Prague and the Conception of General Relativity — Kepler, Mach and Einstein

0:00 All right. Well, many thanks to Yeji and the local organizing committee for putting on such a fantastic event. I will... Oh, I see the clock here going second by second. That's great. There's my title. Let me see if I can do this better than Yeji. Here's an overview of my talk. I'm going to start by arguing that the first great success of Mach's principle was not the motivation to create general relativity, which it gave to Einstein, but in fact in Kepler's discovery of the laws of planetary motion here in Prague. I'm going to say something very briefly about the Doppler effect, which was proposed here by Doppler. Then I'm going to briefly talk about Mach's critique of Newtonian concepts. And then because Mach's principle has generated a great deal of confusion and argument, I'm going to give what I think is really the correct definition of Mach's principle, which I take from Poincaré. I'm going to say about how Einstein attempted to implement Mach's principle but indirectly and explain that this is to quite an extent the reason for so much confusion about the topic and then if I've got time I hope to say something about a direct approach which along with collaborators I call shape dynamics. So let's go straight on to Kepler and I'm going to start by just reviewing certain things about what the planets do, and in fact what they actually do, which is not at all well known. So let me just remind you, first of all, Kepler's first law discovered, he got his first and second laws essentially, I think it was, he finally got them on, I think it's Easter Sunday, 1605, here in Prague. The first law, of course, very famously, is that a planet moves around an ellipse with the sun at one focus. So that's the first law. The second law is a bit more sophisticated, that the area swept out from the sun to the planet, say from the aphelion A, that that area is proportional to the time. So let's go on to the next slide. and I want to ask, does anybody know what that is? Does anybody go to volunteer what that is?

2:30 It is an ellipse. It is actually a representation seen from above of the orbit of Mercury which has by far the greatest ellipticity of any of the planets. The dot in the middle is the centre of the circle. I mean, it looks incredibly circular and with the naked eye it's completely impossible to distinguish. So really it's not really true to say the planets go in ellipses, they go in circles. Now the key to all this, and you will never understand how the laws of planetary motion were discovered or Kepler's achievement without understanding the key difference between the eccentricity, which will be fairly well known, and that's the ratio of this thing over that thing, that's the eccentricity, that is very key. But the ellipticity is the ratio of, it's this little bit up here over the complete bit there. And that goes as the square of the eccentricity divided by two. And as all of the planets have relatively small eccentricities, that ellipticity is very small. and see, now this is Mercury's nearly circular orbit, so wait a minute, the red is actually Mercury's orbit and the blue is the circle which best fits it. And if you take Mars, which is the planet for which Kepler discovered his laws of planetary motion, you cannot tell the two apart. It's a quarter of the size, the difference in that case there. So that gives you some idea of the phenomenal achievement of Kepler and Tycho Brahe's observations in discovering that the planets move in ellipses. But also it emphasizes that the circle was terribly important. If the planetary orbits had not been so nearly perfectly circular, there's no way that the laws of planetary motion would have been discovered in the way they were. So that's it there. Let me just emphasize here, this is the center of the circle. There's the sun and there's the empty focus. And let me just make one point. This eccentricity, the displacement of the Sun from the center is very easy to observe with naked eye astronomy there's no difficulty in observing that but the ellipticity is extremely difficult let's go on to the next one I think the reason why nearly everybody thinks that the the planetary orbits really do look elliptical is this

5:00 famous diagram of Newton where he just had to have a huge eccentricity it's nearly two-thirds the eccentric point six I think Newton had to describe it that way otherwise he couldn't get the details into his diagram and of course the the famous the Bank of England when they put out a banknote right to recognize the Newton's achievement they put the Sun at the center of the ellipse perpetuated in banknotes for eternity so that's that's that one there now I want to come on to something else which is absolutely important and is far less well known than the circularity of the planetary orbits. No, sorry, first of all, I just want to go through the orbital elements for the planet. So you see, Mercury has an eccentricity of about a fifth, and its ellipticity is a part in 47. But Mercury is very difficult to observe of being so close to the Sun. So it played essentially no role in astronomy until, I think it was 1632, Kepler had published the Rudolfine Tables in 1627 and it had correctly predicted the transit of Mercury in 1632 and he got it pretty well dead right. And it was, after his death, it was a posthumous triumph, but it was only then, after Kepler died, that astronomers really began to take him get to Venus. Now Venus has this tiny eccentricity and it's phenomenally circular. Circular to better than one part in 43,000. Incredible. And the Earth is almost as good. And to one part in 7,000 it's circular. And then you get to Mars where it's just possible. It was just possible for Kepler using Tiho Bryce observations to detect the thing there. So let's go on from there. Now I want to tell you about a terribly important thing, which was, in my opinion, the first theoretical model in science. This was Hipparchus's theory in circa 150 BC to explain the non-uniform motion of the sun relative to the stars around the ecliptic. That had been known that it moved non-uniformly. Now Hipparchus far as I know it is the first theory and it actually worked fabulously well. Now what he assumed in accordance with standard ideas is that the sun moves on a perfect circle at a perfectly

