Transcript of a2-l07 ========== _0:10_: Welcome, Richard. 7. This is the We'll finish off Chapter 5. This _0:16_: time _0:17_: would allow us to go into chapter six and seven, which are _0:21_: the more dynamical bits of the of the material. And they will. _0:26_: I hope to get them onto the Moodle today, but they'll appear _0:30_: this afternoon. I expect _0:34_: the couple of things I want to mention _0:38_: which I I think I may have pointed out in the _0:44_: in the in the forum posting. One is _0:50_: in the middle. There's a link to a previous years I think a _0:54_: couple of years ago, _0:58_: and I I say with one, _1:01_: so not this year. Previous years, good one to go to work _1:05_: through because you can do everything in there you can do _1:09_: at present. But the thing that I wanted to draw your attention to _1:13_: was that although the notes of the bottom of that have fairly _1:18_: discursive extra commentary on on the answer and so on, there's _1:22_: also a handwritten version. Might I I did a I I wrote out by _1:26_: hand what would be a great answer. The sort of thing I _1:30_: would be happy at 2:00 in the morning to see popping up next _1:34_: on my screen _1:35_: and that would go, yeah, that's gonna be 16 or 16 but you've got _1:38_: about reading it right. So that's what you want. That's the _1:41_: sort of thing you want to to to be handing in. Now the important _1:44_: points as I as I mentioned in in the model posting, _1:47_: it's legible. It's OK. My handwriting isn't necessarily _1:51_: the most beautiful, but it's it It. I can read it. If you if you _1:55_: want to make a tight version, that's good too. What it's not _1:59_: is photograph or scribbles on your notebook. It's not a mess. _2:03_: It doesn't have to be beautiful. There's a couple of crossings _2:07_: out in here. Bit of Tippex wouldn't have done any harm, but _2:10_: it's not necessary. It is considered. It's a fair copy. _2:14_: This is not your first grasping as a solution. It's a _2:18_: is a a thoughtful description of what the answer is. _2:25_: I I had a little comments to this. You know why such and such _2:28_: is a good thing to see in the answer. I encourage you to look _2:32_: through that. That's what the sort of thing that it is good to _2:35_: see in assessment. If it's not, if it's a mess, then there will _2:39_: marks off _2:41_: because this is part. This is not just an assessment. This is _2:43_: part of the trading for scientific communication, _2:46_: writing papers. If you ever you end up in research, writing _2:50_: academic people to put in by your colleagues. _2:53_: Papers are there to communicate. They are there to show how an _2:57_: idea goes from the basic idea of what the what. The point of this _3:01_: is what the idea was, how it works out, what we discovered. _3:05_: You know there's a structure, academic papers and you're _3:09_: learning that in for example, your lab report. They are also _3:12_: an exercise in scientific communication. And all this is _3:16_: not a paper. It's the same rules apply. It's what I'm _3:19_: communicating. Who am I communicating to and how do I _3:23_: make it clear what I what? I think _3:26_: Remember that the point of initial assessment on an exam is _3:30_: to show that you are smart and you have a clue one way or _3:33_: another. If you can communicate that, then then marks will be _3:37_: found. If something's ambiguous in the question, then see. This _3:42_: seems ambiguous to me because A or B and that's why you have a _3:46_: clue. OK, so if some were ambiguous, you pick one of the _3:49_: two and answer that question and that's fine. _3:53_: They're not supposed to be ambiguous. But sometimes I think _3:56_: it's to get you through that way. So study that _3:59_: and remember the point is communication and and I think I _4:02_: think I mentioned at the end of the _4:05_: but her posting was _4:08_: the main set to be in _4:10_: writing an answer like this. I see one of your colleagues in _4:14_: the class said, I have no idea what's going on with this _4:17_: question. I'm I'm, _4:19_: I don't know which to start. _4:21_: What would you write to them? Someone who'd been to the _4:24_: lectures, bit confused. How would you make clear to them how _4:27_: you've understood it? They would help them. So that's the that's _4:31_: the level you're for. You're not really talking to me, You're _4:34_: talking to one of your colleagues and that's the amount _4:37_: of detail you need to include and exclude. You don't have to _4:40_: see what relativity is. Your colleagues know that you don't _4:43_: have to, but it's not. It's not telegraphic. _4:47_: I could go on and on about that because _4:51_: could be resistant to writing things down. I