Gravity Without Newton's 3rd Law

0:00 As I said, I'm in charge, so I can start without him, and of course, I can do it again, by thanking and George for having me here, and this is the first time I've sort of gone public with these ideas, so it's the first time I've been forced to really organise them, as well as him, as I was saying earlier, it crystallises your thoughts, and I just can't think of a better environment in which to, in which the trial went out, so many, many thanks to George And many thanks to Bachelors for acting as a go-between. Okay, well, the personal background bits, as you've requested, I spent most of my professional career as an experimental power physicist, working mainly at CERN, so I do about seven years of post-doc position I started moving from college to college in London, and then in 1979, I got my first tenure at Virgo, and that was when I met Basil, and David, and eventually Nick, and the team, which was, as you can imagine, quite a thrill to land up in such a place. and it was very soon after I arrived there that I first had the idea that I'll be talking about later this evening but I just never had the time to think about it very soon little bits here and there along the way and of course at the same time in the tea room over here in David and Bowman and the rest talking about was inevitably drawn into all that, and so through that, through all that time, things were a bit to think about, as well as slave away as an experimental particle physicist.

2:30 Around, I'm not sure exactly what the date is, I think it was around Naoki Mountain 4, and I realized that I wasn't enjoying being an experimental part of the politics anymore. It was becoming a drudge. And because of the various political turmoils that were going on in Birkbeck at that time, it seemed opportune to just declare that I was going to stop. And that's not something you're usually allowed to do in British universities. So I did, with the naive hope that I might eventually be able to pitch on to Baselian. Why wouldn't you tell us? I did. Why didn't I listen? Yeah, well of course it was naive, I mean I didn't have a lot of training, my ideas were extremely vague anyway um and funny enough stopping research didn't actually seem to provide that much extra free time anyway because the department was was was contracting and so anyway uh what eventually happened then was in as you probably all know in 97 they decided that it would be a good idea to close down the department uh thereby saving themselves 80 000 There was stationary storeroom, accommodation for the janitors, I mean, a great deal. And our teaching programme at that time was transferred to UCL, where I am now, which is just a few hundred yards on the road. And as part of that transfer I was offered a half-time post at UCL, not to do research, because I'd given up the part in physics which I was doing in collaboration with them. was, in a way, a little bit of a slight, I think, that people there felt that, you know. You can be reassured, Malcolm, that I called the head of the physics department a C. What, his face? Yes. I'm sure that helped a lot. Okay. Okay, so when I went to him and asked him to have the three-year appointment, perhaps extending for a year, his negative

5:00 Anyhow, in the end, this was a blessing in the sky, because it did mean I now suddenly had time. Well, please don't move on to Starry. I'm trying to get you a point. I know you're about to. And it worked. It did. It was at the end of the failed negotiations that I've addressed it, wasn't it? But for the first couple of years, I thought, I've got this free time. I'm going to try and do something completely different when it ends and change my life a bit if I can. for my own entertainment during the first two but as Tim said once you've dabbled in fundamentals of physics it doesn't let you go and I was every day expecting to read a new scientist or something like somewhere that somebody else had had this idea and followed himself and either it was brilliant or it was rubbish or something and it just never happened so in the end I thought well I've just got to bottle down and see where this idea will lead me which led me, yes that's a very good outcome so I'm concerned I'll do for now yeah, okay, so from the title you can see this is about classical gravity so it doesn't impact on quantum mechanics which is what this meeting is about directly but for me at least doing this work has given me a different perspective on physics as a whole specifically on the possible relationships between classical physics, geometry, and quantum mechanics. So I attempt to communicate some of that along the way of just giving you the technicalities of what I've been doing. Right, so... Is that just a misprint of my real co-presenter? Yeah, I beg your pardon? Oh, there's a mis-hose. I misspelt my name. Brilliant. That just shows how confident I am. No, it's an adjoin functor. Your punter will come and open. I did actually have a run-through last week. It's about a Thursday evening. He was represented in what I consider to be the logical order, but he suggested, and I agreed with him, that the tests of the idea, the observational experimental tests of the idea, consequences for ordinary matter.

7:30 Work-a-day sins probably would take a lot of time to get through it, and it's not really what it's speaking about, but ideas, regardless of their practicality. So I'll put that at the end now. So I'll talk about the basic idea, explain that, and focus then in the in-between bit in trying to just give you an idea of the kinds of consequence of this idea. it seems very innocent to start with but it has actually quite dramatic consequences and mainly the more dramatic consequences arise in cosmology now I'm not a cosmologist obviously so I've concocted these pictures in a rather simplistic way but they give you a flake anyway of the sort of implications now in fact for a long time I very naively assumed, because I thought it was a trivial, almost a trivial alteration of general relativity, that I didn't really need to bother with the mathematics, because, you know, Einstein had done that, there's just a different way of using that mathematics. It's just laziness, really. I should have looked into those more carefully earlier, but bit by bit, reading books, reading articles, I realised that there probably is a basic inconsistency here between this idea and general relativity as it's taught. So I needed to look into the more mathematical aspects. So maybe two years ago I started to try to teach myself general relativity. And I really, only quite recently, I would say six months ago, reached the point where I thought, okay, I think I now know enough to start to do something specifically in relation to this. So the mathematical side of it is very, very... It's just a status report, okay? I'm probably making mistakes along the way that all I can do is tell you where I've got to, what I think I need to do next, and I may, well, for new people. Right, and as I said, the amazing idea came to me a long time ago where in 1709 I got my first job at Birkbeck and there for the lectureship tenure post and the first course I was asked to teach was to teach what we called