7:30 uniform speed but it appears to move non-uniformly because the earth is not at the center of that circle. So there is the circle, here is the earth that is not at the center, there is the center and he made observations that the sun from the vernal equinox as observed from the earth from the vernal equinox round to the summer solstice which we've just passed that's 90 degrees as seen from the earth but it took 94.5 days and then from the summer solstice round to the fall autumnal equinox it took 92.5 days And then it was just a matter of geometry for Hipparchus to work out where the elements, and he got this model here, which turns out by the fluke of the value of the Earth's eccentricity to be extraordinarily accurate, to three-quarters of an arc minute per year. But it's physically wrong, because he assumed perfect uniformity. He knew nothing about the non-uniformity of the motion. The eccentricity, the displacement of the Earth from the center that Hipparchus found, was twice what it should be. And that error went through and bedeviled astronomy until Kepler found out, realized what was going on. Now I need to tell you about something which is really hugely important. It's the role of the empty focus. If you hover just above the sun and watch the planet going around the ecliptic, it moves very non-uniformly. of all, it is moving physically non-uniformly, but you're not at the center of the orbit, you're displaced, you're just above the sun, and that doubles the effect. You see the sun, the planet moving very non-uniformly. If you go to the center of the orbit, that effect is, the geometrical effect is cancelled out, and you see the planet moving as it does move. If you go on once more to the empty focus, you see a miraculous effect. The planet moves with almost perfect uniformity. It's just as good as the approximation of the ellipse by a circle. And that effect was discovered in geocentric guise by Ptolemy in about 150 AD and dominated astronomy for 1500 years, that empty focus. That was the key

10:00 thing and it was called the equine. And many people have heard of Many few people realize that it's the empty focus of the planetary orbit. So let's go on. Now, Copernicus actually stumbled upon his idea by trying to get rid of the equant because it... Sorry, I should just go back one thing there. So what you have to imagine is that there's a spoke rotating about this empty focus with uniform angular velocity, and the planet will be at the point where it cuts the circle, and that's incredibly good approximation. Copernicus wanted to get rid of that equant because it meant that the planet on its orbit was moving non-uniformly. And he stumbled upon his system in trying to get rid of that equant. Let's go on to the next picture. Now, when Ptolemy died, he could predict what the sky would look like from the Mediterranean centuries ahead. His models were as good as that. When Copernicus died, he could predict what the sky would look like from anywhere in the he'd worked out exactly, through his hypothesis that the Earth moves on a circle, he had a baseline for trigonometry and knew where all the planets were essentially. And he could predict what the solar system would look like from any of the planets centuries in advance. And it's an interesting fact that that was only confirmed when space travel started, that he was right, because we never left the surface of the Earth before then. Now, the Copernican cosmos, nevertheless, was an incredible muddle and mess. And it all came down to, largely, Copernicus's dislike of the equant. And he didn't give one to the Earth, because there wasn't an equant in that motion, that Hippocrine theory of the sun, so he didn't put one in for the Earth. And so that was, it had an extraordinary effect. He just, Copernicus just inverted the Hipparchan theory, and the consequence of that was that the center of the Copernican system, the center of his universe, was not the sun, but believe it or not, the empty focus of the Earth's orbit. And this is how Copernicus thought that things should be. these are the centers of the orbits for the three outer planets