think that in in _4:54_: schools, people are encouraged filling the boxes. _4:58_: Would that be right? Yeah, bad habit _5:01_: to do that. Teachers should be banned for telling people to do _5:04_: that. I mean, maybe it works for for schools have assessments, _5:06_: but it doesn't work for higher education assessments. _5:11_: Next thing I want to mention is _5:16_: the padlet. _5:18_: Now there is a link to _5:21_: to this on the _5:23_: on the middle _5:25_: is a great this is a useful place, which I think I didn't I _5:29_: I didn't emphasise the very first lecture because it didn't _5:32_: seem to work in Chrome on this, on this machine, whatever you _5:36_: could. I was a human. _5:38_: Anyway, that The thing is, there is a very useful resource, a _5:41_: very good place to ask. _5:43_: Ask questions. I've created it. A couple of folk already mailed _5:46_: me with questions, with good questions, excellent questions, _5:50_: and I've created it with a couple of those questions and _5:52_: the answers I gave. There's section for each of the _5:55_: chapters, just double click somewhere in the right sort of _5:58_: place, type in your question, and I don't get alerted from _6:01_: something appears there, but I do try and remember to look _6:04_: there from time to time. So it's not the fastest way of anything, _6:07_: but has the advantage that everyone else sees. Oh, that's _6:10_: good. I wish I'd asked that question. _6:12_: Or I'm glad someone else asked that quest. _6:15_: You can do a service to your colleagues by asking clever _6:18_: questions. No, no, no, no. Not by asking clever questions. By _6:21_: asking a question that is a problem for you _6:25_: because it's probably a problem for someone else as well, _6:28_: I could talk on, I could blither on this in moderation _6:31_: definitely. I will now close down that window. Just _6:34_: illustrate, there are _6:36_: more chapters to come _6:38_: and I will close down that _6:43_: and go to here and click window. _6:48_: You. _6:52_: Yeah. OK, _6:54_: where we got to last time? _6:57_: What we got last time was talking about the _7:02_: that, the, the range transformation and a little more _7:04_: beyond that _7:07_: and it's taken us 5 chapters to get to this. _7:11_: Now you will remember that you did a bit of special activity in _7:15_: Physics one last year. _7:17_: She doesn't. Vague, sort of slightly nervous noddings. There _7:20_: you did, I think you did in two lectures. _7:25_: You can get to this really promptly from a standing start _7:28_: and you can you can see and the twins paradox exists, blah blah _7:32_: blah. But I don't think you can actually understand much in just _7:35_: two lectures. So what we've done in this, what I've done in in _7:39_: this route I've taken you to get to _7:41_: the Reformation is really roundabout route, which goes via _7:45_: rethink about the axioms, goes to rethinking about lengths and _7:48_: times, _7:50_: goes to thinking about Mikulski diagrams, _7:54_: goes to thinking about space-time and geometry. And _7:57_: then _7:58_: this pops out in a fairly natural way. I hope. I mean, _8:02_: what would, the last time I hope made this make give this a _8:05_: certain inevitability? _8:08_: So that's why we're only getting to this now. _8:10_: But the transformations are, in a sense, the core tool, _8:14_: oh oh oh of of relativity. That expresses the whole, the _8:17_: relationship between a moving and a stationary frame. _8:21_: And the what we're going to talk about this time is two or three _8:26_: of the peculiar consequences of of this _8:32_: and we're going to go on to next time is how do we in the context _8:36_: of relativity topic kinematics, how do we describe motion _8:40_: And after that dynamics, how do we explain motion in relativity _8:43_: context because at that point we can talk about those things in a _8:47_: sensible way with this mathematical tool. _8:50_: So this is not dilly dally. We talked about the velocity _8:54_: transformation, _8:56_: think Minkowski in 1908. So this was only three years after I _8:60_: said thank you 5 paper. Very quickly this became ohh. That's _9:03_: obviously the way of doing things. _9:07_: Um, I talked briefly about I I didn't work through this example _9:12_: because it's quite long. _9:16_: We should have. But I I encourage you, I'll point you _9:19_: towards the section of the notes to to really think through that. _9:22_: It's a useful what example to go through step by step and make _9:26_: things so the next with the watch talk about is kind of the _9:29_: paradoxes or special relativity. Now the word paradox means a _9:32_: couple of different things in different contexts, _9:36_: say contradictory things. _9:38_: But what I