10:00 a preparatory course in those days, which was nowadays it's called an access course which means a course for people, mature students who don't have the qualifications yet to start the degree. It's a sort of revision course prior to starting the degree course itself, and I was teaching mechanics. And I enjoyed it very much, it was relatively easy, it was a good way to get started, except for the bit on gravity, because every time it came to that bit, I felt embarrassed. I was always dreading that one of the students would ask me, Okay, you've presented the law of gravity in terms of Newton's laws of force, but surely, from a modern perspective, gravity is actually about geometry. So, what's happening there? And actually, one year, one student asked exactly that question. And all I could do at the time was shrug my shoulders and say, I know, I don't have the answer to that, really. And move on. But so it bugged me. And I thought, well, okay, what's the answer here? Do we need Newton's laws in order to obtain a law of gravity that eventually doesn't contain the concept of force? so let's try and track through the logic of this just for my own satisfaction so that's how it all started right so although this is all familiar to everybody it helps me if I just go through that the logic behind it now many of you probably know that Herman Bondi in the 50s pointed out to everybody that if we were really careful but they're in principle three kinds of mass. Now that's one too many for me, but just for the record, the three types are inertial mass, that is the kind of mass that comes in force laws, in special relativity, and so on. And then there are two types of gravitational mass in principle, there's what's called the active gravitational mass, which is the charge, if you like, in matter that gives it the property of producing gravity. And then there's the other kind of passive gravitational mass, which is the charge in matter that gives it the ability to respond to a gravitational field. Now, we can very easily get rid of one of those by saying we believe in the principle of equivalence. We believe that two bodies in the same gravitational field fall with the same acceleration,

12:30 which immediately means that what's called passive gravitational mass and active gravitational mass and inertial mass are the same thing. Actually, we just mentioned two types of mass, the active gravitational mass, and that's what most of the time I'll be talking about, and occasionally I'll mention inertial mass, but when I do, I use that word, inertial mass. So this is how you teach it if you're a graduate, the laws of nature, the laws of force, Newton, obtained by terrestrial observation, I mean, everybody, the law of inertia. what I I know this is contentious but I think most people do regard the second law as just the definition of force in terms of mass and acceleration and then the real meat in all of this is the third part where he says given that definition of force then when two bodies interact the force on one is equal and opposite to the force on the other and then using the observations of Kepler of course the inverse square law is deduced, whereby the acceleration of an orbiting body is inverse for four to the square with a distance from the central attraction. And then we put in the principle equivalent, Gallagher's Principle, in combination with the second law, and we interpret this acceleration as the force acting on the orbiting body divided by its inertial stroke passive gravitational mass. So now I have this proportionality. So that's for a set of bodies orbiting one center of force Then you generalize that to any pair of bodies And then you say, okay, well, then we can say As well as the object acted on having a property Which affects this force The source of gravity must also have a property I'm calling that capital M That's its gravitational active charge, if you like And now you compare this relationship and you invert the roles of the two bodies and evoke the third law, then that must mean that this M is of the same nature as this M, in other words, the identity or the equivalence of gravitational mass and gravitational mass. So that's the story as you would teach it to your students. And as I said earlier, from a modern perspective, the way we view that is we say that,

15:00 general relativity says that this is a law not a force of acceleration, are asked where to interpret this acceleration of any body in this gravitational field. Oh, that's interesting. This came out, that one didn't. It's the same. Another little swivel there. Well, it was better than mine. It came out. Or, if you like, in the jargon of general relativity, we say that for a stationary trajectory, that's a body that's held fixed relative to the source, held fixed in the field. It's curvature. Courage of space is given by that expression. So geometry or kinematics now rather than force. But we came to that using the force laws, and as I said earlier, there's no mention of force in that. So how necessary are Newton's force laws to that, to our understanding of gravity? That's the start of my quest, trying to answer that question. On the way, I just remind myself or remind us what we still do mean by force, we still have this concept, and of course it's nowadays expressed in quantum field theories through the Lagrangian or the Hamiltonian formulas. And there the third law, Newton's third law, is built in from the start. We even have a picture of it thanks to Feynman's diagrams. The two interacting particles exchange a quantum, and that quantum carries a certain amount of momentum and energy from one to the automatically guarantees that the law is verified. Now what I'm basically saying then is let's not assume that this necessarily is inertial matter. And the natural question that anyone would ask is, well if it isn't, we would notice. And that's what the, whether we would notice or not is a certainty question which I will not address. But just to point out that if purely gravitational system with no other forces at all, you would not notice. So bodies would still follow the same geodesic orbits regardless of whether the source of gravity is inertial mass or something else. All that it means is that if you compute, say, the mass of the sun by observing the orbits of planets, the mass you compute is my, what

17:30 I might call gravitational charge, which isn't necessarily inertial mass. And since for bodies that are large enough to generate orbits, we don't have an independent measure of the inertial mass. How would you know what the inertial mass of the Sun is? There's no conflict. There's no apparent conflict there. Now, you can contrive very artificial systems where the consequences will be more dramatic, and perhaps the most obvious one, and the one that is often presented when very rarely this idea appears in in the literature is debunked. You say it's a ridiculous idea because you have two bodies, two solid bodies in contact, they have different materials, therefore they have different gravitational mass compared to the inertial mass. When you work out the net acceleration of that body it doesn't come to zero. It has a self- acceleration. If you just let it go, you're going to accelerate away to infinity. And that's clearly unphysical, so it can't be right. It doesn't conserve energy and momentum, that's obvious, but it can't possibly be right. But that is an extremely contrived situation. First of all, you have to assume that this body has no rotation. If it has rotation, it just orbits through. It doesn't actually go off in a straight line. And secondly, we're small correction to something that's already very small. So again, in the slides at the end, I do a little calculation of a sort of example of, for instance, a NASA spacecraft putting in some crew numbers just to see if this idea applied for a NASA spacecraft, what sort of self-acceleration would it have, and that the number comes out of the order of 10 to minus 11 meters per second square, in more understandable numbers, means about a third of a millimetre per second a year. So it's a very, very tiny effect. Okay, so let's continue with the development of the idea. Okay, what am I going to assume that once you start to try to take general relativity to pieces, you want to keep as much as you can because it's a beautiful theory and very complicated. experimental tests of general relativity are all in vacuum. So the precession of the perigene, the mercury, the bending of light, the Einstein, all of that is a verification