12:30 and these are their empty foci and according to Copernicus they should converge on the empty focus of the Earth's orbit and not on the Sun. Copernicus was not the creator of the heliocentric system. He was the creator of the theory of terrestrial mobility and Galileo said, but still she moves. And the clearest evidence that Copernicus virtually no significance to the Sun is that when you look at his diagrams of the motions of the planets they're very hard to understand because the Sun is not shown in them. Now Tiho Brahe came along and had this absolute obsession so far as I can see very early on certainly in Western Europe for accurate measurements and he claimed he could make measurements accurate to two minutes of arc, which is a 15th of the apparent diameter of the moon. Ptolemy and the ancient Greeks had worked to about 10 minutes of arc. So there was this great treasury of observations which Tihebrahe brought here to Prague around 1600. Now Johannes Kepler came here because he wanted to get support for his idea about why there were five planets, his own epitaph that he himself composed was on his gravestone in Regensburg where he died in 1630, but sadly that got destroyed I think at the time of the Thirty Years' War. So there's that very fine epitaph that he wrote for himself and it is so true. He was the first person who in his mind's eye could really wander through the solar system. Those are his, you probably know about his idea, which was completely wrong that the platonic solids explained why there were five planets and the sizes of the orbital elements. And that's why he got in touch with Tycho Brahe. But mercifully, Tycho Brahe put him to work on the motion of the planets. Now I want to come to why Kepler should be seen as anticipating Mach. Kepler was, by the way, this is my own diagram, but it's the sort of diagram you'll find in Kepler's famous Astronomia Nova, published here in 1609. Kepler was hugely impressed by one observation which Tycho Brahe had made of the comet in 1577.

15:00 Brahe had established that it had, from parallax, that it must be in the region of the planets and not up just high up in the sky, in the atmosphere. And Kepler went further, and he had insisted, and it made a huge impact on him, that the comet must have gone clean through all the celestial spheres that were meant to carry the planets around the sun. And he said this, and again and again Kepler emphasizes this, and he comes out with this marvelous sentence, Two, henceforth the planets must find their way through the void like birds in the air. We must philosophize about these things differently. And what Kepler was really struck by was that astronomy up to his time was completely geometrical and empty points, void points in space, played an absolutely vital role. point at the heart of the Copernican cosmos, the empty focus of the Earth's orbit, and there were these equants, the empty, the void foci of all the other planets. These were the key things, these were the marker stones, these were like milestones in space that the planets were supposed to find. And Kepler likened the planets to birds. How the hell could they find themselves, find themselves around in empty space by invisible milestones and markers? So he said, these things differently. So this is what this diagram emphasizes. That red dot is meant to be Mars, and Mars must, so to speak, be looking towards the sun and looking towards the stars to guide itself in its way. And this is very, very Machian. And now let's go on to the next one. The other thing, which is also very Machian, is that he introduced physical ideas. The planets move. They were not being carried by celestial spheres. Something must be moving them. So nine years before Galileo discovered it, he suggested that the sun was rotating and that there were sort of spokes emanating from it which would go round and carry the planets round in their circular form. The strength would weaken so that the planets further out would go slower. And then in addition, he assumed that there was a huge big magnet on the sun and that there

17:30 magnets on the planets and that would alternately pull the planet towards the Sun and away from the Sun, push it away from the Sun. Now of course this was very primitive but it introduced forces and above all it shifted the point of interest from these empty points to the Sun and that was absolutely crucial. He would never have found his laws without that conceptual change of mind. Now one of the really great achievements which Einstein hugely admired was the way he got rid of that flaw which was still sitting in physics going all the way back to Hipparchus's theory of the motion of the Sun because there was an equant in the motion of all the planets that was recognized because Ptolemy had effectively discovered it but there was no equant in the motion of the earth and for this reason, in fact, Copernus and Tycho Brahe were putting the Earth twice as far from the Sun as it should be. And Kepler hit upon an absolutely brilliant idea to work out where the Sun is. So what he did was, Tycho Brahe had made observations of Mars, and he found so many of them, he found three that were separated by exactly 681 days, the sidereal period of Mars. Mars reappeared at the same, was at the same point in space every 681 days and he got three observations and from the earth at these three points he knew the angle to the, between the sun to Mars and he knew that Mars would be at a fixed distance from the sun, three points, so by triangulation he could get three points in space relative to that baseline, three points determine a circle and he could establish that that circle was not where people had thought hitherto and this Einstein said this is one of the most beautiful things in the whole of science and I think it's right it really was phenomenal now I just want to come to how he came to the second law in the second law he noticed that at the absides at the aphelion and perihelion the orbital speed is exactly inversely proportional to the distance from the sun he shifted the point of attention from the empty focus to the Sun. He concentrated everything on trying to understand how the planet was moving

