Einstein and Mach's Principle

0:00 Driftig and even more printing. Well, this is my hope. I do what I can in this direction. I have a list of apologies and thanks and recommendations. Let me begin with thanking you for being here, for the organizer I have, it's important you can understand that, not from once over the night. Okay. And it's my duty and my pleasure to thank everyone and every organization who contributed to me possibly this meeting. First, let me thank the members of the Scientific and Organizing Committee for the proposition of Father Ed. But I must thank too the quite numerous organizations who were accepted to sponsor or to support this meeting. Let me list them. The Center for Einstein Studies from Boston University. The Center National de la Recherche Scientifique. The International Society on General Relativity and Gravitation. The Minister de l'Education Nationale. The Minister des Affaires Etrangères. La Société Mathématique de France, the CERN, is the member of the Société Mathématique de France, the UNESCO, and l'Université Pierre-Marie Curie, verse 6. But last, let me thank you, the CERN, and his director, and his staff, for welcoming us in Luminium. to. I have a list of apologies too, and I had some problems with the dates, and maybe that we would have been even more numerous if other dates would have been choosed, but some of you know that it was not that easy. I'm just thinking to some friends and colleagues who were unable to come due to the dates. But I must apologize, too, for the language. We have a problem with language. Language is quite really a big problem. And I choose the fact of English as the language of the conference. You know, it was, for example, difficult to make two posters,

2:30 one in French and the other in English. At the first beginning, I made two letters, and so I wrote you very often in English, in bad English, I don't know, it's not a nap, someone told me, it's a siesta, you have a siesta just in the afternoon, so I apologize for my English, I hope you will too apologize for your English, because it would be very difficult for us to understand John who is English, it's quite a lot of time, quite a lot of time, I know him. I'm always going to speak with this New Yorker, oh professor, good job. Bonjour. Just me, moi. Please. On a juste commencé une minute, j'étais en train de remercier tout le monde et les participants. Well, so, it will not be a shame, translating from French, Portuguese, and so on. I think there are so many people having good French, and better English indeed, that we must do that, we are not that numerous. and I think that maybe the best thing to do is for everyone to practice his second language. John will speak in French, I will speak in English, and et cetera. Okay. It's a little more value. Of course. What do you choose? Swahili. Okay. But concerning... What's right then? Concerning the material organization, I hope that everything is enormous, that's why it's the same, I was deprived, I was never here before, and I think that everyone is... But if you have any problem, of course, you ask for staff here, to Emmanuel Cesari with helping us in the conference, or to me, anywhere. I hope it will not be any problem. I would like to suggest, if you do so, we are quite an 80s kid, but I would suggest to organize something more, some sessions on special topics, and, for example, I would suggest today, we have just time tonight, to organize a special session on archives. There are some physicists, interesting in this story, who have many people who are interested in archives.

5:00 There's people of the discipline interested in this problem. I think that any people interested in the archive problem must come tonight and do something. I hope that John could organize that, he told me yes, and I think that tonight we'll do some special session on this topic because we have a real problem in the archive, knowing where things are and how to get the archive from general activity. Just one more suggestion, that some library press will be organized just there. if you are, I hope that any people brought with his reprints, books, etc., you put it there, to Emmanuel, you put a list on your reprints, so that we could have a look at what you wrote, and we put our names together. Okay, I think it would be quite a good idea. Well, I'll end my talk in thanking the speakers for all the work I've done, in preparing as well, before asking John Stachey to chair the first session, I'm glad to tell you that we plan to publish on the basis of the conference, very probably, but John will tell you, we will publish the proceedings that come from in the Einstein's data for who we know that how it can be done thank you, I have no more if you have any questions concerning the material organization, of course, please the schedule, for example we'll have a coffee not at 10 as it's written down but at 11 better because then it will be in 20 minutes and we'll just have the coffee now and you will have a tea too at 4 30 just at the bath for people clever enough to go to the can just to my

7:30 Just a couple of years ago, it occurred to me that although a great deal had been said and done and discussed about the specialty of relativity, very little had been done to discuss the development and history of the general theory of relativity. Indeed, as far as Einstein's own work goes, one could almost have a theorem that the amount of work that had been done on the two topics was inversely proportional to the amount of source material that was available. I don't know if that's a rigorous theory, but at least the folk theory. And I decided that since there was some work being done on the general theory of relativity by various people in the various parts of the world, perhaps it was time to get some of these people together and to start to promote more active interest in the history of general relativity. And not that I, by any means, mean to imply that enough or too much has been done on the strategy of relativity. On the contrary, much remains to be done. I just felt that perhaps one should twist the stick a little bit the other way that there's some attention given to the history of general relativity, which Einstein, of course, always considered the crowning achievement of his career. And so I organized what later turned out to be the first conference on the first international conference on the history of general relativity. Of course, I didn't think of it at that time. I didn't have the foresight of Leopold Slezak, who called his first book my collective works. so it was just a meeting on the history of general relativity and now Jean has kindly baptized it in retrospect the first international council at any rate we got together, some 25 of us at Oswood Hill and I think we had a very pleasant and productive week at any rate those there were so much so much enjoyed getting together exchanging ideas, problems, information I must admit a certain amount of social activity that we decided to hold another conference in two years and journalists have very graciously offered to take over the responsibilities for the conference.

10:00 And I think you will all agree he has done a magnificent job of organization. Only those who have gone through it can appreciate it. you might call it ma servitude grandeur de la victoire. He has succeeded magnificently. He certainly has far outstripped my modest efforts in terms of the ambiance, the luxury, the facilities available. And we think in one respect, he's fallen behind. I have not seen the VCR here yet. Some of you will remember the evenings there we spent watching the film. but you have to be remedied this. It's only a suggestion. As Jean mentioned, we have now reached a point where the proceedings of that first conference are in press. They will consider the first volume of a new series called Einstein Studies, which Don Howard and I are editing to find some announcements of this series outside, and also a table of contents and some of the front matter of the first volume, so you can see whether they're in this. We hope you will run out and buy the volume, or at least demand that the libraries buy the volume. We hope indeed to have the second conference as well in the volume, it will not be the second volume, the second volume will be the conference, will be the proceedings of a conference at Osgood Hill on conceptual problems of quantum gravity. But we hope the third volume will be the history of GR2, the son or daughter of the history of GR2. Well, I guess that what Sean Gay went along with this, is that correct? So we're in, we're now ready to proceed with the conference, or we're going to break first, I don't know. I knew that I did, but no, I didn't. I think it's perhaps the world for COVID, but it might be better. Probably not. All right. So, without any further ado, let me introduce Professor Julian Barber, who will speak on Einstein and Moxley.

