Introduction / Twin paradox without acceleration

0:00 My mind did not do to others what is hateful to yourself. Nevertheless, Sean is fine too. What can I say about him? He's been kicked out of his many places. But, okay, so he finished the exams, and so he sat down and tried to decide what to do, because after you do exams, the kids stay, which incidentally is about an hour's drive from the University of Kansas. We have a role that you have to do an oral exam just to read a book to get results. I gave him a book to read and then like two or three weeks later he came in and he had written a paper and it was completely astonishing to me it was a very nice paper and it showed Very impressive technical mastery of a number of arcane mathematical points, which I'm sorry to say I've long forgotten, and a really interesting philosophical analysis of some subtle questions in the interpretation of general relativity and some of the issues, and he has a philosophical bent, so he cranked out a paper in three weeks all by himself, so that's about all I can say about it so far, but I'm hoping for... Much more coming in the future, so I think you'll find this talk, you know, interesting, unless we're doing it in math, where it's, it's, it's quite technically. So anyhow, uh, this is my student. Hi. Well, thank you for, for that introduction, and I thank Walton University for letting me speak in accommodations and everything. Um, yeah. So, yeah, what I'm going to talk about is the twin paradox, and without acceleration, I'm going to talk about a version of the twin paradox that, yeah, I, yeah, so, traditionally, what the twin paradox is, I'm sure probably everyone here has heard of it at some point, you have two twins who, they're born on Earth, they're the same age when they start out, because they're twins, and

2:30 And then one day, one of them gets in a rocket ship and flies off to outer space and explores some distant star and then turns around and comes back to Earth to reunite with his twin and it's discovered that the, well, according to special relativity, if you do it right, you find out that if you regard the stay-at-home twin as an inertial observer and then the other twin as one who... And so, that's represented in the mathematics of Minkowski space-time pretty much by the Minkowski inequality. In Minkowski's space-time, you have the triangle inequality goes backwards from what it looks like. You have space, Genghis drawing, and one space, one time dimension. You have, you know, the length of a, b, proper time of this geodesic going from a to b is greater than the proper time register going from a to c and from c to b. And so, but you know, realistically, you wouldn't think that the rocket would instantaneously turn around. So, so usually people want to draw it like, but then, but then all the differences is you're just doing a pack and a roll. The length of the proper time from here to here is the path integral of this straight line, and this is a geodesic, and this is not a geodesic, the accelerated observer is interpreted as not being a geodesic. I mean, sometimes, yeah, you want to say that the explanation... A very elegant way to explain the twin paradox is just to say that it's not really more paradoxical than you just have two points in space connected by two different paths.

5:00 Why should they have the same length in space? Well, why should they have the same length in space-time? The length of the path in space-time is the proper time measured. Which curve is which? Unless you have some kind of thing that corresponds. You have to know something like this. You have to know the straight one is the inertial one and then the curved one is the non-inertial one. So in a sense the geometry and the dynamics both together explain the traditional version of it. Can we clear it up? Controversy is probably not talked about very much anymore. About the role of acceleration, that acceleration caused the twins to age differently. But if you think in general relativity, you could arrange something where you have two observers. ...who are moving just along geodesics that, you know, one of them goes around a star or something and comes back, and those two agings may not be symmetrical. And, you know, but you see in that kind of thing that two twins are experiencing different metrical neighborhoods in space-time, and in this case, the analysis of that would... You know, be kind of hard and maybe not very enlightening to do. I don't know. I haven't done it. But it doesn't seem like it would pay off to give you an insight here. So, and Brands and Stewart in 1973, they discussed, Brands is the same Brands that invented with decay, the Brands-Dickey theory of gravitation. Or like as an alternative to Einstein's theory. Him and some graduate students, they wrote a paper in 1973 where it was interesting. They discussed a version of the twin paradox where you still have flat spacetime, but it's closed. You just basically take, well, the example is one plus one dimensional. You take...