20:00 relative to the Sun and that led him to suggest that the speed of the planet in orbit as it goes around its orbit was inversely proportional to the distance from the Sun. Very physical again, but he found the mathematics of that very difficult. So what he did was he decided to try and approximate that law by the area law. And bit by bit as he worked on the data he found that the area law was the accurate one. So that's how he found the area law. Now this was hugely important for future development. The next thing that he'd done was now that he was able to show, and he is really the creator of heliocentricity, he was really able to show absolutely unambiguously that the absidal lines of all the planets converged on the Sun. There's heliocentricity for you. And let me just emphasize how important the second law was. Here's Isaac Newton, and if you look in Newton's Principia, the very first proposition is the dynamical explanation of Kepler's second law. And that was the moment when Newton understood that dynamics was created. So that's Briefly, I'm just going to say something about the Doppler effect. I have to say, admit I rely on Wikipedia for this information. There's the portrait of Doppler. But I did see the title of the German paper, which was published in 1842, which is reproduced there. Doppler came to his idea for totally wrong reasons. He wanted to explain why the stars in binary systems have different colors. and he thought it was due to the Doppler effect of the things orbiting going around each other and that this was causing the light to be shifted in colour. So this was actually quite wrong, although 40 years later it was actually confirmed that they were orbiting but not creating the different colours of the stars. His one argument for the Doppler effect in a way that we would recognize was that he said that if you had waves passing a ship at rest, there would be a certain frequency with which the wave crests hit the front of the ship. And if the ship was plying through the waves, that frequency would go up. So that was his argument there. So that was Doppler. His proposal was controversial for many years, decades.

22:30 people were arguing about it the first experimental test of it was in 1844 boys ballad in Holland on a train from Amsterdam to Utrecht had a brass band playing and you could hear the change of frequency as the train went by and of course with police wagons cars now and so forth you can hear the siren change but not in those days they did there was a more peaceful time so here's Ernst Mark. He was here in Prague for a lot of his time, hugely influential figure. I want to tell you just a great experimentalist, also a very deep intuitive thinker. I want to show you this rather nice diagram, which was one of the things which really settled the argument in favor of the Doppler principle. He had a rotating tube, which is here. This tube would rotate in the plane, which is sort that if it was rotating, A would be coming, the top would be coming towards you, then it would swing around, and this A would be going away. So it's rotating in the plane that's going up the passageway there. And there was a whistle up here at the top, and you could blow air through, and that would make the whistle blow. And then you could stand in two places, either at right angles to the plane in which it was rotating, or you could stand in the plane. And if you stood in there, you would hear the whistle change its frequency. But if you were out along here, it would stay constant. And so this was a very clear demonstration for sound waves. And in fact, apparently for two or three decades, it was standard demonstration of the Doppler effect here in Central Europe. So that's quite an interesting achievement of Mach. And of course, he discovered shock waves. This is his famous photograph of a bullet, a supersonic bullet. Let me just read you his critique of Newton's concepts. It is utterly beyond our power to measure the changes of things by time. Quite the contrary, time is an abstraction at which we arrive by means of the changes of things. There are really two Marx principles and this is the first one that he formulates. Time should be derived from change and in a way that doesn't go into the foundations of general relativity in the it should, but you can dig it out and find that it is in there. Then also we have knowledge only of relative spaces and motions. The universe is not given,