20:00 of Einstein's equations in vacuum. The equivalence principle, but more than that, the idea that the geometry of space is given by a four-dimensional metric and so on. So that's what I'm going to keep. I'm going to keep the vacuum bit of general relativity. All I'm questioning here is the way that Geometry couples to matter in the region of the matter itself Which we haven't tested So that's the idea Now, for very understandable reasons Einstein forced this theory To coincide precisely with Newton's law In the appropriate limit The appropriate limit being That your test bodies, your test particles relative to speed of light and that the strength of the gravitational field is not too great. And of course the crucial bit of Newton's law is the inverse square law which again I'm not questioning. So Plathot's equation essentially in that limit any theory has to accord with. I've basically covered those two points. I don't think that the forcing in of precise consistency with Newton has justification either, as far as I can see, I don't have a general principle of Riemann geometry that contains that concept. it isn't necessarily justified from a logical theoretical perspective or from an experimental one but if we make the coupling anything other than to inertial mass because of the equivalence of inertial mass to energy means the coupling is not to energy so from the modern perspective general relativity says geometry is coupled to energy so I'm taking energy that's the crux of it, I'm deciding to take energy out of the picture altogether. So if you like, an electromagnetic field does not curve space-type, would it? In a nutshell. I've covered that as well. The situations, the contrived sort of situations in which you would expect to see the consequences of that, and the fact that those are generally very small.

22:30 So we're taking out energy from the picture. We know that ordinary neutral atomic matter produces gravity, so we have to ask ourselves, what else is it that's in matter that might do that that we can't put under the rubric of energy. And the thing that occurs to me is, well, the fermions themselves, the electrons, the photons and neutrons, or from a more modern perspective, the quarks. So you have to be a little bit more specific in order to make any progress, and this is my working hypothesis, if you like, that it's intrinsically the fermions that are the sources of gravitational fuel. It's a scalar coupling. That's probably just... I don't know whether that's got physical content or not, that idea. Because ferions are, to all intents and purposes, located at a point. It's a property of that point. So it has to be a scalar type of property. What I mean by, I've used this blooming Planck scale, haven't I, which I don't like. Okay, so what I mean by sort of the point like downward Planck scale is, for any purpose that concerns me, I can regard them as being point-like. Not necessarily singularity in a mathematical sense, but small enough to be considered as points. So the fundamental fermions are obviously on a mass scale much, much smaller than the Planck mass, therefore the Schwarzschild radius for these things is less than the Planck length, so it's way out of sight. I'm saying I don't need to worry about that. Now there are many consequences, as I said, the most dramatic ones come in cosmology, and the reason for that is that once you take energy out of the picture, suppose that gravity is positive definite. The positive definence of gravity,

25:00 that all matter attracts and never repels, is due to the fact that energy is the source, and energy is always positive. Once you take that out of the picture, that is no longer a requirement, and if you associate what's left with fermions, then to my mind, it's very natural to suppose that anti-fermions could produce negative gravity. And in fact, just recently I found in a paper by Einstein in Feldman Hoffman exactly that statement. Once you take energy out of the picture, there is no natural requirement for gravity to be attractive. It's purely an added assumption in that scenario. Incidentally, these papers by Einstein and Philip Hoffman are the only ones that I managed to find in the literature that come close to really addressing this type of system, because, of course, everybody, virtually everybody, is taking general activity as their starting point. That particular paper, which I'll refer to again, or those three papers that I'll refer to later, were concerned with a purely particulate system. of singularities is what Einstein refers to it. What he was trying to prove in those papers was that the assumption that test particles move on geodesics is not a necessary assumption for general relativity and that it could come out of the vacuum field equation. So I'll come back to that later. But in order to prove that in a general way he had to consider a system of singularities as the source of gravitational fields. Okay. So the next bit is just phenomenologically. As I say, just to give you an idea of the sort of consequences that this idea has, the sort of changes it makes to our perspective of that. Now, I've put this in square brackets because I'm sure most people here could work this out for themselves, but in case you're feeling a bit dozy, just make it quite clear that this is not like electrostatics at all. In electrostatics, like charges repel and unlike charges attract. What I'm saying is I'm keeping the principle of equivalence so matter attracts everything,

27:30 other matter, radiation, antimatter. Antimatter repels everything. Other antimatter, other matter, radiation. And radiation is gravitationally neutral, does nothing. That's basically the picture I'm considering here. So to just play around with this idea within the realm of cosmology, following the traditional picture of the early universe as full of basically hydrogen, electrons, protons, and now a precisely equal amount of antihydrogen. That's the natural assumption, of course. In order to arrive at the universe we see now, it's usually assumed that, for some reason, there was a slight excess of matter over antimatter. Now, I'm not going to do that. I'm assuming they balance precisely on the large-scale average. So, just as in the conventional picture, you can imagine that hydrogen and hydrogen scatter and annihilate. So, gradually, the amount of baryonic and antibaryonic matter goes down, the amount of radiation builds up to annihilation, and so you're approaching a limit where, in those regions where there is a slight excess of matter, you end up with just matter and radiation, and in those regions where there's a slight excess of antimatter, you end up with just antimatter and radiation. Now what would happen, the matter regions are going to contract again, in a way very similar. Right, great to meet you, hope to see you again too. Good luck, yes. Sorry to interrupt you. I hate you, Edwin. Good to hear you too. So we get atoms, molecules, dust, stars, galaxies, black holes in the usual way. But in the antimatter region, there's expansion. that atom has been moving away from itself so there's no condensation at all we probably even don't get atoms we don't get any dust or stars therefore no heavy elements no supernovae, no cosmic rays or anything we clearly inhabit a region dominated by matter and so the first result, if you like, of this idea is that it can potentially explain why