12:30 Let me just begin, if I may, with a tiny little bit of autobiographical material. I'm always a bit of a problem at meetings like this because I am in fact independent not attached to any particular university although I live near Oxford I translate Russian journals for my living if you read JETP or Rusbyaki you will find my name at the bottom of quite a lot of translations I was rather busily writing up my last few graphs when Jean started his talk but as I came in I heard him saying something about we should all speak in some other language Right, well, having scared you to death, I hope I've got you in the right frame of mind to accept what I'm going to say, I'll carry on with just a little bit more autobiographical, if I may. Many years ago I got interested in Marx's principle, and I first of all read book, and then I read Einstein's papers, and I was very amazed, because it seemed to me that what Einstein was trying to do when he said he was trying to implement Marx's principle fought very little resemblance to what Marx had said needed to be done. And the more I looked at Einstein and the more I looked at Marx, I felt, well, there's a huge problem here. Einstein didn't even address the problem. One day it didn't come out right, his idea so then I thought well what one should really do is go back to the first principle really try and work out what the Marxian problem is what Marx said one should do and try and construct some theories that one could quite clearly see were Marxian and I worked on that for several years some of it in collaboration with Bruno Bertotti and we developed what we thought were genuinely Marxian theories and we were progressively making these or we thought more realistic in getting closer to the point where we would have a theory which would be a genuine alternative to general relativity. But at that stage, we started collaborating a bit with Karol Kukash, who, as many of you know, worked a great deal in the problem of quantizing general relativity.

15:00 And he gave us really, he finished off our work in a sense, because he pointed out that the principles which we developed and were convinced were what made a theory Martian were in fact actually already in general relativity. That in fact general relativity was a theory that was Martian precisely in accordance with the principles that we were developing. So, Bertotti and I were, so to speak, likes and forms, late and rather reluctant converts. I think in the end I accepted a bit more happily than Frodo but really what I'd like to talk about today is what I see are the problems that really should be solved when you come at it from Mark's point of view say a little bit about what Einstein seems to have been trying to do at a conscious level where, in my opinion, he was really confused all his life. I believe Einstein really was confused all his life about what Marx's principle could and should do. And then try and show, if I've got time, that at the deeper level of the structure of general relativity, it is actually profoundly Marxian. That's sort of my overall scale. Now, I don't know whether today I'll get right through to going through all the precise details of general relativity shows that it is Marxian, certainly for the case when you're considering a closed universe. Perhaps that will be an opportunity for discussion privately if I don't get through. And I could also say that recently at Bologna there was a conference, Marx's Principle in Retrospect, on which I spoke on the part that Marx's Principle played in Einstein's creation of general relativity and relativistic cosmology that the stimulus it gave him to create these fantastic theories. I've got a copy of that talk there. The talk I'm going to give today will not really overlap with that at all, but if anybody is interested in that, I have a copy, as we suggested we should put these things out there. Right, I think that's enough. Now, what are the problems, really? Can I see if this thing works? It fits. The white one. Let's see.

17:30 Let me just also say one anecdote. I was talking many years ago with Donald Lyndon Bell about Mars Principles. He wrote a beautiful paper on it. And he was telling me what Marx said, and Marx said you must do this, and Marx said you must do that. And I said, excuse me, if you don't mind me saying so, what you've just told me is not what Marx said. It's Dennis Sharma's interpretation, of Einstein's interpretation of Marx. And he stopped from there and he said, you're quite right, I've never read a word of Marx. Which was very gracious, however, I thought. But, to me, I mean, the essence of Mach, it's to do with the relativity of motion. It's problems created with the relativity of motion. Now, to me, one of the most striking things, when you read Einstein's papers from 1907 onward up to 1918, and right through to the end of his life, the expression, the relativity of motion, I've actually only found that expression once in all of Einstein's papers. 1914, published in the Giorno Scientia. Otherwise, Einstein always speaks of the relativity of inertia. He always speaks of the relativity of inertia. And by inertia, he does not mean Newton's first law of motion. He means the inertial mass of a body, the resistance of a body to acceleration. And he sees that as the supreme problem at his conscious level. Now, that was no problem at all to Mark. had given his own beautiful definition of inertial mass, his operational definition of inertial mass, where you see the accelerations which two bodies impart to each other when they interact, and that the masses are by definition inversely proportional to the accelerations they impart in each other. Mach was not in the slightest bit concerned with the problem of the inertial mass, and I don't think it was really a problem at all. And quite to what extent that's in general relativity anyway is another matter. But it's my firm conviction that Einstein introduced a totally bogus problem when he talked about the inertial mass, the problem of the inertial mass. I don't think that was it. That's at a much higher level.

20:00 It's not at the really deep level that the problem really resides. And I've wondered quite what the cause of this was. I think at the end of the day, Einstein, he was a victim of a semantic confusion. Inertia today means two things. It means the law of motion, the first law of motion, that everybody continues in a straight line with a uniform speed. That's actually a much more recent meaning of the word inertia. It originally meant the resistance to acceleration. And I think, if you look at Einstein, he uses inertia for those two really very different concepts quite indiscriminately, and he doesn't really seem to distinguish between the two things. And I think this is one of the main reasons why Einstein is confused about the problem of Marx's principle. The second one, I think, where he is, well, I won't say confused, well, I think there is also an element of confusion in this, is that, as I hope to show you, I think really the problem of Marx's principle is about the structure of dynamical theories, It's how dynamical theories should be constructed to overcome certain very basic problems. And the most natural way of looking at those problems is to see to what extent a theory can predict the future uniquely from observable initial conditions. It's something to do with the initial value problem, the way the theory is structured to enable you to predict the future from what is observed at a given instant. And I think this is an aspect of general relativity that really came in rather late into the proceedings. And in fact, when you read Einstein's papers up to the creation of general relativity and beyond, he seems to be treating general relativity as if really he's got elliptical equations, as if, so to speak, there is a matter distribution which is defined in space and time, and somehow from that you use the field equations in Poisson's, with Poisson's equation to find the gravitational potential from a given mass distribution. He seems to think that you then, from a given matter distribution, will determine the geometry. And he thinks of it in those terms. Now, I think the more recent work, which I think was initiated by Professor Bergman on quantizing general

22:30 relativity and seeing the structure of general relativity, showed that it's much more important to look at it from the point of view of the initial value this is something that came in rather later, and that the theory would have made much more sense, and the Machian aspect of it would have been revealed much more clearly, I think, had, from the start, Einstein thought of it in terms of an initial value problem with hyperbolic equations, but he didn't. Now, that takes me straight into, now, what I want to do then, if I can, is try and establish was when Marx first formulated these ideas in 1872, show how you can solve them with a simple theory. Then I want to show how those problems should be reformulated in the conceptual world in which Einstein was working in 1912. It was an incredibly rich period between 1872 and 1912. Conceptions about what the world contains, how it works, had completely changed in those 40 years, but I assert that these basic underlying problems with which one's concerned actually go through all this period, but they just need to be reinterpreted in terms of the later view of the world that Einstein had in 1912, and then, as I say, if I've got the time, I hope to show that the general relativity at the level of its basic dynamical structure does address these problems, and in fact I think solves them as well as one may reasonably ask. The basic problem is to do with the relativity of motion, and for me the most illuminating writing really on this problem is in Poincaré's Science and Hypothesis, which appeared in French in 1902 and in Translation into English in 1905. There's two problems. I'm going to describe the problem that Poincare came upon in my own terms, but it's essentially the terms that Poincaré was using. In fact, there's two problems. Poincaré only pointed out one, but there's a closely related second one, so I'll do the two together and I'll use my own language.