am taking paradox to mean that what it's useful sense _9:42_: of of that is a scenario which appears to be wrong, appears to _9:47_: be self contradictory but has not been working out why we're _9:51_: the contract. By looking closely where the contraction appears to _9:55_: be working, be precise for the contract appears to be, and _9:59_: discovering why there isn't in fact a contradiction, you end up _10:03_: with being propelled into thinking about the the structure _10:07_: of what you're learning about. Paradox is in the sense in the _10:11_: presence I'm referring to them are things which sound wrong but _10:16_: aren't _10:17_: OK, and the best known of them is called the twins paradox. _10:22_: You probably heard about this from you in popular coach _10:25_: relativity or on TV programmes or whatever. It's simple but it _10:29_: has caused more not a noisier and more fundamental _10:32_: disagreements about relativity over the last 100 years, but _10:35_: people who sometimes ought to know better. _10:40_: So first a non relativistic version. _10:45_: But if she is _10:46_: leaving, Troy battles, guard was to go home, get a cup of tea, _10:50_: heads off toward Ithaca across the Mediterranean, right in a _10:54_: straight line, because he has a clue. _10:58_: He gets from trying to Ithaca. _10:60_: Problem solved. That's what actually happens. He takes a _11:03_: detour _11:04_: and goes ticket dog leg route from Brightcove here. Yes, I _11:07_: see. Now which takes longer? Which of these roots is the _11:11_: longer, the longer route, _11:16_: the dog? The dog later, Yes. So very obviously there's nothing _11:20_: complicated here going going that the indirect route is _11:23_: longer _11:24_: if you if you don't on the on the same with Misery and touring _11:27_: this Taffrail log behind him, this little propeller, it will _11:30_: turn more times. If he goes the indirect route, then you go the _11:33_: direct route. _11:35_: No problem _11:36_: right now as imagine _11:40_: a species version of that we're at this time. Odyssey stayed at _11:45_: home and Penelope heads off to a star which is 25 light years _11:49_: away at half speed, right. Turns around, comes back, _11:54_: ends up back at Earth. _11:56_: OK, that's our dog legroom. _11:59_: No _12:01_: penalty. Is travelling at hospital late _12:04_: and that's a gamma of I think 1.15, so you know, a bit more _12:08_: than one. _12:11_: So _12:13_: Oh yes, yes, watching this or people or people in Odysseus _12:17_: frame watching this will see Penelope's clocks run slow. _12:21_: Let's take more bass going out and coming back. So in both _12:25_: cases Penelope clocks are running slow and and so Penelope _12:29_: is younger _12:31_: when they're just use where did you get the twins? That's what _12:36_: they call the twin paradox is younger than Odysseus. When she _12:40_: gets back back home to earth _12:43_: and V equals half Gamma is 1.15. She is 87 years old when _12:47_: Odysseus has aged 100 years because it takes 100 years going _12:51_: hospital like 25 years out, 25 years back, whole thing 50 years _12:55_: there, 50 years back, 100 years. So that 100 years is why that's _12:59_: that's why that's 100 years. _13:02_: OK, no problem. That's nice and straightforward. Now you _13:05_: understand about relativity and time relation, that's not a _13:07_: problem, _13:09_: but then some breaks, Park says. Ohh, but from from Penelope's _13:12_: point of view, _13:13_: it's audacity is moving. _13:15_: He's moving half of late in in in between, which is true. _13:20_: So this is a clock will be running slow _13:24_: on the way out and the same is true on the way back _13:28_: through this year. It's got running slow so but then finally _13:32_: gets back home. It'll be he'll be 100 years old and it'll be or _13:36_: some some age. And it's _13:39_: Odysseus who were younger than her. _13:42_: And that's a problem because while it's while two people in _13:46_: different frames can measure the length of a of a clock of of of _13:49_: of a of a rod, _13:51_: and they mutually measure each other's roads to be shorter, _13:54_: like let's length contracted. And we discussed in chapter _13:57_: three, I think it was whenever it was _14:00_: that that that's not a problem big partly because you're _14:04_: talking about extended things. And in that same scenario we _14:08_: discovered that the two observers on the trains could _14:12_: look at each other's watches and discover and work out the time _14:16_: we've been deleted. The time was moving slower symmetrically in _14:20_: the two things. That doesn't isn't a problem _14:25_: here. _14:27_: We're looking at two people with two clocks arriving at the same _14:31_: time, and that, and