be caught in a 3 plus 1 way this goes back to Simon's question and to kind of Bergman-Komar type discussions about recovering change in terms of the scale of variance and the other question how important is it in the nuts and bolts of those modes of representation was it that your third mode you brought out explicitly that frames aren't individual to this specific model mg it was important for you that they were actually a family of dipheomorphisms sigma cross r to n a family one for each n right at least one for each n that represented a given little w it seems to me that's pretty hairy for a frame to to be a family like that and i don't think it's needed for what the drift of your thought well i actually disagree of a frame is to be thought of in direct physical or operational terms. A frame is something to which you have direct access by virtue of your powers as an observer and experimenter. And so a frame is something which is fundamentally located in the world. It's not located in any particular model. And since the model's a diphtherically equivalent, the representations of this frame were obviously also diphtherically related. Now, with regard to the point about and our common observables and locating change in some descriptive way, that seems to me to be just taking as it were an external perspective on things, whereas

2:07:30 change is a notion which has given its significance from our internal perspective on the world. And so a frame is, from our point of view, to cashing out what it's like to be an observer in the generalistic world. Now, one can abstractly, without any attempt to pin down a frame in physical terms. However, of course, we are actually occupying a frame only if all these Comor observables are doing their thing and realizing our occupation of the frame by there actually being physical fields, which make us, as observers, be in those frames. So there are two sides to the same issue. From the internal perspective, what I've been emphasizing, think. The frames are what really count, and the fact that you can define the structures on the space-time is what matters, irrespective of whether or not any matter happens to be occupying them in some nice way. On the other hand, for us to be able to observe change, we have to be in one of those frames. And for us to be in one, then matter has to be doing the right thing to put us in them. And that's the external perspective. I think if you restricted yourself purely to the external perspective, the notion of change so perhaps that's I'll try and be a bit more radical, that last remark I'm actually not totally convinced I would say that I do believe very strongly that is a problem of time in classical general relativity but it depends on how you perceive the canonical form in the two different ways the way you've talked, you've taken it for granted there's a space-time there always starting with space-time simply look at the canonical version of that. Now, if you do that, then there isn't a problem of time, but there's the notion of an ambient space-time, a space-like evades, and so on. But that's not, in fact, what people talk about the problem of time being. I'm talking about your name, and then, because having heard you now, well, I have a Greg Jones paper, that is exactly the classical version of the first-informalism. And why it's a problem is this, is that you're asked there to start with a purely spatial freedom and to try to recover the space-time picture. comes in. But that's as much present in the classical theories in the quantum theory in fact you read the literature on the problem at the time most discussions are fact done in the classical context because you can't be called it relatively right so you can say classically it's about saying so I think that's the difference he's talking about

2:10:00 what happens when you start off with economic performance then you have these discussions about things like intrinsic time extrinsic time some things Carl talks about but of course if you start from a space type perspective then yes okay but my reasons for saying you have to start from a space type perspective is that without that you don't even know what you're talking about Well, then that's what the problem of time is, right? Okay, okay. But what I'm going to make is you can discuss it classically, so the state of things, the quantum quantum phenomenon, I mean, very different. In the quantum theory, if you don't have a space, you have no option. Whereas in classical GI, you have an option. But if you forget that option, and just say from the beginning, let me deal with the spatial three numbers, then you do meet maybe what we call the problem of time, and we just don't do that. Okay, so I will accept that if you take this, apparently the worst perspective on the classical theory, and forget about the fact that change is an essentially frame-dependent notion, then you get yourself into trouble. So that is that there's a problem of time. The simple example is why this is important, because this is how the Koukart points out, is that most of the definitions of time that would be used by early workers in quantum gravity, intrinsic times, did not correspond to any space-time model, even classical. They didn't realise it. And Koukart points this out very, very of time counting, so it is important that one has this class I'll leave right June I'll keep it quick, but I'll ever first Okay, well I've got a few questions, so I should say all but one to the T and just ask, I mean in favour of Simon's view that actually the four dimensional frame independent language might have a role, if you take just empty Minkowski space-time, there's a sense where you can slice it up in sort of what would seem an unnatural way and say there's change going on. There is, whereas intuitively you might want to say, well, there is a difference between a space-time like empty Minkowski space-time where, you know, intuitively you might think nothing's changing and one where you've got different regions of space-time which are different characterised in four-dimensional frame independent terms happening in regions which are time-like related and you might think that means that however whatever frame I choose there's going to be change whereas in the Minkowski space time case relative to some frames you can get rid of this

2:12:30 change which makes the change appearing in other frames look a bit sort of of an artifact. Well I want to re-educate people's intuition so that they'll see change all over the place in Minkowski space time and the reason it's there is because all changes are independent and there are lots of frames with respect to which there's change in empty and costly space I mean that's the radical tinge to this paper and you start off by saying the intuition is well I'm hoping to change people's intuitions and if they don't want to then they have to give the wrong intuition as a reason for not doing that do you mean costly space geometry is changing all over the place in different textile range of frames is the last question we do as it's a comment I think we should have thank us Thank you. Thank you. Thank you.