25:00 Now suppose you just have, go back to the world in which Mark lived, in which there were material particles moving in Euclidean space. That was the conceptual picture still just about in 1872. Newtonian theory was still so dominant in everybody's mind. Now suppose you had a universe, an island universe, which just consists of n-point particles in Euclidean space. And suppose you imagine that you take a snapshot of those particles at one instant, and you will just find a certain pattern of dots that represent the relative positions of these particles. You will just get that one snapshot, which will show you only the relative positions. Now, that's the relativity of position. see is the relative distances. Conceptually, that's really all you should allow yourself. Now, the real problem is not to do with one snapshot. It's to do with two snapshots. Suppose now these particles move a little bit so that they move into a slightly different relative configuration. And they will then, each of them move to a certain position. Now, suppose I take a second snapshot, just a fraction of a second later. Now, I want to ask myself, is about the laws of motion. It's setting up laws saying how bodies have moved. But how are we going to tell how these bodies have moved? If I take my second snapshot, how can I find out how the bodies have moved? If I put it on top of the first snapshot in some position, that will give me one little set of motions, infinitesimal instruments, which suggest that the bodies have moved in that direction. But my placing of the second snapshot on the first I could put it somewhere else and for each position I put it I get a different set of motions so it would seem that you can't define where objects have gone so you would seem to be in a really great problem and in fact if you go back and look at Newton's unpublished paper De Gravitazione written in about 1670 you will find that that's exactly the problem that forced Newton to introduce absolute space that motion is relative, and the only motion is relative to other objects. And when Newton looked at that, he said, well, then I can't formulate any laws of motion. It's quite impossible to construct a theory of dynamics, and that's why he introduced

27:30 absolute space. And that is the central problem, I believe. That is what I call the first mark problem, the fact that that is, you lose, when you take the two snapshots, when everything is moving, you lose all connection between the two time slices. And that is the first mark problem. Now, Poincaré pointed out that it's very interesting to look at Newtonian dynamics, celestial mechanics, shall we say, the n-body problem of celestial mechanics, in terms of this fact. And he says, we've got use from all these two or three centuries where we've been doing dynamics, we've got used to the idea that if we know the initial positions of a system of particles and their initial velocities, we can predict the future uniquely. And he says, is this the case with Newtonian dynamics? And he points out that it's very nearly the case, but not quite. And it's very interesting in what sense Newtonian dynamics doesn't quite succeed in predicting the future uniquely from observable initial conditions. Because if you... Suppose you have, first of all, just one snapshot. And suppose you know the masses of these particles. Poincare says, suppose we know the masses of these particles and we've got the relative distances between them. If we know the masses, we can calculate where the centre of mass is. Let's say it's there. Then we take a second snapshot. From that second snapshot we can also calculate the position of the centre of mass. So when we're worrying about where we put the second snapshot on the first we know at least we can put the centre of masses on top of each other and then we've got them in the centre of mass frame. And if you've got them in the centre of mass frame that's alright. but the really insuperable problem which Frank Perry points out to which there is no resolution is the fact that even although you can make the sentence of mass there's no way you know how much the one is turned with respect to the other and in fact what that means is you cannot get at the total intrinsic angular momentum of the system and if you can't get at the total intrinsic angular momentum

30:00 of the system, you don't know how There is a one parameter in this simple two, in two dimensions, there's a one parameter arbitrariness in the future evolution. Just think of the Kepler problem. If you have a sun and a planet, and in absolute space the planet has no velocity at all, it's just at rest relative to the sun, it will just then fall straight into the sun. But if it's got a component at right angles to the direction of the sun, it will go around in a Kepler orbit. So there's indeterminacy. And as Poirier points out, that indeterminacy is very curious indeed. Suppose you have, it's somewhat more sophisticated in three dimensions than it is in two dimensions, and that's to do with the non-commutativity of angular momentum. But essentially, if you've got the n-body problem in three dimensions, so we show you for a globular cluster, which has got half a million stars in it, that are billions and billions of relative distances and relative velocities. And if you had two snapshots of a globular cluster, you'd have billions and billions of initial data, but you would just lack three bits to be able to predict the future uniquely. You couldn't arrive through it because you haven't got that angular momentum. And in fact, in three dimensions, you have to know at least one acceleration. without one exception. That's all you lack. So that is the first mark problem. And that's highlighted, as Poincare pointed out, in what's called the reduced two-body problem, with which I'm sure you're all familiar, passage to the Routhian, where you eliminate the azimuthal coordinate, which is a cyclic coordinate. You finish up with the Lagrange function I've got at the top, which contains only the relative distance, distance, its rate of change, and because of the unknown angular momentum, it appears there as the so-called centrifugal barrier. That angular momentum, that's the intrinsic angular momentum which appears there. And in the observable dynamics, in the observable data, that is, so to speak, a mysterious constant which you can't calculate in any way from what's observable. You have to put it in. Now, Boncari says, if you're considering an island universe, the whole universe,

32:30 you can go over to a description like this. But then you have a curious situation with this angular momentum, this M, this mysterious constant. If you've only just got the entire universe, what is the status of this M, this capital M? Is it part of the fundamental which has, so to speak, an extra term in the potential represented by that? Or is it the reflection of the fact that there's really an absolute space and that's the angular momentum in it? But if you have the entire universe, you can't tell. Now that's the two-body problem in two dimensions. If you go into three dimensions, let's say because of this particular property of angular momentum, you actually can't do that. You have to have a form of the equations where you have a second derivative Poincaré says what is the status of this constant n this is what the first Mark problem that I call this device now there's an exactly similar problem which Poincaré doesn't mention which is to do with time now suppose we're considering the entire universe, we say it's just n particles in euclidean space how do we how do we think about time? According to Newton, we've got a clock ticking away on the side, and fine. But if you haven't got a clock ticking away, you've only just got the particles and their relative motions. There isn't in this, so to speak, any clock. And if you think about this, if you just have our two snapshots here, and no knowledge about the passage of time, you'll realize that from those two snapshots, you also, if you want to predict the future, missing some vital data again. The vital data is that you don't know how fast these particles are going. If you've got no plot to tell you how far apart in time these two snapshots are, you're missing, so to speak, how fast they're going. Even if you can solve the first Mach problem, which is how the second snapshot is placed relative to the first like that, you still can't solve the second Mach problem, which, sorry, that will then tell you, so to speak, where the particles are moving in their configuration they're moving, if you can solve the first problem in some sense. But you still don't know how fast they're moving in that direction. And that is what I call the second mark problem.