30:00 there's apparently no antimatter that it exists, that it's very far away. There are no energetic processes in that region which would reveal its presence to us. They just look to us like voids. Now to try to construct a picture of the larger universe, I use the word fractal rather casually, I did. But to give you an idea of what I mean by this, I'm saying that we're in the matter that there should be, on average, an equal amount of matter and antimatter, we must be embedded in a larger scale antimatter region and that on an even larger scale in matter and so on, ad infinitum. So, around us, we are in a big bubble of antimatter, which of course will be there for overall expanding, where the sitting is condensed. So, at that crude level, there is potentially an explanation for the expansion Not only the fact that it's expanding, or our region of the universe, but it's expanding and accelerating. This is a very recent result that you've probably all heard about, that once you look at a very distant supernova, you see that the speed of recession has been increasing with time, rather than slowing down, which has been super long time. This is the origin of the concept of dark energy. This dark energy is supposed to be an all-pervasive repulsive component, which is not like ordinary matter which is invisible and which is causing this accelerating expansion. So in this picture you don't need anything exotic to explain that, it is simply the diffuse antimatter around there. Now this is one of the big questions I need to try to address. I can't do it on my own, I just have to keep searching the literature until I find somebody who's tackled a problem similar to this. What sort of, is this geometry. Can you have a geometry in four dimensions where you have patches alternately contracting and expanding? The very picture of that. You might say it's like a foam where the gas regions are the anti-matter pushing out, the liquid regions are the matter, surface tension is drawing them together. They come into

32:30 Equilibrium in a foam, would they do that in Riemannian geometry? I don't know. Or possibly a two-dimensional picture of this would be like a quilt where you have the sewn parts which keep it pinched together and in between you have the stuffing that's popping it up. That would be the anti-matter part. So that is somehow looking into it. So I'm thinking of a periodic metric of something with alternate positive and negative parts to it. because nobody out there is studying such metrics because they're regarded as unphysical so I don't have much hope as I'll dig anything up that'll help right, next result then from this picture is when you think about quantum vacuum fluctuations according to one field theory the vacuum is full of virtual particle and antiparticle pairs which carry energy and so people who know how to do this can compute the energy density due to these quantum processes and it turns out to be far too great for the universe even to exist Evidently, in this picture each particle-antipartic pair has a net zero gravitation from charge hence the smallest of the cosmological constants within cosmology this effect is encoded in this cosmological constant is it has a very natural explanation. Okay, so from there then, my next natural question is if you now try to picture an even larger scale, what's happening in the limit? Does this fractal behavior go on forever or do we approach a limit where everything's perfectly smooth? And at that limit, is it in perfect balance, stationary? the net of contraction, what do the field equations say about that? From a Newtonian picture, obviously, it would be dynamically neutral, and all that would happen would be that over time, the matter would fall together, and then that would draw in the antimatter, and then it would all annihilate, and you just end up with nothing. But possibly From a geometrical model That's not necessarily the outcome

35:00 Technically of course This idea I'll come back to this again later Violates Just about any symmetry principle you can think of In particular There's no Matter-antimatter symmetry As there is in all In all conventional field fields Just one or two more of these cosmological things and then I'll get on to my attempts at formulating a field theory that fits these ideas. So possibly some of these non-linearity in the geometry might produce a net acceleration or net contraction. But in fact, thinking about it, there's actually a much more straightforward of why there might be an overall expansion. There's radiation everywhere, which tends to equalize the temperature in different parts of the universe. Now, as matter contracts, you're going to get atoms formed. And so, the mass of the elementary particle, therefore, is increased. You don't have free electrons anymore, they're attached to protons. so if they have the same temperature the matter region has the same temperature as the anti-matter regions but the basic unit of dust so to speak is heavier than at a lower speed and in almost any version of this sort of theory there will be a speed dependence of the effective gravitational density so the matter region would have relatively less gravitational density than the anti-matter regions and therefore there would be dominance of the antimatter regions giving rise to an overall expansion to the universe rather than contraction. So, at least once you get to the era where atoms form, there's a natural explanation of why the universe is expanding as well as why it starts to expand even at that epoch. I'm not able to go back earlier than that epoch. before that I can't really, I can't consider them out. Just one little extra point of interest

37:30 that I think within this scenario black holes do not evaporate as Hawking and others have suggested. For the simple reason that near the horizon where you get these virtual particle antiparticle pairs again, the conventional notion is that one of them falls into the black hole carrying negative energy, which therefore tends to cancel out some of its mass. Now, a negative antiparticle, negative energy antiparticle, in my scenario, effectively sends in positive energy, so there's no net change to the internal mass of the black hole. So black holes appear then to be permanent in this scenario. And finally then, in this section, can we explain the so-called dark matter? The stuff that's supposed to hold galaxies together and the clusters of galaxies and so on. Now, in principle, you could think you're free to ascribe almost any gravitational charge you like to neutrinos. You don't really have a constraint on that. So maybe neutrinos on their own would be sufficient to do this. Or, even fully ionised hydrogen, so there could be more hydrogen in the universe than is usually supposed. If it was fully ionised due to photons from anionation and so on, it would be virtually invisible, and that alone might be sufficient to hold everything together. Okay, so, you get an idea of the sort of consequences it from that rather simple beginning. Now, just to make a statement that this is what gravity is, it's just microscopic singularities, and as you get closer to singularities, you approach some sort of Schwarzschild type of metric, and that's it. It's not a very useful statement of a theory. You need to be able to convert it into a sort of field equation for dust or fluid or something, which would enable you to do calculations. And that, in this perspective, means I need to find a way of averaging over singularities, in other words, convert the geometry in a region of a cluster of singularities

40:00 into some smoothed-out version. Now, in the absence of non-gravitational forces, we can indeed, just really, if we wish, just take over Einstein's equations, because they're perfectly consistent with this scenario. So I'm, okay, now at this point I'm not sure whether to explain what all these symbols mean, but let me see, who have we got here? I don't need to explain anything The Ricci Tensor equals 0 That's the vacuum equations So I'm assuming they're valid In the presence of matter This has to be therefore something else So I'll just recap what Einstein did He constructed this quantity Made out of the Ricci Tensor And the Ricci scalar And the beauty of that object is that it has zero divergence intrinsically, or identically. So you can equate it to this thing on the right-hand side, which is the energy momentum tensor, which by law conservation of energy momentum also has zero divergence. So it's made a perfectly consistent set of equations there. You can invert this around and put the region tensor on its own on the left-hand side, and then you have this function of the end of the Atlantic tensor. If you postulate that the matter is in the form of what the cosmologists call dust, which means dust particles without any interaction, so the dust has a four-velocity u and a density rho. That is, if you like, the rest mass density in the rest frame of the dust. And then you can write this energy momentum tensor in that form. So the contracted scalar determines the density in that situation. So this was my naive view that that would do. All I'd rather do, really, is to say, instead of rho being the inertial rest mass density, it's the density of something else. And no problem. So it was just a scaling, if you like, for each species of matter relative to inertial mass