those two clocks they arrive at the same _14:34_: point at the same time. They can't both be both. _14:38_: What? What one can't be behind _14:41_: and ahead of the of the other that that can happen _14:44_: because you're talking about two things are happening to _14:47_: temptation happen at the same place at the same time _14:50_: because we are talking about the time dilation in the trades _14:54_: moving past each other scenario. _14:57_: There, _14:59_: the observations being made are are, if you go through it _15:03_: carefully, be observations of different clocks _15:07_: at _15:09_: at different times. _15:11_: So you're making a statement about the passage of time in the _15:14_: other frame, as opposed to making a statement with passing _15:17_: time on a particular clock. OK, _15:20_: that's a point. That's the point which is important, but which _15:22_: bears a second thought. So, so we through the notes, read _15:25_: through notes on this. _15:27_: So what's happening here that this seems to be just wrong? _15:31_: That there seems to be a paradox, I think, which can't be _15:33_: true here. _15:35_: And this has caused us see this has caused people who were _15:38_: professionally engaged in thinking about relativity to say _15:41_: overall, who must be wrong, Oh my God, what's happening here. _15:45_: That there is a deep problem. It's not, because _15:49_: the thing that we haven't _15:52_: stressed is that _15:55_: Odysseus Penelope are not vague about who's been out and who's _15:59_: turned back there. The situation is not symmetric because only _16:04_: one of them has turned around _16:06_: at the remote star. _16:09_: In order to go and come back, _16:11_: Penelope had to go to half up to hospital right? And then slow _16:14_: down and then and then turn round. So she is in no doubt _16:17_: that her who is slow would slow down and and and and and and and _16:21_: and and and and turned round. _16:24_: And that in a way _16:26_: breaks the resolve. The paradox the the the paradox of it comes _16:30_: because we think that the two observers are symmetric and so _16:33_: and so and so the whole time relation thing has to happen in _16:37_: the same way for both of them. It doesn't, because only one of _16:40_: them _16:41_: to actually turns around. There's no. There's no ambiguity _16:44_: there. _16:46_: And _16:48_: another way you can think of this. _16:50_: Well, so, so, so the the and the key point here is that there is _16:53_: no inertial frame that Penelope stays in throughout the entire _16:59_: adventure. _16:60_: Odysseus is standing still in his inertial frame and stays in _17:04_: that inertial frame the entire time. But now P is an inertial _17:08_: frame on the outward journey _17:10_: and in an inertial frame on the inward journey. But they're not _17:14_: the same inertial frame. And that's where in a sense, the _17:18_: problem fails to appear. _17:21_: And you don't even have to worry about the the slowing down and _17:25_: turning round. Because what we could do _17:28_: is would you guess to the star she knows there's someone is _17:32_: heading inward back toward earth at the same speed you _17:35_: conveniently And she and she says, OK, here's my log book _17:40_: could you take that back to Earth or this is the time _17:42_: showing my clock You have to send a radio wave or something _17:45_: to see this time a mic lock. Can you reset your clock to this and _17:48_: keep track of how much time there? _17:50_: So the fact that there are three inertial frames happening here _17:54_: is for Brexit, Brexit. There's symmetry. Another way we can see _17:57_: that _17:59_: is by _18:01_: if we're looking at the relevant Minkowski diagrams. _18:07_: So _18:20_: I'm not sure what the. _18:24_: I'm not sure which one of these is the one that is _18:28_: right. I think that's the one that appears on _18:33_: what do you call it E360 rather than this one. So not on that _18:36_: basis, that's one will be recorded in the OHH, just just _18:39_: parenthetically. And the other thing I want to mention, _18:44_: which I think I did mention in posting, is that the _18:48_: sound recordings of these lectures are available in two _18:51_: places, 1 via E360 and there is a link from the Moodle page to _18:55_: that collection and also my own recordings. The other ones are _18:60_: also available at a podcast linked to there. I think the _19:04_: the equity 60 ones have video attached to them. You know of _19:07_: the slides or or or this. I think the my ones are better _19:11_: audio quality because I don't really know which microphone _19:14_: that was supposed to be using. So whichever one works for you _19:17_: is fine. They have the same content. I would be interested _19:20_: in any problems with either either of them. So I'd be _19:23_: interested if you which one works, because if the 61 works _19:27_: fine then I didn't bother making my my own recordings _19:30_: anyway. So I believe the the the the pad _19:36_: is _19:39_: is that vegetable? Yes, yes Good. OK, _19:44_: so in the course diagram there is. _19:50_: Odysseus is free _19:52_: X&T _19:54_: and Jesus goes from _19:56_: origin where the the two twins separate. _20:00_: To the endpoint, just straight that world. His water line is _20:03_: along there. Nice and simple very simple. Water line stage at _20:07_: equal 0 as time moves on. So that's the sequence of points _20:10_: that that creates the world. Like _20:15_: Penelope heads off _20:18_: in this sort of direction, _20:20_: so her _20:22_: if we draw her. _20:26_: So that's the world line. Hope not the world line. _20:31_: And if we if she's stationary in her frame the the prime frame, _20:35_: then she is moving along the T frame axis of her frame. _20:40_: Nice and simple. And so that's what the the the the the mycotic _20:45_: diagram of her frame on. _20:51_: On the _20:55_: Ohh, this is free. Looks like _20:59_: we get to that that that's I think let's call that event one _21:03_: when she does the turn around. _21:06_: OK. _21:07_: And at this point _21:10_: she entered a different frame, _21:13_: should change his frame into A-frame which is not moving in _21:16_: that direction to hospital light, but moving in that _21:18_: direction hospital light. How do we draw that on the screen _21:21_: diagram we draw? Our world line looks like this _21:27_: in the in the in the in the frame. So she's moving at a _21:30_: constant speed in the negative X direction. As time goes _21:34_: increases, she's moving the negative X direction. So in that _21:37_: other frame that's the T double framed _21:40_: frame and there will be a a _21:47_: X double primed axis there. But the the right _21:54_: now _21:55_: one thing. So I I what I'm getting at here is that _22:00_: I have sort of resolved the paradox by seeing that she _22:03_: changes streams and that they're not symmetric. What I'm doing _22:06_: now is is is trying to give you a bit of a clue to where this _22:09_: extra time went. _22:10_: Why is it that there's thirteen years or whatever? It is _22:13_: certainly missing from Penelope or from only one of the two _22:16_: other participants. _22:18_: So at the point where she turns around, _22:23_: that time there is simultaneous _22:28_: in Penelope's freedom _22:30_: with an event which happens _22:35_: atrocious at event two. _22:37_: Why have I drawn it at that angle? Because that's parallel _22:41_: with X prime axis. The X prime axis is the _22:45_: the collection of events which all happened at T frame equals _22:48_: 0. _22:50_: OK, _22:51_: so if I were to draw the _22:56_: diagram _22:58_: X primed, T primed. _23:01_: All of the events which happened along the extreme axis happened _23:05_: at frame equals 0. All the events which happen along a line _23:08_: parallel to the extreme axis happen at the same _23:12_: with the same T frame. _23:14_: So there _23:15_: the events which happen _23:18_: event one and two _23:21_: because the panels the extreme actually happened the same T _23:23_: frame, so they are simultaneous in penelope's frame _23:27_: but not simultaneous in _23:30_: or disease free. _23:32_: And then in the new frame the Penelope jumped to _23:37_: the X prime. X double prime axis _23:41_: comes along here and through the event 3 _23:45_: which is simultaneous with event one in Penelope's new frame. _23:50_: So event one, _23:52_: it's simultaneous with event 280 Penelopes Outward Bound frame _23:55_: and simultaneous with event three in her inbound frame. _23:60_: And so that's sort of where some of the times gone missing, if _24:02_: you're looking for where's the time gone, missing where the 13 _24:05_: years gone, it's sort of there. _24:07_: If you I'll be thinking about this _24:10_: is if every birthday _24:14_: Odysseus sent out a happy birthday message to his his twin _24:18_: sister. _24:19_: Then they would these by radio, and these ***** greetings _24:25_: would be sent out at regular intervals at the funeral, 8:00, _24:29_: so at 45 degrees, _24:31_: and would _24:36_: but the same benefits halfway there. She's only receiving the _24:40_: ***** greetings from the first half _24:43_: or just 100 years _24:48_: so. So from her point of view they appear to be slow, they to _24:52_: be to be to be late _24:55_: on the way back. However _24:59_: I you have to sort of think I think this through is not _25:02_: completely obvious. Audacity is still sending austerity at _25:05_: regular intervals, but on the way back _25:08_: to help you seeing them at faster _25:10_: than than than once a year. _25:13_: So she sees the busted busted meetings being slower than once _25:16_: a year on the way out, and see the buffering being faster than _25:20_: once a year on the way on the way in. _25:24_: So you're still seeing a hundred of them, _25:26_: but they don't add up to 100 of her years. The I haven't shown _25:30_: that that that nod not adding up on the on the diagram, but this _25:33_: is the way you can think about how the different perceptions of _25:37_: the passage of time go in addition to frame and penalties. _25:41_: If that last bit doesn't make a lot of sense, don't worry, I _25:45_: think I mentioned it in the notes. This is the. The point _25:48_: here is just that I'm _25:51_: showing that you that again building up a rather messy, _25:53_: rather complicated and costly diagram step by step. _25:57_: I'm sort of want to trying to suggest you where the _26:01_: with the. With the extra missing years have gone. _26:06_: I could keep talking about that, _26:08_: of course, more or less indefinitely. But _26:10_: other other things that you know this requires you know, going _26:13_: and thinking about the things that are puzzling right off the _26:16_: top of your head. _26:18_: What things you're holding a little bit shell shocked _26:22_: or or or big size or or or or something are the things that _26:28_: our seem obviously amiss. _26:34_: Walk it through again you hate ohh yeah question. _26:36_: Just wondering from the diagram as when the _26:40_: like back and forth, are they meant to be a specific angle _26:43_: relative to there are 45 degrees, OK, but they're all _26:46_: radio signals so so they tried to speak light which is 1 light _26:49_: metre per metre. So, yes, they're all 45 degrees. _26:54_: Yeah. _26:55_: Yes, that that's not, I think, completely differently. That _27:00_: would be clearer _27:03_: question. This double prime of X double prime frame, is that like _27:08_: that's some that's the fear of the way back. I'll be back, _27:11_: yeah. And it's the point it starts at is like when they turn _27:14_: around. That's right. Yes. So that's a good point. Yes. So. So _27:17_: if the frames if the frame S and frame S primed are in standard _27:21_: configuration, _27:22_: frame double frame is not in standard configuration with _27:25_: others, because why? What? What's? What is it that makes it _27:29_: not in standard configuration? _27:33_: Yes, _27:34_: so the the origins of of of of the of S&S prime coincide _27:40_: at at the origin. The origin of the of's double prime, which is _27:44_: is. It is event one which is doesn't coincide with anything. _27:48_: So you could you could use the range transformation to get from _27:52_: frame to frame S prime, _27:54_: but you could not without further algebra, _27:57_: you the rest transformation to get naively from frame XS primed _27:60_: to double framed because they're understanding figure. _28:04_: That's a very good point. _28:08_: OK. _28:11_: And the other diagram drawn somewhat _28:14_: politically, _28:23_: it's in the note. _28:27_: The other point is worth mentioning here _28:29_: is that _28:31_: the dog leg in the original Mediterranean version, Penelope _28:34_: Odysseus in this version is clearly takes a longer route _28:38_: because he goes a dog leg root and as we all know, a straight _28:41_: line, the shortest distance between two points. _28:45_: We all know that, don't we? _28:47_: But this is a good point to _28:50_: mention _28:52_: that _28:53_: in the in the geometry of the _28:56_: of Minkowski space, _28:58_: a straight line is the longest distance between two points, _29:02_: It turns out _29:04_: and there's an exercise in the. One of the exercises in attached _29:07_: to this section allows you to work through that and and and _29:11_: and part reassure yourself that that that that that particular _29:15_: case. So in the Makovsky space, with the right intuition _29:20_: it. Well, I don't know whether intuition but at some point it _29:23_: will become it becomes obvious that a dog like group is going _29:26_: to be shorter because it's not the straight line. _29:31_: OK _29:33_: and and I I emphasise that when I talk about the exercises, I _29:36_: mean the exercises attached to to to to to these notes which _29:40_: are in the direction notes folder. The tutorial handbook _29:43_: has exercised attached relativity. They are good _29:46_: exercises _29:48_: going through those, but not tightly keyed particular _29:51_: sections as my exercises are. So just to avoid ambiguity there. _29:56_: OK, moving on _29:58_: the other famous _30:01_: and _30:03_: I would draw that. I'll blow that up. The other famous _30:08_: paradox is the Polebarn paradox, or the ladder in the barn _30:12_: paradox, or rather the variance of that. And you have a barn _30:18_: which is 10 metres _30:20_: in length. _30:22_: Are you a farmer? _30:23_: The pool which is 20 metres in length _30:27_: and the agile young farmer who would run at speed where gamma _30:32_: is 2. _30:34_: So the farmer is running through the farm _30:37_: at whatever that's a .86 of the speed of light. Gamma is 2 and _30:42_: so the pole is length contracted from 20 metres to 10 metres. _30:47_: So getting into the, into the, into the barn, the farmer's wife _30:51_: slammed the door shut and to the poles entirely inside the barn _30:55_: because length contracted _30:58_: with athletic and all that. But you're sort of familiar with _31:01_: that idea from other things about land contracts. _31:05_: But then you look at it from the point of view of the farmer _31:09_: running. _31:10_: In the farmer stream, _31:12_: the farm is moving at .863 late in the direction, so the pole. _31:16_: So the barn now has its length contracted from 10 metres to _31:21_: five metres. _31:22_: So it's ridiculous to suggest that the pool is going to be _31:25_: able to get into the barn because it's 20 metres long in _31:28_: the farmer frame and the barn 25 metres long in in in that frame. _31:32_: So what's happening? Does the poor get into the barn _31:35_: or not? _31:38_: So you see the problem. _31:41_: What do we how do we draw that _31:44_: in Minkowski diagram? _31:49_: Let's have _31:51_: the _31:56_: so I'm not _31:58_: and the world lines of the front and back of the barn _32:04_: are _32:05_: nice and simple because the barn isn't moving in the barn stream. _32:09_: The barn isn't moving in the barnyard _32:12_: so the the world lines are nice and simple. The the front of the _32:15_: barn so the the the the pole is going to be going in this _32:17_: direction in front of the barn. _32:21_: It lined up along the keyframe access, but we assume that it _32:24_: could be at X = 0, the back of the barn. Another nice simple _32:28_: water line. _32:30_: OK, _32:32_: now let's have the water lines of the _32:38_: of of the front and back of the pool in the running through _32:41_: that. _32:43_: Let's have the word line of the back of the pole, something like _32:46_: that. _32:48_: And that's the back of the pool. And we've chosen our origins so _32:53_: that the _32:55_: at time equals 0. The back of the pole is at the front of the _33:00_: barn. _33:01_: OK, _33:03_: so this is back in the pool. This is the _33:07_: front of the barn, _33:09_: at the back of the barn, _33:12_: and we're the world line of the _33:17_: front _33:18_: of of the poll. In this case, well, it's going to be at A _33:23_: and _33:24_: it's going to be moving at the same speed at the back of the _33:27_: pool, so it could be at the same angle. So it'll be a line _33:31_: more like that _33:33_: and I'm going to sort of _33:36_: but by thinking through the consequences of this I, I, I, I _33:39_: know that it we're going to be talking this be length _33:42_: contracted. So I'm going to draw it at an angle somewhat like _33:45_: this, like this, _33:51_: and uh, _33:57_: no, not not. No, I'm not. _33:59_: You know that _34:01_: the way this works out _34:04_: somewhat like that, _34:06_: so that's the _34:07_: front of the pool. _34:10_: So this is going to be our prime axis. _34:13_: So it means our X prime axis is at the same angle in the other _34:17_: direction. _34:19_: And this is a Makovsky diagram drawn for the right speed. How _34:24_: do I know that? Because in this diagram _34:28_: the _34:30_: the pool fits exactly into _34:33_: the the barn. So the angles aren't right for for for for _34:36_: that speed and and and and that gamma. But the the the the the _34:40_: relationship is right. So this this shows _34:43_: the case where the _34:47_: pull it length contracted _34:49_: to just the right length of the barn. _34:54_: What does it mean to say the pool is then contracted to the _34:56_: right side of the barn? What What it means it's when we _34:59_: measure the length of the pole, it comes out to be the same _35:02_: length of the bar. _35:03_: What does when we measure the length of the pool mean? It _35:07_: means having two observations of the front and the back of the _35:11_: pole at the same time in the farmyard frame. In other words, _35:15_: at the same time t = 0 in the farmyard frame, there's an _35:18_: observation of the back of the pole being at the front of the _35:22_: barn, and at the same time an observation of the front of the _35:26_: pole _35:27_: be at the back of the barn _35:29_: and they are simultaneous in that frame. Therefore that is _35:32_: the