35:00 You don't know how fast they're going. You can't get at the kinetic energy, and you can't get at the total energy. And again, you get another very characteristic one-parameter arbitrariness in your initial value problem. And that is reflected very interestingly in remind you of Jacobi's principle Jacobi asked the problem trying to solve finding the general solution of Newtonian problem can I can I first of all find the path that a dynamical system will take in its configuration space first of all find the path and then afterwards find out how fast it moves along that path and you can do that, you arrive at a geodesic principle in configuration space. And this is here, Jacobi's principle here. The Jacobi action is a geodesic principle in the configuration space, where you have this curious product structure, where T is very nearly the ordinary kinetic energy, E is a constant, and V is the ordinary potential, the ordinary potential. Now, the paths in the configuration space are labelled by an arbitrary parameter, lambda, which can take any values. There's no time along this configuration space in the formulation of the Jacobi principle. It's just a path label. And so it's with respect to this path label that you find the dr by d lambda in Jacobi's principle. Now, this constant, E, turns out to be the total energy. what you do in Jacobi's principle is that you specify some total energy for the system. We say the system has some total energy. Then you solve the problem in its configuration space by Jacobi's principle and you find out the path it takes to the configuration space. Then you use the energy theorem where you say the total energy plus the potential energy must be equal to that constant E. And this requirement effectively fixes your lambda to be Newton's absolute and that then tells you how fast you integrate this condition and that tells you how fast the particle moves along its part in the configuration space in Newtonian absolute time but again, if you're considering the entire universe, I don't know I've never read Jacobi I'm afraid

37:30 on this but I don't know whether he thought about this problem but if you're considering the entire universe once again we have no external time you haven't got it And this constant E appears exactly in the same way as the constant angular momentum appeared in the Poincare problem. And it gives you another one parameter arbitrariness in the evolution. So you get these two central problems in constructing a theory of motion when you're trying to apply it to the entire universe, if you consider the universe to be, in some senses, finite. I just don't know how you transfer all these ideas to a genuinely infinite universe. So there should be the proviso. All that I'm saying refers to a universe that, in some senses, is finite. Now, basically, my central message is this. That if you address these problems, I construct dynamical theories which overcome these two very basic Mach problems? What sort of theory must I construct to overcome that? You can do that in the conceptual world in which Mach was originally thinking about inertia and these problems. You can very readily find theories to overcome these problems. And then my assertion that general relativity at its fundamental level solves these problems too, is really the essence of what I'm trying to say. That if you actually follow through, you look how you solved the problem in the world of 1872, see how the problem must be generalized to the world of 1912, and you then find that general relativity actually addresses exactly the same problems and solves them in exactly the same way as Bertotti and I found for these much simpler models for the world of 1872, knowing nothing at that stage about this reformulation of general relativity which brings that out. And looking at the plot, I don't think I'm going to get to that final stage, but that's the basic message. And there'll be reprints of the papers in which Bertotti and I did this, so you could check up on that if you're interested. Let Let me just outline the basic general framework that Totti and I are loved.

40:00 The basic concept is the relative configuration space of the entire universe. We take Marx seriously, and Bishop Barclay to that matter. There are only relative distances. If you have a configuration of particles, all that is meaningful is their relative configuration. So that there's a point, instead of having an ordinary configuration space which gives the positions of the particle an absolute space, you say, I'm only going to allow relative distances so that if I have two configurations which can be brought to exact coincidence by placing one on top of the other, and then they're exactly congruent, I say those are identical. of course, Leibniz's principle of the identity of indescribable. And so that's the basic concept of the relative configuration space of the universe. So that is sort of taking on board the underlying picture to do with the first mass problem. And the second one is how we think about time. We say there's no such thing as some external time ticking away, no absolute time. all that we've got in the universe is that we've just got this universe of a finite number of bodies who are going through their motions they're going through a succession of relative configurations and as they do that they trace out in the relative configuration space some path and this is in the relative configuration space and we assume that it's meaningful to say that we can imagine snapshots so to speak taken of each successive relative configuration as it moves through the relative configuration space. But that's all we can do. There is no sense in saying that this is five minutes later than that one. Because if you're considering the entire universe, if you try to speed it up, all the objective things that happen as you go along there are just the same. It doesn't matter if you run the film through twice as fast. I mean, you can run the film through twice as fast because you're sitting in the cinema and there's a clock on the wall telling you how fast the film is running. the entire universe. You can't do that. It doesn't have any meaning. All the objective relationships, everything that happens, remain just the same. So that's the basic idea, that there's no time and you just have a succession of instants which you can label by an arbitrary

42:30 label parameter lambda. And that's the underlying principle. So the history of the universe is a continuous path in its relative configuration space. Now, Bertotti and I found two basic ways of setting up theories which overcome the two muck problems that I mentioned. I'll briefly talk about the first one we found because that seems to be more transparently muckian. It's more readily muckian. We spent quite a lot of time working on it. We abandoned it for two reasons. out that you have an effective anisotropy of inertial mass in such a theory. And that means that, in fact, the motion of the planets in the solar system is sensitive, for example, to the galaxy, our galaxy, which is a large concentration of mass. And the presence of that galaxy, of the galaxy, would show up in the motion of the planets. There would be quite anomalous advances of the perihelia that are really relatively large, much bigger and completely rule out any such theory. And the second reason we abandoned this approach was that we couldn't see how it could be readily generalised to feel theories and to come up to date, so to speak, and produce more realistic theories. So for those two reasons, we abandoned such an approach. But I just want to mention it briefly because it is, in my view, so transparently marquing. And all of this, I believe, it's just a pure, that theories like this were not discovered in the 19th century. If Muck had not been such a reluctant theorist and hadn't so despised variational principles, I think he could easily have produced a theory like this in the 19th century. I think it's a pure historical accident. But I think one ought to be aware that these theories can be constructed and in a sense, I believe, completely sold the Muck problem as it existed at the end of the 19th century. And this is done by, if you've got just the relative configuration space, and let me remind you again that I can specify an initial position in the relative configuration space. That's shown by my one snapshot. And I can specify the initial direction in which the universe is moving in the relative configuration space.

45:00 I've got the direction. What I haven't got is the speed at which the universe is going along this particular direction. Now, what sort of dynamical theory must I construct to nevertheless have a dynamical evolution, a well-defined path in the relative configuration space? Well, the most obvious way and easiest way of doing it is to set up some sort of geodesic principle. I need some definition of a metric in the relative configuration space geodesic problem. And if you've got a geodesic problem, if you have any two points in the relative configuration space, then there will be a unique path joining them. And alternatively, if you specify an initial position and an initial direction, you will also get a well-defined distribution. So the solution of both the Muffian problems is simply to set up a variational principle, which is, well, that's the easiest way of doing it, which is the geodesic principle in the relative configuration space. You can do it in a more sophisticated way, but that's the simplest one. And this is the variational principle, which we worked out in detail just to see how it happens. It has, incidentally, just as in the Jacobi principle, a characteristic product form. You have two factors. Now, the first factor looks just like Newton's, sorry, The first factor, sorry, this is really the guts of the theory. This is something analogous. I put in only relative distances. This is, so to speak, to get, this really realises Mach's principle, this factor here, because what it does is say that the law of inertia arises, and also the inertial math for that matter,