25:00 is not twice given with an earth at rest and an earth in motion, but only once with its relative motions alone determinable. And let me say about Marx principle, I would say it is that the universe in its totality determines the local inertial motion of bodies. And let me also say that Einstein, brilliant as he was, is the generator of huge confusion about Mach's principle. One of the main ones is that he claims Mach wanted to have a derivation of the inertial mass of bodies from the effect of the universe. This is totally and utterly wrong. Mach gave a wonderful operational definition of inertial mass which he regarded as an intrinsic property, and he said any of each body, and he said any attempt to go beyond it will generate confusion, and he's right. This was a complete mistake on Einstein's part, and I just have to think it was something to do with a semantic confusion between two totally different meanings of the word inertia. There is inertial motion, and there is inertial mass, and they are two distinct things. Let's go on. Now, Einstein did not attempt to implement directly but indirectly, and the main reason was to do with time and special relativity. And in 1898, Poincaré had identified two issues relating to time. The definition of duration, where he proposed the solution which the astronomers had put forward and was essentially what Mach had been requiring. But then he said, but there's another problem which people haven't really noticed, and that's the question of how do you define simultaneity at spatially separated points. So that's what Poincaré pointed out. So Einstein came along and he solved the simultaneity problem absolutely brilliantly as we know in 1905 and in a way with more aplomb and with more enthusiasm that Poincaré had done it simultaneously. So Einstein, perhaps somewhat unfairly, gets most of the credit for spatial relativity. But Einstein did not make a direct attack on the problem of duration. And I've looked through everything of Einstein's I can get hold of, and I can't find any evidence that Einstein ever really seriously addressed the problem of what is duration. What does it mean to say that a second a day is the same as a second tomorrow? Let me go on.

27:30 Let me just emphasize that there are two totally different meanings of relativity, which are all too easily confused. The Einsteinian relativity, which was then taken so much further and so brilliantly by Minkowski, where relativity means that the division of space-time into space and time is relative to an observer. And there's the Machian meaning, where the position of each individual object in the universe is thought to be defined by its distance from any other object in the universe. So this is a completely different meaning of the word relativity. And I would conjecture that actually the Machian one could turn out to be the more fundamental one. So let me just mention that in 1918, Einstein explained why he had not made a direct attempt on implementing Mach's ideas. He said, we want to distinguish more clearly between quantities that belong to a physical system as such. I think he's clearly thinking of the separation, the four-dimensional separation between two neighbouring points. That's the thing that belongs to space-time, the system, and quantities that depend on the coordinate system. One's initial reaction would be to require that physics should introduce in its laws only quantities of the first kind. However, the scientific development has not confirmed this conjecture. It cannot dispense with coordinate systems and must therefore make use in the coordinates of quantities that cannot be regarded as the results of definable measurements. And let me just say how Einstein did attempt to implement Mach's ideas. And I would say Einstein's work was a combination of two things. It was a very powerful motivation that he got from Mach to get rid of Newton's absolute space and to create a theory which was generally relativistic, hence the name of his theory. But he, and the way he thought he would do that came already in 1905 with the special theory of relativity where he realized that uniform motion through space could not be detected. first sign that maybe absolute space is not detectable. So Einstein's argument was that if

30:00 you can't detect absolute space, if you can't detect motion through absolute space, then it doesn't exist. So this was his indirect way of attacking Marx's ideas. So that was the first step. Then two years later, with the equivalence principle, he realized that uniform linear acceleration would not be detectable. Then in 1909 he started thinking about uniform rotation. There's a letter to Sommerfeld where he said it's very important to try and show that uniform rotation is not detectable but then you have to take into account the whole universe already. He's beginning to think about that and then he progressively generalizes that to the idea that any accelerated, any coordinate system should be as good as any other and that's and then he got this idea of the requirement for general covariance and then his recollection of what Gauss did with generalized coordinates occurs to him and he's away. He's off to see Grossmann in Zurich. So that was there. Let me just say, and this is really complementary to what Yejia has been saying, so his heuristics were general covariance and the equivalence principle. Now what went in? I mean, it's some years since I really went through the Einstein papers, I'm just absolutely amazed by them and Einstein's correct claim that in a way general relativity was discovered 50 years before it should have been. And I'd just like to throw out, suppose quantum mechanics had been discovered when historically it ought to have come before general relativity. We might have totally different conceptions about how things ought to be. But so what were the inputs? So he combined this general push to find a new theory motivation was to combine certain things and and reading those papers of his you see how brilliantly he did it and he had certain things which he were very very secure and he used them absolutely sure-footedly like special relativity and Minkowski space the implications of the Michelson-Morley experiment the search for a gravitational analog of Maxwellian electrodynamics that was hugely important. Energy momentum conservation, he was very aware of that. He could easily have, and the realization that there would be non-linearity, he could easily have discovered, if he'd known about unique spin-two

