Introduction / Twin paradox without acceleration (contd.)

0:00 What is the difference? Well, it must come somehow from the topology of the space-time, because that's the only thing that's different from the previous version, so it's got to come from that somehow. And I'm going to come back to that and point out, though, that the fact that there are physical differences between inertial observers also raises another issue here. It seems to apply in the face of the principle of relativity. As the strictest way of stating it, which is the way Einstein said it in 1905, he said that the laws of physics do not distinguish between two inertial reference frames, or if you want an exact quote from him, or actually I guess this is an English translation of him, from 1905, his relativity paper, he says, he says the principle of relativity is this. It's not, you know, Lawrence groups or anything. This would be for that. The laws by which the states of physical systems undergo change are not affected whether these changes of state be referred to the one or the other of two systems of coordinates in uniform transitory motion. Okay, I just like to think of it as the laws of physics are not distinguishing between two universal reference frames. And so as a consequence of this, there can't be any physically distinguished privileged reference frame, like an ether frame or something like that. So the laws of physics do not single out a state of absolute rest. So this is a toy universe, of course, so you don't even necessarily need to worry about it too much. Well, the twin paradox of this, this Magellan version of the twin paradox, where one twin goes all the way around the universe and comes back, it breaks the principle of relativity, as stated originally. But it's not really so devastating, because as Bronson Stewart pointed out in that paper, that special relativity, special relativity to which the principle of relativity applies, does not need to hold any more than vocally in general relativity. And by playing with the topology of the Minkowski spacetime, we found ourselves standing with one foot in special relativity and one toe in general relativity. But it gives us some small lessons of general relativity.

2:30 So I said that it's okay, because we just need a principle of relativity to hold locally. So what does it mean, though, to say that something's only true vocally? Well, in the present context, there's sort of two distinct notions of locality that would come to mind. There's a paper by Buchnitz in 2004 where similar ideas are expressed in a similar context and I recommend the paper. It relates what this talk is about, the Sagnac effect. It's an interesting paper. But anyway, there are two basic kinds of locality that we could talk about here. Where we would say that, alright, there is a geometrical sense of locality where we would say that something is local geometrically, a geometric locality, we would say that something is local in the geometric sense if it happens in an infinitesimal region, infinitesimal region of space-time. And then there's another, well, what that means, I'm not necessarily going to go into what that means, but that's probably where the topos theory might come in, what does it mean for it to be, or you can maybe just talk about tangent spaces or something, but, and then topological locality, which is really what I'm going to use in this talk, is about just saying that some physical systems local, In the topological sense, if they can be contained in, and we're thinking of sort of space-time as just being sort of this piece of background for physical systems, I believe that, but that's just what we're calling it.

5:00 So physical systems, local and topological sense, they exist wholly in a single coordinate, single coordinate patch, where you're thinking of coordinate patches that define the manifold. The coordinate patches are open subsets that are amorphic to the Euclidean ball or Euclidean space. Why do you know they're subsets? Why do I need coordinates? Well, I'm just taking the definition of a manifold to be a set that has A certain collection that has an atlas, I'm just taking that as a definition, and the coordinate patches are just the, you take the maximal atlas, and all the coordinate patches are just those sets from the maximal atlas where, I want to define it so that they're subsets of, they have a topology of the Euclidean. This manifold that he's studying is locally isometric to Minkowski space. So anything that happens in one coordinate patch is isomorphic to something that would happen locally in Minkowski space. So if something stays in one coordinate patch, then it's equivalent to an experiment in special relativity, and the principal relativity would have to halt. Now, if you replace homeomorphic by diffeomorphic, you don't have to say single coordinate, that will follow. An open subset, diffeomorphic to Euclidean ball, does have a single coordinate, in fact, induced by the global coordinate system on your covering surface. Well, it can be many, it can be many, and we want to make it independent of whatever.

7:30 I just wanted it to be something that was just like a region of Minkowski space-time. Since an infinitesimal, whatever an infinitesimal engine is, it should be contained within this idea, this idea that it's contained in a single coordinate patch. And so one of these is strongly Yeah, and so that means that geometrical locality implies topological locality. Something that's not local topologically is not local geometrically. So the idea is that this idea of the topological locality is something stronger. We want to show that something is not local. It's enough to use this notion of topological locality. This is what I'm going to talk about. And finally, the local laws of physics don't distinguish between two inertial reference frames. And by the way, we're also going to think of the inertial reference frames as the local things themselves. Yes, you didn't mention that in this thing you could distinguish one reference frame from another. Yeah, that's what I'm talking about. Okay, you're going to do that. Okay. Yeah. Well, first of all, you can distinguish... Then, from the twin paradox, you can also use license going around and show that. Do it explicitly. So, we're thinking of, here's another copy of A here, here's another copy of A, here's G, we have other copies of G.