47:30 arises through an interaction between bodies which has got a 1 upon R dependence. This is sort of a physical 1 upon R dependence in there. But the really important thing is that I'm putting in here the rate of change of the relative distances with respect to my arbitrary label parameter. And the reason why we have to have this very characteristic square root is that I've got to have this overall variational action to be quite independent of the label parameter I choose. I've got to be able to re-parameterise. I've got to be able to go from one lambda to a completely arbitrary other lambda subject to just sort of continuity and polyphobicity. So I have to have the Lagrange function has got to be homogeneous of degree one. That is the one really important thing which will solve the second math problem. That means I'm not using any external time. No absolute time is going to be. It's an entirely intrinsic theory. And so this forces... I've got to have at least a quadratic in here to get a non-trivial variational problem. If I tried to put in just a great sum of linear terms, I would just have a divergence and I wouldn't have a non-trivial variational problem. So I've got to put in quadratic terms, but then I've got to take a square root to get the re-parameterization invariance. And then, to get something that looks like Newtonian gravity, I have to multiply by a potential function, which I take to be something very like Newtonian potential energy and gravitational theory with one important difference that again to get the re-parameterisation invariant I have to multiply these two together because if I just added it I would then have a V d lambda term this isn't changed by re-parameterisation this changes under re-parameterisation and I would not have an invariant action forced to make a quadratic, a product form there, I have to take the square root and I have to use a product to satisfy the, to resolve the second Mach problem, to ensure that I'm not using an absolute time. And I'm solving the first Mach problem by just putting in relative coordinates from the word go. I'm not doing anything else but relative coordinates.

50:00 Now, I'm not going to work out that theory It's worked out in detail in the paper that Bertotti and I did And I've got a copy of that here if anyone's interested I just want to say what you can do with this theory The first thing is You get a very When you find the equations of motion You get a very characteristic thing that happens you get these two factors here occur on both sides of the equations of motion you've got one that way and the other side on the left hand side you've got that and on the right hand side you've got that you've got this characteristic ratio here of these two factors which have gone into your Lagrange function and when you do the variation to find the Euler-Lagrange equations you have to keep your time parameter absolutely arbitrary but once you've done the variation and you've got your equations of motion which in the general case turn out to be really quite complicated for this Negronov function you get these two factors that occur on the two sides of the equation and they suggest quite unambiguously a particular choice of the lambda which will simplify the equation and give you a so to speak a canonical form where they take on their simplest form and that is quite obviously achieved your lambda to make gamma equal to psi, because then both of these factors become equal to unity. And when you do that, this gives you a quite unique lambda, and that lambda turns out then to be exactly like Newton's absolute time. It is exactly like Newton's absolute time. And that is exactly as Poincaré said. Poincaré said time is chosen to make the equations of motion take on their simplest form. It's a thing of human disposal. You decide what the time is going to be. But it's not totally arbitrary because you've got quite definite equations of motion. It's those equations of motion that suggest that particular time. Now that's the first sort of characteristic thing. You recover something very like Newtonian absolute time. You also recover Newton's first law to a very good approximation. considering some sort of cosmology where you imagine that the bulk of the universe consists of, shall we say, a lot of particles in a spherical shell

52:30 that are expanding, moving apart. So that you've got a background cosmology where all the particles, where the universe as a whole is expanding. And then you consider, in this background, how a set of particles near the centre, a small subset of particles, will move relative to that background. And when you work out from that Lagrange function, when you work out the things, do the various approximations, you find that in this frame of reference, these particles move more or less in accordance with Newton's laws. You recover more or less Newtonian gravity and inertia. But that space in which the objects move along straight lines linked to the cosmology of the home the background of all the other matter in the universe so essentially you recover Newton's laws you recover them with corrections there are certain correction terms which characteristically call cause the perihelion of the planets to advance which looks quite nice but I think it's a little bit, I don't think these are really realistic theories but for quite some time we were a bit sort of excited by this and thought ah we'll get the perihelion of Mercury high. But the trouble is, as I say, that's all right here, but if you've got the galaxy here, the centre of the galaxy there, you find, as I say, that the motion of the planets in the solar system is quite sensitive to the presence of the galaxy there. There's a big hunk of mass there, and it causes quite anomalous perihelion advances. So that's such a theory that's really ruled out. But what I think is nice is that this shows quite explicitly how Marx ideas can be implemented, you can go a long way to recovering Newton's laws, show that the scheme is realistic and that it's viable. Now, so much for that. We then, as I say, abandon that approach because of the difficulty of the anisotropy of mass, of the inertial mass, and the fact that such a picture didn't seem to generalise the field theory. And it was in trying to work out how we could develop similar ideas in field theory that we hit on a second basic approach, which bit by bit

55:00 we discovered was then actually more or less identical to the one in general relativity, though they're at a very much more sophisticated level. And the idea of this is just very simple. Suppose I'm considering a scalar field theory. You have exactly the same problem with the snapshots in field theory as you do in point particles. Suppose I imagine my scalar field to be a pattern of intensities. And I take a snapshot and it will show me a pattern of intensities. that's my first snapshot then I take a second snapshot another pattern of intensities how have the intensities changed field theory is about how the field intensities vary in time you've got exactly the same problem if you're considering the entire universe and you deny the existence of an isolated space you don't know how the intensities change from one snapshot to the second so this is why problems, and you've got exactly the same problem with time as well, these very basic problems go through to however sophisticated a theory that you're trying to construct. Now, how can you get over that? Well, necessity is the mother of invention. In desperation, I said, well, look, suppose we do the following. Suppose we take the first snapshot and put the second one on it in any arbitrary position, And that will give us, in that given arbitrary position, I'll call the intensity phi, then I'm going to get a d phi, which is the change in the intensity between the two snapshots for an arbitrary positioning of the second snapshot on the first. And I can then calculate over, let's do this in two dimensions, the x squared, I can calculate that for my given configuration. but then I can move the second snapshot relative to them I can shift it around this is a positive definite quantity and I can find where this quantity is minimum and then I can say ah that is when I've got actually what has happened that is so to speak the intrinsic variation between the two snapshots that is so to speak the irreducible minimum I may not be able to say how much

57:30 vary, but I can say this, so to speak, is some measure of the minimal amount by which they vary. This is an irreducible quantity. This is, so to speak, what I call the intrinsic variation. Now, it's a global, it's globally defined, but it will lead to a d phi, so to speak, at each point, once I've got the one snapshot on the second. And we call this the intrinsic derivative. Now, you can put that in, I've got till about 10, 2, no, only 5, 10 more minutes, haven't I, before, to 50 minutes. You can put this in a more general form. I just want to put it in a general abstract form, because that then brings it very close to what you have in general relativity. Suppose I define an ordinary configuration space like I would imagine it in standard dynamics, a dynamical system, where I've got certain dynamical variables, Q, which define a position in an ordinary configuration space. And suppose I have what I call a Q metric. This is a sort of familiar thing which one would normally do, where you have some sort of metric in that configuration space, which is represented by this quantity here. And those are my two Q1s and Q2s. Those are, so to speak, my two snapshots, and there's the bit between them. Now, if this, in the way we've been talking, we've been talking about the group of motions of Euclidean space, where I can move one snapshot relative to the other in any way I like. snapshots there. I've got a group of motions there, and I've got a Lie algebra which generates those motions, those rotations and translations. And that will be represented by some Lie algebra of which these O's will be the infinitesimal generators. So that if I've got Q goes to Q prime, when I'm moving my second snapshot on top of my first, when I move the second one, it will generate a sort of a variation, which is due to the fact that I'm moving the second snapshot. I take a coordinate system and fix it with the first snapshot, and lay it down on the first snapshot. Then I take the second one and move it around on the top of it, and that will give me