32:30 representations, he could have discovered the flat space approach to general relativity that was developed from the 50s and 60s. And then the mathematics of Pseudo-Riemannian geometry. So those were all the things that led him to the of brilliance, no question on it. Let me just say I've got about two minutes to talk about an alternative way to this. Those of you who are interested, this is a culmination of work I've been trying to do for 50 years nearly now, of a direct implementation of Marx ideas, and in the last 13 years it's been developed very strongly by Nilo Muraku and myself and various collaborators, and one of them, Tim Koslowski, will be giving a half an hour or a parallel session on Friday afternoon. So those of you who are interested in that, please feel free to go along to that. And otherwise, just sort of look up shape dynamics. Let me just mention the key idea in shape dynamics. It goes right back to Riemann's famous paper in 1854. Riemann said in that paper, measurement requires that I actually bring my measuring rod up against the measured interval. that they are definitely the same length. You can see that my hands have the same length. But he says, I'm going to base as a hypothesis for the foundation of geometry that if my hands are far apart, they still have the same length. And it's meaningful to say they have the same length. And that assumption is just as suspect, if you stop and think about it, as simultaneity at spatially separated points. And the idea which is really underlying shape dynamics developed in the last 13 years is to say that scale is gauge in the three that we're talking about a dynamical theory of three-dimensional geometry, Riemannian three metrics, and saying that only the angles are physical. The conformal geometry determines angles between curves that meet at a point, and that is physical, and that the determinant of the three metric, the scale and that's what's underlying there and I think at that point I won't attempt to go on to any more because I see I'm down to 5 minutes and 12 seconds so perhaps we go on to questions from here

35:00 Thank you for that remarkable trevice of time from Ketler to the Bay If you all don't mind, I have one remarks, a remark concerning so-called Mach Principle. In my opinion, this is incorrect and has no experimental confirmation. I remember that in a very distant past, English physicist and mathematician has tried to construct local physics starting from cosmological speculative assumption he wasn't successful in obtaining local physics from cosmological speculative assumptions. I think your main statement was that it doesn't have any content because there's no experimental verification Let me just say that if you take, there's a huge difference from the Machian point of view between a spatially infinite universe and a spatially closed one. Einstein was very aware of this and that was behind his 1917 cosmological model. If you have a spatially closed universe and you develop a Machian theory of dynamical geometry, a Machin theory of dynamical geometry and matter interacting with it it leads unambiguously to Einstein's theory in its entirety with its field equations and in my opinion every single verification of Einstein's theory if, and it's a big if if the universe is spatially closed is a perfect vindication of Mach's principle

37:30 the field equations of Einstein's theory follow from Mach. And in fact, it's the capital G0 and G0I field equations of Einstein's theory, which express Machian relativity in a very perfect way. And as Carol Kukash, who was mentioned earlier, points out in his lectures very beautifully, if you have a space-time, for example, in which the, in the Hamiltonian formalism those first four Einstein equations become Hamiltonian constraints if those Hamiltonian constraints hold on any space-like surface then the space-time satisfies Einstein's theory. That's just one way of showing how deeply Machian ideas are embedded in it I will insist it's the very heart of general relativity and you can see it in the case of a spatially closed universe you can see it absolutely explicitly How essential was Marx's principle for Reince and Dickey for deriving their theory? Because they argue Marx's principle played a role in their derivation. I did talk to Dickey about that and he said that, he said he was a bit embarrassed by claiming it was Marxian. It was at most a bit of motivation, he said. I met him once. To go on from the question that was asked, could I ask the question, have we decent evidence that the universe is indeed closed? And can you formulate Marx's principle in a way in which it still works if the universe is open? Well, I mean, I think the majority opinion among cosmologists is that the universe is spatially infinite. I think this is perhaps, I don't know, probably people here know much better than I do. I suspect this is something to do with belief in eternal inflation. As far as I know, the observational evidence is absolutely tiny, marginally in favor of spatially closed, but it is so statistically insignificant I see Yoshi smiling at that. Now, the situation is just like it is actually, I would say, in Newtonian mechanics. If you have an island universe in Newtonian mechanics,