10:00 And you can think of, let's say there's X, and we've got... This line is just described by, this is just me and Alan, you know, say if they, say if a light signal, so this first one's on here, the light signal is sent in this direction, it's going to go all the way around and it's going to come back to A at this time, but it's going to come back to B a little bit later. So it goes out, well we're identifying like this now, so it's going to come back around, and then it's going to... It's going to hit the two observers at two different times. It would actually do it actually. And then if you send the y-single in the other direction from here, it's going to come around and go around. It hits B and then it comes to A. At the same time, the other lysing will do. So if you send a lysing on, you can draw it over here, and it's going to hit you right here, but it's going to hit, it's going to hit, well, I could, maybe it is not going to, it's going to hit, so it's going to hit the, so the two lysing will go in opposite directions, they're going to hit A at the same time, but they're going to hit B at two different times. For A, he can set up a system of synchronized clocks, according to the Einstein convention. They can be synchronized all the way around, but the other observer can't.

12:30 And you can see this also in the, if you do look at the transformation, remember I said that there's a coordinate dependency way, the way I wrote down that coolant duration before. If you transform the coordinates, kind of like I started out with this. So that gives you a physical experiment other than this. The time, the twin paradox again, which allows you to distinguish between the two inertial determinants. If you look at this, this is the equivalence relation again. You can see that if you apply the Lorentz transformations, because I think it's pretty likely to be one, and so in a boosted frame like over here, if we were to use this. If we were to try to change coordinates up here and look at it from B's perspective, we would have an equivalence relation that was like this, and I looked at, I was speaking to him all the way, or in B's, B coordinates, A coordinates up here. When I say geometrically, that just means that you have this family of powers, this large power to your A of everything. There's a circle, a circle which is orthogonal to the other one, whereas for the family B roots, there is no orthogonal to the other one. It's a helix, yeah. So A can synchronize its clocks globally, and B can do it for a while and when it comes around it's a contradiction and you can never fix it. Somebody did a staircase earlier. It's an objective contradiction. That's right. Yeah, it's something that you can do, like, in 3D sort of thinking. But this is kind of the philosophical word there, if you can really do, if you can still do 3D things to understand 4D. And you can, because you can do things like this.

15:00 But again, in a small locality, even if it's not a point, but it's small, you do have that opportunity to go all the way around. Yeah, it's only when you go all the way around. So this is where this topological locality idea comes in. Using two light signals going all the way around the universe and then coming back is not local in this topological sense that I've defined because in order to describe that experiment, you're talking about you're going to have to use two patches, at least two patches, one for this side of the sun and the other one for the other. Patches being the ones that are homomorphic to So the two inertial planes are locally equivalent but not globally. Yeah, that's right. Experiment to distinguish a global topological factor with a physical name. In a more general setting. You're going to do the electrodynamics, Dan? No, actually I was going to skip that. I had some other questions. That's sort of nice, actually. So now suppose you put a charge here. And then you'll discover that the form of its field... So you could say, you could do an electrostatics experiment, and that would distinguish the two fields. And they say, aha, here's the contradiction to the principle of locality. Go ahead. Yeah, that describes a paper I read that made me mad because I was like, no, you can't have a local experiment to distinguish them. Because I just proved it. I mean, you have, you know, it seems very simple. Certainly at least it's topologically local, in which case it's not, it's just the experiment in Minkowski's space-time, and we know the principle of relativity holds in there, so how can there be any local experiment involving electrostatics, which tells you that you're done. But the problem is that that experiment is not really local, and maybe this is just semantics and playing with words, but I think semantics are important because you can't eat steak with a hammer. So, yeah, I mean, you have an electrostatic field which goes all the way around the universe now, and maybe a charge, if you want to think of a charge, and the way they describe the experiment, there's a charge there that generates the field.

17:30 And the charge has been there for all time. Yeah, and it's been there for all time. And so, you're talking about, even though you're just measuring the little part of the field, well, where does that field come from? It's got to come from the way the topology was described. So, I mean, if you really describe the experiment fully, I think, or I... I'd like to maintain or shout out that it's not a local experiment. Because the propagator for the electrostatic field includes parts that go around, and so it's not a local experiment in that sense. And also the guy who did the calculation made a mistake summing the propagator. He got the propagator right. Yeah, he messed up with the complex analysis part. It doesn't affect the main argument. But it was a very interesting paper. I mean, I thought a while about it before I could figure out, well, what's wrong with it? I think there's something wrong with this paper. I don't know what it is, but it was a very clever paper. I thought it was very magic, but I disagree with it. So, yeah, we have... This light signal effect, which you can use to distinguish, to observe, and by the way, this is very similar to an experimental thing called the Sagnac effect. Sagnac. Sagnac. Maybe I'm not pronouncing it right. Sagnac? Sagnac. Sagnac. Sagnac, okay. Can you explain what that is? Yeah, you have like a disk spinning around and then you have an emitter and a thing that can emit and absorb the light that, while it's spinning around it, it shoots lights off in both directions and then receives it, and well, since it's spinning around it, it receives one before the other. In a word, it's a rotational microconvoluting experiment. We do find it doing that. Yeah, that guy... If he's spinning a bucket of water, he's got a really good eye. Yes. But he can do it with electromagnetic fields, yeah. Yeah, the connection to the Sagnac effect is also pointed out in that paper I've mentioned before by Woodness, 2004. I think that's a good paper. I don't know if it's been published, but it's an archive, and I think it's been submitted, but I don't know if it was published right, and I think it should, but, you know, whatever, it was my opinion on it. Um, so...