1:00:00 that will generate this is the second snapshot here the value is Q there, they will go to a new Q, Q prime which is my second snapshot there, plus the effect of the generators working on my second snapshot, and because these are infinitesimal parameters here, and I can take either the Q, the first snapshot, or the second snapshot here, because they're assumed to be close together and only to differ infinitesimally. So this sort of spurious variation which is generated by moving the second snapshot relative to the first is represented in this way there. And formally, you can minimise this by taking these things here and having a variational problem where you say, I move those two things around, which is done by trying all the epsilons, all these infinitesimal parameters which say how much you're displacing a thing, and move them around until you minimise that. And that will give you a certain intrinsic variation, indicated thus, which is where these things give the minimum. These are the values of epsilon which minimise your quantity there. And then you've got this intrinsic variation. That's just a mathematical expression of this procedure here. But let me just take that off and then show how that this then leads to a second type of theory, which is what we call intrinsic dynamics I assume this is just so to speak up here a Q metric which is in the ordinary configuration space as if we had absolute space there which is just the little bits that the particles move going back to the particle picture I'll illustrate this in the particle picture this is going back to the idea that the particles move just a little bit and the second snapshot has got sort of a dr. I'm going to take the same potential as I did before and I'm going to do a variational principle which is now in the relative configuration space using the intrinsic derivative where I take again this characteristic product form. Here is the quantity that I'm going to vary to find two things. I'm going to first of all find where the intrinsic derivative is, where I get the two snapshots to closest degree.

1:02:30 I'm going to do, on top of that, the ordinary variational principle. With these quantities that have been found there, they're going to determine how much the things have changed. And then I'm going to consider an entire path and find how the action is minimized along the entire path. So the variational principle has two parts. First of all, you have this part which is moving the one snapshot to the other at any one point to find out what is the intrinsic variation. and then you sum up the intrinsic variation along the entire path and then you say that the actually realized path is the one that minimizes that intrinsic variation along it. And that is expressed there. So again, I have to take a product form. There's the thing that's going to give me the intrinsic variation where I've got auxiliary, you can call them gauge variables if you like, auxiliary variables which are going to depend on how I put the two snapshots relative to each other. Here's a potential function multiplying it, and again I take the square root to make sure I've got homogenous degree 1. And if you do I'm now at this stage here supposing that I've actually done the variation these are the intrinsic variations here I've done this shifting business and solve that part of the problem there. And I now get the Euler-Lagrange equations, which are here. Now, this quantity here I'm going to take to be, again, very like the Newtonian kinetic energy. In fact, identical to it, except that this is a completely arbitrary label parameter at this stage. But my potential function is going to be the same. These are the Euler-Lagrange equations that I get. And here you get this characteristic ratio that I was telling you about. the potential and this t term, which is, for the moment, got the arbitrary lambda, and then it's inverse on the other side there. And here you can see quite explicitly and very simply that you're going to get the simplest equations of motion when you choose the lambda, which is always, and you can always do it, always make the t to be equal to the v. And then, when you do that, you find that you then just get Newton's equations of motion. and you get it though with two very important differences because you require this

1:05:00 you find that the time that you've got this Newtonian absolute time that you've constructed has the consequence that actually what you recover in Newtonian theory is the solutions with energy exactly equal to zero and because of the intrinsic variation where you've shifted the two snapshots relative to the other you find that you get the intrinsic angular momentum is exactly zero. So you recover Newtonian theory, but with those two differences. There. Now this was the thing that rather surprised us. Now, it's getting to the end of my time, so I think I'll just say just a tiny little bit about how this generalizes to GR, and then say if people want to ask me or read the papers, we'll do it that way. If you think of Clifford. Clifford, I think it was in about 1870 when he translated Riemann's papers into English. He came up with this idea, he says that there's no matter. All you have is a Riemannian space and waves of curvature, a variable Riemannian space, and matter and the motion of matter is nothing but waves of curvature moving through this in this variable Riemannian space. Now, if Clifford and Mark had got together and Mark, in my language, had pointed out this problem with the two snapshot problem, how would they have set up a dynamical problem as Clifford imagined it? Mark would say, if you take two snapshots of your... Let us imagine two simultaneities. one Riemannian space with genuine curvature in it, and one that differs from it a little bit, how are you actually going to define the change between it? How are you going to set up a dynamical theory? How are you going to get equations of motion that tell you how in a lawful manner this curvature varies between the two things? And if you think about it, you're forced to go to a generalization of this two-snapshot idea of that I described it, but this time it's vastly more sophisticated. You have to, so to speak, try each point of the one Riemannian space with all the other ones. So whereas before you just had the Euclidean group of ordinary rotations and displacements,

1:07:30 in this case you're going to actually have, so to speak, something like three-dimensional general covariance, three-dimensional general covariance, where you have to, so to speak, test every way of matching one point other points in the subsequent snapshot, and then do that for each of the ones all over the snapshots. And in fact, and I'll stop there with the assertion, that this is actually exactly what happens in general relativity when you're trying to formulate it in terms, as Wheeler does, of geometrodynamics, where you ask, how does a three-dimensional geometry evolve as you go forward through space-time? What is the dynamical law which geometry evolves. And in fact as reformulated in geometrodynamics, you find that this is exactly the process that's going there. And there is also something very closely analogous with the time as well. But I think time has come to an end so I'll stop there. Thank you very much. This is not the famous end of time. Well, it's been a very provocative time. I'm sure there were questions and comments. I would like to suggest that there's one point that you have not generalized, and that is Yes, thank you very much. That is the question I was hoping for. Thank you very much. This again, I must say that my debt to Carol Kuhlhash is very great. it was Carol who pointed out to us really how this comes about if you, it's exactly the thing, when I say that general relativity is machian in the sense that we require that's true, but it's machian in an incredibly special way and it's precisely to do with this tilt invariance that Professor Bergman has just mentioned you, the way we attacked special relativity was to say well, let's not worry about it let's sort of construct