42:30 So you can show that if we suppose that these things are fermions and therefore they have a vector-type conservation law on that current You take the divergence of the age of momentum tensor in this form Find it out, then this bit is zeroed by that law And this bit can be written just as the density times the four acceleration And if you're in a purely gravitational system Then the particles of the dust follow GD6 And therefore their four acceleration is zero And it all works what I've actually discovered of course is that once you allow other forces to act on this and those forces don't themselves have a gravitational effect, then the forward acceleration is no longer zero and therefore the divergence of the energy tensor is not zero and therefore it can't possibly match up to the Einstein tensor which has an identically zero divergence, hence I've got a mathematical problem here And I went round and round and round for a long time worrying about this. Do I therefore need to do something like what Einstein did and what Maxwell did, but instead of using this object on my right hand side, which I'm going to take a divergence of, I should be using this vector object on the source side. Now that, you just can't do that. You can't four dimensions say that the matter vector density equals something which is going to make out of the metric. Because the metric is a rank 2 tensor so by differentiating it twice, contracting it, it still remains a tens of even rank. Your antisymmetric tensors of even rank in four dimensions. There's nothing you can do to produce for yourself a vector-like object, as far as I can see. So I thought, okay, therefore we now have to move into five dimensions. I'm very reluctant to do that. In five dimensions, the so-called anti-symmetric tensor has five indices, so in principle you

45:00 can then, in five dimensions, generate objects of a vector type. I didn't follow that up. we should eventually, as the city does leave anywhere, but they were my original thoughts, so I started reading all these pages about five dimensions and so on that took a lot of time couldn't see anything obvious I was hoping, oh there it is, on the page for me, no, it isn't there so I'll put that aside for now and just, okay, just go back to Einstein's approach I could call it that and try to mimic that but generalise it a bit now with hindsight that eventually doesn't work I thought well why isn't that working then what are the general principles then for putting together a field equation what are we supposed to do most of us are not in a position to invent a field equation that was done by a very few people what are the principles here you've basically got three things a conservation law, you've got Lorentz invariance of various objects, and you've got the field equation. And they seem to interrelate in a very complicated way. And I just couldn't find anything in most of the textbooks that I've read to really elucidate for me what's going on here. The one beautiful book that has helped me enormously, what I can't remember the title, is Wolfgang Rindler's postgraduate-level introduction to general relativity. He really does touch on all the things that I need to be touched on. And he shows, he uses the example of electromagnetism to give an idea of how all this works. So I'll just quickly run through the argument that's given by Rindler. So you presuppose in electromagnetism that you're starting with a field tensor of rank 2. There are certain possibilities and this is about the simplest one you could suppose. So here's your field tensor and you're going basically write down a version a covariate version of Newton's second law and this is the way it

47:30 turns out A and yes A and U are the four acceleration and four velocity of the test particle which this field is acting on. So those field equals mass times acceleration transformed to a general frame. Now, if you multiply this by another u then the right hand side becomes identically zero because the four velocity is always orthogonal to the four acceleration for any trajectory. So the right hand side is necessarily zero. This bit is obviously symmetric, but the only way that that can be identically zero for any u is if the field tensor is identically anti-symmetric. Please forgive me, this is all obvious to you. It was not obvious to me. Can you repeat, why would u a u be zero? This is identically, kinematically, this is identically zero in special relativity. You contract the four-velocity of a trajectory with its acceleration. Because its energy is constant. It's because of the norm of a velocity vector is minus one. I abbreviated here I didn't want to spend more than one slide on this So I'm afraid I've abbreviated here So if you now construct a field equation using this thing And it's antisymmetric Basically there's only one way to go You take the divergence of that thing twice Sorry, you take the divergence of that and equate it to Then you get a vector You equate it to your source vector If you take the divergence again the double divergence of an antisymmetric tensor, again, must be identically zero therefore, yeah, so I'm saying that that's going to be our field equation

50:00 and because this is antisymmetric this is a symmetric object, so again that has to be zero therefore the divergence of J must be zero and that's conservation of charge so it's unavoidable at this level you can't really escape a law of conservation of charge if you say that there's a there's a field metric of this simplicity which is going to be related to a current it just falls out now if we use that as the model that people use that as their model for everything else that you do. And clearly Einstein has mimicked that. He's written down an equation which is identically balanced in terms of the divergences. But geometrical gravity is a much more complicated theory than electromagnetism. The relationship between the metric and the connection to the Riemann-Temps, the Kyrgyz-Temps, are mathematically much more complicated. So I just don't see why you necessarily have to mimic this prescription of having a field equation where basically the divergence is on each side. One divergence is zero because of a physical law, nothing to do with gravity. The other is identically zero because of basically analyticity condition I just don't see why you have to do that So this is how I try to I don't need to worry about that anymore I'm just going to try to construct a tensor here to assign to the Ricci tensor what are the invariants I've got available well it's just the scale of density here and the four velocities of the fluid and the gas there's only really one general way of putting those together to make a rank 2 object so you basically need to generate a you can put two velocities together like if you have two indices where you've got the metric itself, which gives me another way of making a two-index object.