20:00 Don't hit yourself on the head in public. Let others do it for you. So Brons and Stewart say that the space-time, M here, that we've constructed, it has a state of absolute rest, which is exhibited by this A in the example, the exclusive thing I've written up there, in which, you know, it's a state of absolute rest in the sense that a global system of Einstein-synchronized clocks can be defined, and they state that... The absolute rest frame comes from the topology in their paper but however I wanted to say that I realized this last night and thinking about the Godel universe and I don't know if you noticed that but if you just do this this construction in it just with a simple modification we can we can still keep a cylindrical topology just like what we have and have no state of absolute rest. But the price we pay is that we get the existence of closed non-spacelike curves. So we get whether a closed normal curve or a closed non-spacelike curve. It's very easy to do. All we need to do is introduce a large time shift from the start. Like, you know, I define it this way. This is how we define M. The quotient here to get M. Well, that's a formal formulation. Change it where, let's change this definition now. We'll define, we'll choose, first of all, we want to choose k and then equal to, and then on that, we still have, we still have something a little bit more comical, but we still have a cylindrical topology. I'm going to call this m tilde, so let's give this m tilde as a quotient of, you know, it's...

22:30 1 plus 1, Minkowski space, time, Ma and Al find this. And so we have, we define it like that. Well, and if we look at what this, what this looks like in some arbitrary boosted framework reference, what does this transform to? If we just apply the Lorentz transformations like I did before, it's going to be like this. X prime, I did this, you know, Now since beta is less than 1, if we're going to be physically realistic, let's say beta is less than 1, well that would mean that minus beta is greater than minus 1. All of these terms are equal to or equal to zero, but there's not going to be any observer where that's equal to zero. The observer is always moving at less than the speed of light. So no observer can set up a global system of synchronized clocks. So there is no privileged observer and no absolute verse frame in this space-time. Here it would have just twiddled it a little bit. And yet you would still have inertial twins that could be periodically in age at different rates. So therefore it is incorrect. As others in the literature, I think, have said, that the unexcelerated twin paradox, which is, you know, disgusting, some references I could give you, in a close-based time like this, is explained by the existence of an absolute rest frame, or even a preferred one. The right answer is really just that there's just a difference between the inertial states of motion. There's some, you know, like they send out light signals and then they get them back at different...

25:00 Separated by different amounts of time. There doesn't have to be one where it receives them both at the same time. Just that they receive them at different times or something like that. I mean, if you were going to define a privileged observer as one who can set up a system of Einstein synchronized clocks. So, you can't have a, you can't, you know, the idea is that an absolute, you can also have a space-time where you have infinitely many. If you just add some linear non-compact dimensions to the one we started with, you would have lots of privileged frames there and that were all moving with respect to each other. So in this M tilde, there is no privileged frame. M tilde is sort of like a baby version of the global universe. Any questions? I could go on a little bit. I think I'm almost out of time. Yeah, yeah. A lot of questions were asked during the lecture. Okay, well, thank you. Okay, so what were we doing now? After all, the plan was for us to have a short break and then to begin the general discussion. The 15-minute break? Well, let's call it 15 minutes. You kept talking in the middle of this. You talked out loud. You thought you thought out loud. Yeah, I mean, I think you need to write lectures.

27:30 You need to write them out, because otherwise you get flustered and because you had something perfectly good to say, but you didn't just flat out say it, and also there were a few points where you skipped some, you know, and that's why you need to write them out. It takes a long time. You're much too nervous, and you can't yell at somebody to stop being nervous. And you'd really better work from a technical point of view, but I mean you had something perfectly reasonable to say, that wasn't a problem. Well, it's only been three points. It could have been a short time. But it was easy. It's very straightforward. Well, actually, I'm wondering, do we actually want to have a table there? I'm just going to say, we don't need to actually set up a table there, do we? I mean, normally I'd set up a table, just move that table over there and set it up somewhere. Any more questions?