1:10:00 along these lines, and then let us see whether it has got additional invariant properties which at the level of special relativity reflect Lorentz invariants, or in the case of general relativity, reflect the general tilt invariants. Now, what in fact happens, and I think this is really the work that Kukash and Hochmann and... Sorry, I forgot the third one. Geometry Dynamics regained. What happens is if you, if you construct, you've got a, your Lagrange functions has a certain characteristic structure, but there's still a great deal of freedom in your particular Lagrange function. Now, what happens in geometrodynamics coming from general relativity is that you get, well, shall we say, first of all, the most general theory of the Clifford Mach type would give you some theory which would give you a succession of three geometries which you can then stack together to make a given four geometry. Now, in the general case, your Lagrange function would be very general and there's all sorts of Lagrange functions you could have. But you get general relativity in the following way, a fantastically special theory is that having solved the problem once, so to speak, you start with two snapshots and build up a space-time, you can then, having constructed that space-time, which is analogous to constructing Newton's absolute space-time, you can then re-slice it and slice it completely different surfaces of simultaneity. And the thing that is so special about general relativity is that when it has been re-sliced in any quite arbitrary that the dynamical law which is governing that evolution is exactly the same dynamical law as governs the original one. So you've got, in addition, this extra tilt-in variance, and it's quite true. That is what is so distinctive about general relativity. I think it's a fantastic theory, and one could go into a great deal more about this. If anything, it made me feel rather stupid and humble, it was sort of realising this. I mean, there are at least two gentlemen here who heard me speak ten years ago at Eiche, about a thousand kilometers across the Mediterranean, Professor Papapetro and Bergman, before I realized these things. And I expounded, essentially, the same sort of talk I had here. And it was only after that that Kukash really showed me how general relativity does this, but does a whole lot more.

1:12:30 And it's really fantastic. So thank you very much for the question. Professor, can you tell me? My thought is just in the same way, the same plan. First, I will say that Einstein probably had consciously some misunderstanding of the math principle because I think he has first learned it through purpose. And again, the other thing which is, I think, relevant is he didn't think of the relativity of motion, but the relativity of spirituality first. And so the relativity of motion comes in his mind little by little. And so when he takes the general relativity, he thinks of relativity after acceleration. But the fantastic thing in that is that by thinking from relativity of simultaneity, He gave an operational point of view which makes general relativity first a local theory and then leaves global problems, put global problems in a differential manifold so that you can't confuse snapshot and maps. Snapshot being something which is brought by light and which is information coming to a local observer and maps is just some representation of the events which we do using those snapshots.

1:15:00 When you do the differential geometry, you use maps, which is just some points, some events given by numbers, the coordinates. But these coordinates are just a construction which has been done using our local observations. Maybe the observation of galaxies or any other things, but made by a local observer. And I think that the very interesting thing in general activity is to put the problems, the physical problem on a local point of view and then take a kind of global by slicing, if there is slicing, slicing the space section and other things. And so, I think the main idea of relativity remains in the relativity of simultaneity rather than relativity of motion. I'm not sure I entirely agree exactly about the math, but your main point about the relativity of simultaneity, I agree wholeheartedly. I think the, I don't think there's any doubt, having gone through those Einstein papers as I did recently, a few months ago, to see how terribly important special relativity was and the relativity of simultaneity. This was the driving force that was pushing him forward, in a sense, and was dictating a great deal as he developed general relativity. He had this machian psychic stimulus, but it was the relativity of simultaneity that was really the driving force, and he found it that way. find so marvellous is that you get this incredibly sophisticated realisation of these ideas in this roundabout way from the Machian point of view it was a very roundabout way but let me say also that I think Einstein took the correct way, he would never have arrived at the particular structure of geoveterodynamics from developing these ideas because this fantastic very specific structure that GR has could never have been guessed in that way, I think you're absolutely right on that, so So, I mean, to me, it's just another wonder of general relativity that it does all that and these other things as well. I wonder if I could just take the opportunity to say that I am, at the moment, engaged in writing a monstrously long work

1:17:30 on the absolute relative question, which I think is going to involve me in writing quite a bit about the history of general relativity and how it was found. So if there's anybody here who's actually engaged on a similar project, I would be very interested to get together, possibly to perhaps avoid unnecessary overlaps or just exchange views and so forth there. So I'd like to take the opportunity to say that. Hubert Gernic? Taking a very operationalist on the field for the moment, I have difficulties with two concepts. Within your formulation, Marx's original position and Einstein's course. That is, I mean, it's always taken for granted that if you describe the dynamics with regard to relative distances then you are on a good positivist or operationalist stand but this is only true if you can identify your partner and i'm not quite convinced that you can identify your partners without concepts of absolute space and time i mean how do you identify these various You have to have some internal variable and the only one you have is the mass. And the other concept that creates difficulties to me is, say, the entire universe. Again, you cannot identify the masses that are supposed to be formed in the entire universe. And that's because of these two points I always worry about what this kind of formulation means. Yes, I think you've got a very valid point. I think what I could say is that this sort of analysis is at a sort of conceptual level where you form a concept of what the contents of the universe are like and then you say how could I find a dynamical theory which most naturally reflects this sort of concept of the universe that I formed but again Kukash has pointed out to me that the whole question of how readily you can determine what anything is gets very difficult already in Maxwellian field theory when you're actually, to get the field strengths, you actually have to observe accelerations because, I mean, the field strengths are defined operationally in terms of accelerations, which again presupposes that you've already got an absolute space there. And I would say that there's a terrific amount

1:20:00 of work that would have to be done really to get this on a is really muck in in the sense of going from direct sense perceptions right the way through to the dynamical structure and how that all is traced. And I think there are a lot of questions there that still need to clarify. All I would say in my defence is that we sort of started off in a picture where it seemed natural and one seemed to know what one was doing. If you imagine that the speed of light was infinite or you could imagine that you were looking at a globular cluster from a great distance where all the information, a more or less two dimensional globular cluster where all the information comes to one instance out here where you would have something very closely resembling these snapshots and you would have a dynamics which would evolve in that way and then from that picture where you fairly clearly formulated how you're going to attack these dynamical problems you then would have these conceptual ideas that actually the contents of the world thought, how are we going to modify the theory to make it still have these properties that we want and I think you could then argue in that way you would arrive at general relativity but I think you're quite right the step from observation to the distinction is a very long one and I don't suppose it's worked out at all and if you talk about globular trust we're taking account of statistical methods and not with these the greatest question touches closely on something I wanted to ask perhaps I can use my privileges to share and ask it now Indeed, if you want to take the point of view, which I can emphasize the operational aspect of this, if you want some information you really have, and you keep talking about snapshots, you really then should not be talking about space-like hyperservices, because the information you get from a snapshot is made to go on the characteristic surface of Maxwell's equations, or whatever equations you want to use to describe your electromagnetic fields. Unless you make the explicit assumption that the velocity of light is infinite, which you don't even need to do in a Newtonian theory, of light, then you should really be analyzing your data on characteristic services. We're talking about how characteristic services fit together. And if you want to go a step further, of course, and look at the fact that what you really observe is only the, so to speak, the sphere we see. So you look out at the night sky, you see a sphere, and you have problems with conformal invariance as well, because that invariant is a conformal transformation. So you really are in the, it seems to me you've been very selective. Oh, absolutely. And I could add one other relativity that you have omitted, which I think, I don't