52:30 So I have a parameter alpha such that I get Newton's law in the low-speed limit. That's Newton's law written in terms of the Ricci text, so that's the thing that you have to satisfy in the low-speed limit. by parametising it in this way alpha drops out at this level so you guarantee you satisfy that law. In general relativity alpha is 1 I'm saying maybe I can make my theory work, maybe I don't have this conflict between the divergence of the left and right hand side if I pick a different value for alpha Right, that condition the left-hand side of the Einstein tensor, that condition that this, the divergence of this is zero, for those who don't know, it's called a Bianchi-identity, or a contracted Bianchi-identity. So that's the thing I'm fighting against here, a way of getting around the Bianchi-identity. in a minute, but first of all, it is of interest to know what the velocity dependence is going to be of what we call the effective, or the Newtonian, density. Going back again, what that means is, if you're in the rest frame of the matter, and you imagine some matter passing through, a test particle moving through, it will experience an acceleration, so we We have a measure of an effective density by the trajectory of that slow-moving particle which is given by this relationship. So what's the, and that's the rho here, or over the page, I'm calling it rho n. This is the effective Newtonian gravitation density which is written in terms of the timeline component of the Reacher sensor. So I'm working all that out with my parameter alpha, following it all through, it comes and then this is the Lorentz factor so I'm approximating to second order in the speed just to get an idea and I get this result here now in this result the M is the if you like the rest mass

55:00 the gravitational rest mass or the gravitational charge if you like but NO is the density of particles in the observer's frame not now in the rest frame of matter. So that means that if I integrate NO, I get a constant, because I'm talking about a fixed number of particles integrated over the volume. If I integrate this effective density, I get the effective mass of this cluster of particles. Now let's look at what happens when we choose different values of alpha. As I say, when alpha is 1, that's general relativity, and you get this form, where the effective mass is this scalar quantity times 1 plus 3 halves V squared. This is one of the weirdnesses of general relativity, that the effective gravitational density is not the same as the actual energy density. If you look in the books to find out why, they say it just comes out of the equation. You see it does. It's a sort of pressure effecting effect. So if I put now alpha zero, and work it through, then I do get that the effective mass is just the energy density, the density you get from special relativity. If I put alpha minus a half, then I get a Coulomb law type density, a density that doesn't depend on the speed of the particle, so it's now like an electric field source. And another interesting choice is minus 1, where you can't end on the write down, really, an effective mass. You now get a thing, a density that's behaving like the cosmological constant itself. I'm calling it a totally invariant scalar. I don't know what else to call it. It's just something that's the same everywhere, in any framed. That's all I mean. It's a pure scalar density quantity. So I'm quite pleased with that. It means I can tune in the way that experimenters do. They like to tune theories to fit what they want. I can now pick the value of alpha. Okay, so relationally then I think, okay, let's take that expression and now again check it against the Mianchi identity. So there's my conservation law. This is my supposed divergence of my field equation, working through, I won't go through the detail with you.

57:30 What you finally come down to is a relationship between the time dependence, the proper time dependence of the Ricci scalar and the proper time dependence of the density and alpha. But over the previous page, I think I skipped over it Where was it? Yes, here Starting from here, if you can track that equation You find an expression for the Ricci scalar In terms of the density and the parameter alpha So that's fixed And if you want to make that consistent with this result What you have to do there you find alpha and then we hold 1. So it has to be general relativity but my theory conflicts with general relativity in the presence of non-gravitational forces so that equation doesn't work I needed some clues at this stage well it doesn't work if I'm going to insist that the Bianchi identities So I've sort of convinced myself that maybe I don't need to do that. I've sort of skipped over this point, but I think it's quite an important point. What I'm trying to do here by inventing a field equation, a continuum field equation, is average over singularities. Now, I think that's quite a dodgy thing to try to do. and so maybe when you're doing that you're sort of patching your space-time to make it smooth where there are really singularities so if you hear a fully accurate description of this system then it's vacuum everywhere and you wouldn't have a problem it's because of the averaging so maybe in the process of doing this averaging it is inevitable that I'm going to violate the anti-identity hunt around for an explanation of the physical meaning of these identities. I found two things to help. One is in the famous book by M. Myers-Rothorn and Wheeler. They do it a couple of ways. I don't understand this book, or maybe 1% of it, so I just have to pick out bits that seem useful.

1:00:00 And this is where they're explaining the physical meaning or deriving of the anti-identities using an integral approach so they take an infinitesimal 4Q and this is a quote from the book in all the integrals over the sign of this 4Q the star it just hasn't come out as a star this is a star is treated as a constant and taken outside of sign of integration the reason for such a simple treatment is the duality operator the star involves only the metric and the matrix locally in constant throughout this infinitesimal 4 cube over the boundary on which the integration extends. But of course, if the little cube contains a microscopic singularity inside it, then that is not a valid assumption. So that was my first clue that maybe I don't have to insist on the anti-identities. Then, in another rather nice book by, what's he named, Schutz, Berners-Schutz, I think, on differential geometry, I find a more physical explanation that the meaning of the anti-identities, that basically, if they're satisfying, it means that the connection of the geometry is symmetric into its indices, which in turn means that you imagine two nearby particles following geodesics, and you join them together with a vector, then as that system is allowed to evolve, that vector will remain attached to both geodesics, however far you track it, provided the identity is valid. Now, if you think of those two geodesics going near to a singularity, and I decide to remove a piece of space-time and plug the hole with something smooth, and this linking vector passes through that region, it seems to me plausible that it might, or probably would, detach from one of the two vectors if you parallel translate it on the other vector. So this encourages me maybe that is the way to go, to focus on all of the anti-identities and try to get an idea of, convince myself that I don't need to religiously hold on to them and try to get a feel for what their physical meaning is. Another tack that I took and that I was inspired to this approach by another very nice book by Poisson, his first name, called

1:02:30 The Relativist's Tool Kit. Just a book of all the things you need to do to be a relativist. clearly explained. Something called the Comar mass formula seems to be the only formulation there is that gives you any kind of local definition of mass in general relativity, as pointed out, but and Nick. In general relativity, the concept of energy is intrinsically a global thing. But in a rather special case, that is a stationary space-time, a rather severe limitation, there is this local That's obviously sort of a circle, which says if you take a local region of space-time, you enclose it by a two-dimensional surface, and then the mass enclosed by that two-dimensional surface, given by this formula integrating over the surface element of the derivative of something called the Kine vector of the metric, a stationary space-time just means a vector like this, zero spatial performance, and the time-like component is one. And you can use Stokes' theorem to turn that surface integral into a volume integral, so we're now at a double derivative there, and then there's a lovely little theorem that relates that Poisson-like expression, Poisson's equation-like expression, to the Killing matrix itself and the Ricci tensor. And so I can use that then to define an effective vector current density of them. So I can now identify these two things and and that gives me a kind of field equation, valid in the case of the stationary spacetime, it gives me a vector is equal to, well, I haven't written it down, have I? No, I don't seem to have written it down. Anyway, my field equation would be J alpha is r alpha beta psi beta.