40:00 you can actually determine uniquely an inertial frame, the center of mass inertial frame of that island universe. If you're in an infinite Newtonian universe, you can only determine local inertial frames of reference because they may be moving in accelerated motion relative to each other. This was already understood before Einstein came along by people like Seliger around 1900, that there was no global definition of an inertial frame of reference. I think it was understood by Newton. To some extent. Well, he used this concept in the satellites of Jupiter. Oh, yes, quite true. This is his corollary six. That's quite right. But Newton posed the problem of how do you determine an inertial frame of reference. It's what I call the scolium problem. And Newton, interestingly, says that he wrote the entire Principia to demonstrate how you get the absolute motions from the relative motions. But he never returns to the problem. And it's a hugely neglected problem. It's very interesting. Very, very few people have been concerned. Ludwig Lange, here in Germany, is the person who introduced the concept of an inertial frame of reference. How do people... You read any textbook on dynamics, and they say an inertial frame of reference is one in which Newton's laws held. tell you the important thing which is how do you actually determine it that's the key thing and it's when you ask how you determine it that you come to how you should formulate math's principle so it's a great uh thing that is missing in in dynamics textbook they don't tell you what time is or what a clock is for that matter either the most fundamental concepts are not addressed To my knowledge, one of the reasons why Einstein favoured the closed universe was that one could not bring a body to an infinite power away from other masses. Because then, because of Marx's principle, it would have followed that this particle would have no inertia. So let me repeat again. And one of the reasons, I think, to my knowledge, was that Einstein favored the closed universe was that no particle could be transported to infinity far away from any masses because then, because of Marx's principle, this would have implied that this particle would have no inertia. Is that correct? Absolutely, yes. And this is actually, I would say, the most notorious example. I mean, it's brilliant, the 1917 paper, but it is actually the most notorious example of Einstein. failing to understand what the problem was. It was to determine the inertial frame of reference, not to determine the inertial mass.

42:30 And there's a beautiful paper, 1923, by Hermann Weyl, a dialogue between Peter and Paul, in which he points out that it is completely hopeless to try and determine an inertial mass because you have to have something. There's got to be some sort of charge of the things. I mean, it's an infinite regress if you try and do that. Weyl clearly shows that that was a wrong idea. And unfortunately, Weill then concluded from that that Mach's principle was completely wrong because he trusted Einstein to do it. Donald, may I tell a little story at your expense? Please. Yeah, I met Donald in 1974 and he'd just written a paper on Mach's principle and he was telling me what Mach had said. And I said, if you don't mind me saying so, what you've just said is your interpretation of Dennis Sharma's interpretation of Einstein's interpretation of Mach. And Donald was very, very, a real gentleman. He immediately responded, you're quite right. I've never read a word of Mark, he said. However, he certainly has in the meanwhile. So, well, I think we are in agreement on that. Thank you for giving me permission, Donald. I'm getting a bit late. One more question. only one word of about reference to the so-called internal inflation, because there are actually many confusing statements regarding it. And so actually the real thing is when speaking about spatial geometry, one should keep in mind that about which hyperservice are you speaking? and in particular where a type of surface of X, a poor approximate, how about Genki in our universe is not a cache surface of the whole space time. So in particular, the fact that it is, in some sense, it may be non-compact, does not contradict the possibility that the cashier surface is still copper.

45:00 So the only, I would say, right statement is that in the so-called internal inflation the structure of space-time at spatial and time infinity is very complicated. So it has a completely irregular structure. So it's even, I suppose, that it has no sense to speak about it in some regular terms. Yeah, I can see that there are all sorts of issues and the question of hypersurfaces is very important. Let me just say, in a way, it's really quite simple-minded, our approach. The simplest way to do it is a spatially closed 3 to S3 spatial manifold. And what is interesting is actually that we are led through Mach's principle straight to constant mean curvature foliations, and we actually have a dynamical derivation of York's method of solving the initial value problem for general relativity, including the key role of scaling of the various components that come in the Lishnarevich-York equation. The gauge group is not quite full three-dimensional conformal transformations, but volume-preserving conformal transformations. Otherwise, you can't have an expanding universe. So, interestingly, we are forced to that, and we actually get a new result in the mathematics of general relativity, where we derive York's laws of scaling that he put into the Lishnarevich-York equation without any proper justification. by trial and error, but we actually get it out of a Machian derivation of the theory. But the point is taken about how one must be careful about these things. Thank you. Thank you, and that's the end of this session. Thank you.