1:22:30 I don't know what Mark would do it, but I think Mark Rebison would wear it, but that is a relativity of scale. Euclidean space is self-similar, so without even violating the possible Euclidean underworld through 90s space, I'm still the problem. How do you know what that Rij is, and should your theory be invariant on the scaling? So there are two aspects which seem to be very selective in what is really operational and what isn't. Can I say, yes, we were very aware of this, and actually Bethotti and I did explore quite a lot of possibilities to try and do something about that. You know, there's a whole lot of things, and I would say, I mean, my feeling is that this whole language of two snapshots and simultaneity is not really the appropriate way to look at it, because it still is too much geared to that way. And I think that, also, I think this may relate to the problem of the possible infinity of the universe, that one probably wants a different way of thinking about these things, which takes into account all those things, and probably quantum mechanics as well. start working on it, is all I would say. But I think this is interesting, at least it follows the logic that, I mean, the only claim I would really make for it, it does sort of follow the logic of Mach at his time, so to speak, and shows what could have been. I think if you look through the history of physics, you find that time and again, theory marched ahead of observations and wasn't at all aware of sophistications of these operational definitions and things, and very often discovered and then a long time later was really properly interpreted and maybe in quantum mechanics we're still waiting for the proper interpretation or something like that so that's sort of the only excuse I give. You were just touched on most of the things I wanted to say but it seems to me that the assumption of space is measured at relative distances is also sensitive which should be taken into account Automatically, we need to measure what you're saying, and also, as a second term, the notion of constraint dynamics comes out directly from the introduction of the arbitrary programming of time, and the follow-up essentially to rocks ideas at that point would find that the introduction of something that's prior to measuring clock, or clock measuring, would

1:25:00 be something that wouldn't be arbitrarily introduced. by being, you know, saying that energy should be at times. But that means that's the case, and that's sort of magic, isn't it? That particular notion is what atomic clocks tell us. Yes. First, you'll actually write about, again, about the assumption of Euclidean geometry. Though, of course, geometrodynamics start to come to grips with that, of course, because it assumes, it doesn't assume that. as regards to time yes that's again this immediately actually brings one into problems of bond and gravity and so forth could I just say that I think if these simple models that Bertotti and I did have sort of any value it's possibly as the very simplest dynamical models that have all these problems with constraints that you have in general relativity trying to quantize it this re-parameterization variance is really the real problem, and in fact I'm pleased to say that Jim Hartle is just about publishing a paper where he sort of is looking at it from the point of view of Berthotti and my models, because all these problems of time in quantum gravity in the canonical approach are all contained really already in these very simple theories, so that all these problems of quantization, I mean we have in the simplest possible form, the basic constraints that you're struggling with in general relativity I believe again, Professor Bergman was one of the first to get at these constraints and start coming to grips with these problems is that not correct? I would like to see some credit too so I mean yes it's just the simplest place it's the characteristic structure that I would like to get across I think that goes through this whole thing and we just happen to find I think the very simplest dynamical theory is where this characteristic structure first appears I just have two brief historical questions you mentioned the point of quantary science and hypothesis where he discusses an arbitrariness with respect to angular momentum I think you missed his emphasis on that I wonder if you could say a few words about where quantary knowledge is at and secondly, since we know Einstein I think you could make a few comments

1:27:30 on how it affected Einstein's sort of general with regard to the second question I get the impression not at all except possibly in 19th I don't think it's there where he says about defining time there is a little bit in there no I don't think that it had any influence all. I've got with me an extract from Science Hypothesis where I think I can pin that down. There's two places in Science and Hypothesis where Poincaré talks about this thing. One is where he talks about the generalized law of inertia, which is fairly early in the book. And later on he's talking about astronomers who can observe the positions of the planets, but there are no fixed stars beyond them. And he says, he puts it in terms of the area constant in the angular momentum in the solar system. But it's later on in Science and Hypothesis. Obviously, I've brought it out for my purposes. It was there where this aspect of the problem dawned on me. But I think I can show you actually the explicit passage. I don't know, but probably it's a copy in the library of Science and Hypothesis. If you were interested, we could change it up. But I found Poincare very illuminating, I must say. Perhaps I speak in French. May I ask for some translation? I will recall just a small thing. In the Islander 6th or something, a Frenchman's name is Ledoux, you know? Just probably probably from the University of Cal, I am not sure. have given, in the context of Mach, something in classical mechanics, which is strange, but perhaps it is still historical. He considers classical, in other words, dynamics, but he considers the universe with convergence on all, and the global kinetic energy of the eulogers. And we search for frames for which this energy is minimum.

1:30:00 And with respect to this frame, we have many good properties and in particular, the principle of equality of action and high action is assuring. with respect to these claims, and also with respect to the Galilean corresponding claims. So, it's time-shing, and it proclaims that it is completely in agreement with Spar. Do you not know this? His parish in the company of French Academy, approximately in 1906. Yes, 1906, if I may say. 1906 is Chalmers. I think it's amusing. Historically, that together with Einstein, you have this approach to justify the strange choice of the famous in classical dynamics. Föbel, of course, did something quite similar. Minimum of the energy of the world. But I think, did he not publish, go on publishing, I mean, I think, I have checked up on one Frenchman, certainly, who began with an L, who in the 1920s was saying that time should be defined in this way through the energy law, and I think that may be the same. Also, Zandstrup is a very interesting work. Ledoux. Ledoux. Ledoux, yeah. I'll check that. And I have a feeling that I may have come across it. Certainly on the time, I think I'm familiar with Berthotti. I think it's a safe source here. Comprendu? Comprendu. I'll check that. Thank you very much. But it's strange in this way. But may I just say, these ideas keep on coming back. I mean, Berthotti got in contact with me because of a charming Italian fellow called Padula, 16 was taught, at school, was taught Newtonian mechanics. And he said to himself, there's something wrong here. And he went off and he thought about it and he thought about it. And during his military service he came up with a theory which is almost identical to this one that I produced with the relative distances. Not the time part but the relative distances. And he came to Bertotti and he said, I've got this theory.

1:32:30 Look, this is Mach's principle. He'd learned about Mach in the meanwhile. And Bertotti said, well, what a darn pity. Barber just published a paper in Nature two weeks ago with the same ideas. So, I mean, they keep on recurring, these sort of ideas. I think they're very natural. Sorry. I don't think we'll have to go ask questions. Just a historical point, sort of in the sense of Einstein's reading of Marx, and whether he got Marx right on the score, and I think it depends on what interpretation he gives to Marx's principle. You've been giving a more or less physical interpretation of Marx's principle, but there's just as much and a systemological interpretation of Mock's principle. Basically, no dynamical effects could be charged at anything non-observable. And that's certainly in Mock, and it's certainly a version of the principle that Einstein himself emphasized it over and over again. So if you look at it from that point of view, I would argue that he has understood Mock's principle, that he's being true to Mock. I think, I would say he was true to the general thrust. and this business there's this very characteristic mark of his about things should not have an action if they're not subject to a reaction Thank you.