1:05:00 Yeah, no, sorry, there it is. There it is. Now, that is superficially very like the equation that I was thinking about a minute ago. I multiply this by the velocity vector of the fluid, then, where are we? that's the that's J there and this is the region tensor times U so it's the same except that here I've got U and there I've got the killing vector I want to do something with that but I don't know what to do with it unfortunately because I can't make any physical sense of this thing called the killing vector because it's a constant it's a constant vector in its upper index form And so the norm is not a scalar, depends on what the metric is. So I don't know how to make a physical sense of that quantity at all. And is this something to do with the fact that in effect I am saying, or general relativity is saying, there has to be some energy to hold the particles in place. If this is a stationary system, there is energy there. You've got to have extra energy to hold them in place. And if you take account of the gravitational effect of that energy, somehow these are consistent. And because I'm not taking into account, there's inconsistency. I really don't know. So I'm stuck there. There's one other avenue to follow. So, as you see, I'm left with a lot of mathematical questions here. I just want to say a few more words about these papers by Einstein and Jordan Hoffman covering more than a decade they first published this work in 1938, then they revised it in 1939 and then they revised it again ten years later in 1949 what they were trying to do was to demonstrate that the assumption that test particles move on geodesics is not a necessary assumption, and that the vacuum field equations are sufficient to determine that.

1:07:30 Now, if that's true, if literally the field equations say that singularities must move on geodesics without any qualification, My theory is doomed, because I'm assuming, as in Newtonian physics, that in principle you can grab hold of a planet and move it around arbitrarily, and the geometry of the space-time will somehow adjust to it, without the space-time needing to know about the forces that are doing the moving. and that is the point that I wanted to get in here somewhere and I couldn't find any way of putting it in the slides my sort of intuition that the perfect match in Einstein's equation between geometry on the left hand side and the match equation on the right hand side that totally encompasses the law of conservation of energy momentum is just unreasonable the idea that geometry should know about the laws of physics all those field equations, those Hamiltonian laws, which produce for us conservation of energy momentum. What have they got to do with each other? Why should they be necessarily so perfectly compatible? It just doesn't sound physically plausible to me. As I say, that's really the early papers that I've seen that really addresses this problem of the system of singularities is, uh, uh, contributing to the, to the geometry. Um, so I'm just going to, and lots of other people have done this, it's not very interesting to have quotations, uh, read out to you, but it's very tempting, uh, so I've got a few here. Um, okay, from the first paper, the only equations of gravitation which follow without ambiguity from the fundamental assumptions of the general theory of relativity are the equations of empty space. It's important to know whether they alone are capable of determining the motion of bodies. The answer to this question is not at all obvious. Okay, so that motivates what they were doing. And just to give you an idea of how it looks as though my idea is due in the light of what they did, I'm just going to try and give you an idea of what the idea of the investigators was and what the input to this. Essentially what they're saying is, the fear of the singularities here, you don't know how to deal with the singularities, so you enclose the wall in

1:10:00 little spheres, and on each sphere you have a circular, a closed linear path, and from the metric, from the general expression of the metric using the first derivatives, they they were able to write down a three-vector, a three-vector quantity, so if I call this surface S and that path C, then they just do Stokes' theorem in three dimensions and that says the, yeah that's right, so the lining to go around that closed loop, and equals the surface integral of the curl of V, yeah, so that's the surface of the vector. So this is just obtained from the metric, okay, they show that it has the transformation properties of a three vector obtained from the metric, the general expression So then we have Stokes' theorem, and then they, I've never seen this done before, I was amazed to see this done, maybe it's a trick that all theoretical physicists know about. Then they allow this line, this closed line, to contract to nothing. So they then say C goes to zero, effectively. So that this surface, S, now approaches a closed surface. So since C has approached zero, this thing on the left hand side must approach zero. That becomes a closed integral, and the integral has to be zero. condition, and says that the integral of this curl over this closed surface is zero, whether you've got singularities inside or whether you've got them outside, provided they're not on the surface. So by insisting that that thing is zero, you're saying that any singularities inside must stay inside, so that the locus of this surface has to match the locus of singularity in science. Is this a well-known trick?

1:12:30 Well, Angus, you've seen that? It looks dodgy, doesn't it? There doesn't seem to be any real input to it. No, and then it starts hearing words from the surface of order, and they do away with the border by letting it shrink. Yeah, Weaver's famous statement, the, what is it, the boundary of the boundary of the zero. This is another example of it, obviously. What happens is, the nearest I can understand this is it's like the rate of change of electric flux. Okay, if you think of this as an electric charge, then as long as that charge stays inside, the total flux is constant. And as long as the singularity stays outside, the total structure is constant, if you make that singularity cross that boundary, then, in time, this thing is going to have a moment when you don't have analyticity or something like that. I wonder, what you just say, quote, boundary, boundary, ceiling. Isn't that something coming from topology? Isn't that they come very general? Well, I've heard a difference with geometry rather than topology because I don't have a problem. It's in the foundations, isn't it? Yeah, I think. Topology. Now, what does it mean? I don't know. Delta square is zero. Delta square is zero in this boundary. It's obvious. It's obvious. It's just what I've just done here. It's so obvious. Yeah, in that sense, I guess it is topology. Yes, it is. If, for boundaries to be a boundary, it can't have a boundary. to say. We don't seem to think this is the magic that does everything. Okay, so that is such a simple input. If that really does it, what they claim it does, I don't see there's any way around it at all. Then it must be what they say. There's no room for anything else to put into this, into the equations, unless you allow that other thing to also contribute to the curve of check, is what they say is true. So that is another thing track down and understand their paper better more quotations and this seems to contradict what they're just claiming our equation of motion do not restrict the motion of the singularity more strongly than the Newtonian equations now