7:30 No plus one Minkowski space-time and you you just wrap it around like a cylinder so that now twin doesn't have to turn around you can just he just goes off in a constant velocity and then he turns it well he doesn't have to turn around just oh so I'll draw maybe I'll erase I'm trying to draw everything over here because I think this covers everything up so I realized actually when I was preparing this that there's a lot of things, so I'm trying to go fast because there's a lot of things I want to say, but I'm going to go up too fast. There's actually, I couldn't go on about this so I realized it. Okay, I don't want to waste time saying that. Okay, so yeah, you can take Gronkowski's space-time. This is 1 plus 1 Minkowski spacetime. And I'm just thinking of it as coordinates, you know, and you have a metric. The metric is, well, let's just say x and t because x is space and t is time. I think I'm starting to like this convention, you know, where you write the minus in spacetime. Anyway, so you take this and then you do a quotient. You quotient out. You make an equivalence relation for every integer with this. So what you end up with, so basically what this is, is it's just a quotient, it's just a quotient space of R2 as a cylinder. So you just end up with the space that has a topology S1 cross R, you know, looks like this. Or if you want to think about it as a covering space with a projection on it. You know, you have, you know, like all of these, these are all the same line, spaced out at one, and if you had some observer over here, you know, this, if you have an observer going like this, well, it's going to, it's going to look, it's going to go around, this maps to this helical, so you have a, um,

10:00 You have a cylinder space talk. And then this is interesting because in such a universe you could have, say this is twin A and this is twin B, they're just moving at constant, well you can sort of translate things in the covering space to things that are going on here. Here's twin A, here's twin B. Let's just say that it's God given that this is a universe with just these two observers and they're just going around in a circle, that's all they do, and you see they're both completely, there's no acceleration involved in here at all, right, this quotient space just inherits the same metric of Minkowski spacetime. Now, this equivalence relation, though, of course, was coordinate-dependent, and that's something I'll get to, I'll be addressing later. So, yeah, the points in this space can be thought of as equivalence classes of points. You can think of, you know, this is a point in here, so I'll put points in this spacetime. But let's call, let's say this is, you know, the cylinder spacetime. And if points or events are equivalence classes like this, or in other words, this quotient map, you can say that you can think of this quotient map takes you from a coordinate x, t here to, you can take a representative from each of these equivalence classes where there's, yeah?

12:30 Well actually it's stronger than that. The transformations that you're modding out by are in fact isometries. So the thing picks up a metric, it inherits a metric from the main space because all the different things are isometric, all the different pre-images are isometric. So locally it's isometric to me. It's flat. It has no, as far as you can tell. It's just all in, it's only a global topology that's different. Thanks for answering. Sorry. Okay, so, so yeah, philosophically this is very, very simple construction. Mathematically it's probably not that interesting, but philosophically it's interesting because now there's no dynamical distinction you can make between these two twins, or even a geometrical distinction really. Language. We can account for the asymmetry. I mean, these two twins, they're going to age at, they're going to meet each other at different times according to their own clocks, and maybe it seems, well, it seems that this is only accounted for by going to the fact that they're just different paths in space-time, and there's not a, there doesn't seem to be a... Any dynamical or like immediately physical distinction to be made. Now although it might seem like that's the case, I'm going to say that's not the case. I'm going to explain why there is a subtle dynamical, well maybe not dynamical, but a concrete physical difference between the two.

15:00 Okay, I understand the problem is when they meet, they've aged by different amounts. Yeah. And before you could say it was because one was accelerated. But now neither one of them is accelerated. They just both travel freely and they're just banging to one another and they've aged by different amounts. Yeah, you can say with this construction, like let's say if using this, using those coordinates that I just erased, you can say like twin A is given by, just to make it explicit, just to show you the calculation. A is given by this four line, where this is the point at M, and B is given by this. This is in the least coordinates, where beta is something but that's less than one, and maybe I always take the speed of light units where the speed of light is one, and if you do that, well, and B is, I guess, more beta than greater than zero, too. So beta, so B moves at a uniform velocity of beta with respect to A. And if you were to do the math here, you would find that according to A, the twins meet periodically with a period of one over beta. It's a simple calculation. And according to D, they meet with a period of... It's a kind of a combination factor in there. Gamma. We'll use gamma as tradition.