1:15:00 Newtonian equations provide slots where you can play any force you like there's a place to put that second law but in their treatment there's no slots to put an arbitrary interaction, there's no way to put it. So maybe that's just a rather sloppy statement they're saying there, or they've just counted up degrees of freedom or something, it doesn't really mean what it seems to say. Now that this is the quote I referred to before, the representation of matter by means of singularities does not enable the field equation to fix the sign of a mass, so that, so far as the present theory is concerned, it's only by convention that the interaction between two bodies is always an attraction, not a repulsion. Well, that's just to give me some support for Einstein's theory. And then, in their conclusion, they say, we have therefore obtained the Newtonian equations of motion from the field equations alone without extra assumptions, such as was hitherto believed to be necessary as supplied by the geodesic lines or special choice of any utensils. So that is saying that they achieved their goal. and who am I to question these great men more questions what is the right way to average over singularities remaining geometry somebody must have done that and tackled that question in a general way and possibly their paper provides a clue are there really geometrical constraints At dinner, Angus was telling me about some constrained equations that the loop quantum gravitists are quantizing. So there are perhaps other constrained equations that I don't yet know about. Well, this is a slightly different thing. When you formulate GR in a 3 plus 1 version, you break the sympathy. Well, I am. If I've got singularities, I have to have a restrain for it. Yeah, sure. But what you then do, you have to see how does the symmetry group of theory, the decomposition group act. Then you see you have, well, you have basically projection of the green surface. because you have one projection of the difference in the other bar in the maximum parameter time.

1:17:30 This turns out to give constrained equations. You don't think it's necessarily relevant. Geometrodynamics is the key. Okay. I've concocted a few points in relation to quantum mechanics just for the purposes of this meeting so they're not very well thought through obviously extreme speculation first of all I see some parallels if you like always suspicious of parallels but nevertheless I'm having to somehow smooth out my space-time over singularities is that in any way like what we do with the wave function the position of electrons in atoms and cells. If there is any connection between these two things at all is the fact that in doing the smoothing out, I'm forced to accept a connection with torsion, which means I've got a rotation. As you move along a geodesic, can I associate that rotation with a phase? And if I have a closed trajectory and I want that connection to be single-valued, does this give me some kind of quantization which is in any way similar to the quantization of what's done in quantum mechanics? I'm trying to average out something that contains kinetic energy. Is that at all similar to the averaging out of kinetic energy that the quantum potential in the bone model does? Another sort of straw to grasp at. Now, if this was to be correct, if this was to be a correct classical theory of gravity, what would it say about the program to quantize gravity? Because this is not a conservative system. You cannot write down a Hamiltonian. the solid object that goes sailing off to infinity as we're accelerating, shows you that at base level, there's no conserved energy, there's no conserved Hamiltonian mechanics. You can't do it

1:20:00 using Hamiltonian mechanics. If you try to marry this with idea, at least, with quantum mechanics, how would you do it? Could you do it? What sort of consequences would there be? The only way I can see of possibly doing it is using the method, where you say you eventually identify the quantum phase with the action which you obtain from a, by integrating Lagrangian. So you, I think you basically can write down a Lagrangian, it wouldn't be a conservative Lagrangian, but you could write down a Lagrangian. It would presumably contain an extra bit, which would be something like the potential energy, the gravitational potential energy, something like that. So I thought, well, just for the hell of it, let's do what Kenrose did, and work it out, see what it, see what get. Now, Penrose assumes a mass of apparatus size. I don't have any justification for doing that. So I'm taking a mass of quark size inside a hadron as my example. So I'm assuming a mass of the order of GV on C squared and separation of the order of a pheny. And I'm saying the visit of the order of the mismatch in the inertial to gravitational mass, for the quarks, strange quark for instance. So this is going to give you a phase shift, sort of a normal phase shift, of the action over h-bar, which is going to be the perturbation of the Negrosian times time over h-bar, so you'll get a perturbation of the phase of per unit time of this put those numbers in and you get that figure 5.6, 10 to minus 8 radians per year so in the lifetime of the universe this could be significant on the scale of the time it takes to do an experiment not very significant well, except, I don't know what is significant in this context if we think that this what I call phase shift is going to somehow disrupt coherence or collapse a wave function or spoil your bell state, how much you know random shift to the phase do you need to do that? I have no idea at all, I don't know if anybody does that, that's just just to sort of

1:22:30 dip my feet in the water of what that principle would say for quantum mechanics. Now, in a more sort of philosophical vein, in quantum mechanics everything is non-local, so energy is non-local. But in general, you take energy as the source for this locally defined continuum. Is that reasonable to think that energy is the right thing for is locally defined, the curvature. Maybe that mismatch between the non-locality of energy that quantum mechanics insists on and the requirement for a classical limit to your gravity theory that have a locally defined geometry is, well, maybe it's obvious that but anyway, slightly different way of putting it I think maybe that's one of the different reasons that we have this difficulty in matching quantum mechanics and general relativity fermions on the island are if there is such a thing as localisation they are the most local things now, Nick doesn't believe in fermions he thinks they're an abstraction in quantum field theory but I believe in those classical objects quantization, they have a meaning. So I'm saying first quantization is closely connected to classical physics. So if something has a meaning in first quantization, I take it to have a meaning. Is that right? I understand that you understand everything, right? again, back to this, my worry about energy is having this being, you know, the controller of geometry what is energy, what do we really mean by energy well my view of energy is and always has been the potential to produce kinetic energy I don't see we have any other operational definition of what it is in particles it's only when it manifests as kinetic energy in particles it becomes well defined and localized otherwise it's just the potential to do that I think we've come to think of it as a thing in itself just because of the conservation law

1:25:00 you think it's okay it goes away and it comes back again, it must have been somewhere but if I'm saying Newton's third law is to be challenged in gravitation we don't have that conservation law anyway so in that context energy doesn't even have a meaning other than as kinetic energy even outside gravity in principle you might say