length of the pole in that frame. So when we talk about _35:36_: length contraction, what we mean is when you do that sort of _35:39_: observation, when you measure the length of the moving object _35:43_: by that means you get _35:45_: that distance, the separation between those two worlds being _35:48_: the same with the length of the pool. _35:51_: But _35:53_: notice _35:55_: that in the farmer stream, _36:00_: who's the farmers is stationary moving along the the T frame _36:04_: axis. So for the farmer I'm covered for these chalk. For the _36:08_: farmer, _36:09_: lines parallel to the X prime axis are at the same T prime _36:14_: coordinate. _36:17_: So for the farmer, the event there of the front of the pole _36:21_: hidden in the back of the barn _36:24_: happened _36:26_: had a negative tapering. _36:28_: So the back of the pole hitting reaching the front of the barn _36:32_: happened at time T frame equals zero _36:34_: happened at the origin that we will set this up. _36:37_: But that event there happened at negative press, _36:40_: so it's already happened _36:42_: by the time this the back of the back of the pool reaches the _36:46_: front of the barn. _36:48_: So for the so those two events that we the separation which was _36:52_: our length of the pole in the form in the form of frame, _36:58_: I don't do each other _36:59_: and from the point of view of the farmer _37:02_: because this event happened before. So, so the point where _37:04_: the farmer very reasonably _37:07_: from their point of view the back of the port, _37:10_: the front of the pool rather were beyond _37:13_: the back at the back of the barn at the time when the back of the _37:17_: pool _37:18_: was. _37:20_: So that that's that that event there is simultaneous in the _37:23_: farmer's frame with that event there and that event there is _37:26_: happening beyond the back of the of the barn. _37:30_: So that makes sense from the farmers point of view _37:33_: because at the point where the _37:36_: back with the pool hits the front of the beaches, the front _37:39_: of the barn, _37:40_: the front of the of the pool is well beyond the back of the _37:42_: barn, as it should be of course, because it's longer than the _37:44_: barn. _37:46_: So the so the, the the, the the the. Paradox disappears when you _37:50_: think carefully about what kinds of simultaneous. _37:53_: And when you _37:55_: are talking about the length of something, you are implicitly _37:58_: talking about simultaneous events, because that's what we _38:01_: mean. When we talk about the length of something and we talk _38:04_: about simultaneous events then that's A-frame dependent thing. _38:08_: So if ever you are looking at our a description of relative _38:12_: doesn't make sense because _38:15_: online or whatever _38:17_: what you look for to see why someone has confused themselves _38:21_: is we have the implicitly thought, so we're the implicitly _38:24_: thought. Something, I thought Simpson 80 doesn't actually _38:27_: apply _38:29_: civil society. That is where it all breaks down in in the _38:34_: and we can draw that _38:37_: paper. Actually _38:40_: yeah _38:42_: that's that threat thing just drew and that's the _38:45_: the same the same scenario drawn in the _38:50_: farmers stream where _38:53_: going to there and the event of the is very clear that the event _38:57_: of the front of the pole hitting the back of the barn happens at _39:03_: keyframe negative it's already happened but the same the back _39:06_: the back of the pool hits the front of the. _39:11_: Another way you can think about this _39:14_: is that if you were to try that, if if there's bound didn't have _39:20_: had a, you would vote against a a mountain or something. So _39:24_: there's a solid back wall. So the whole thing is just 10, you _39:28_: know, 10 metres, then rock, _39:31_: then what would happen? Then _39:35_: the poor can go beyond the end at the end of the the the the _39:37_: the end of the barn. _39:39_: In that case the front of the pole would hit the back of the _39:43_: barn and would stop. Say it's an immovable mount or some, _39:48_: but the back of the pool doesn't know that's happened yet because _39:51_: the information _39:53_: that the front of the pool has stopped _39:56_: has to get to the back of the pool _39:58_: by a shockwave. _40:01_: Which can travel faster? The speed of light. _40:04_: You work this all out the information. Even if the the _40:07_: shockwave were to travel through all the atoms in the in the pool _40:11_: at the speed of light, it couldn't get to the back of the _40:15_: poll before the _40:16_: back of the of the of the pole got into the barn. _40:20_: So that's two different explanations of why this all _40:22_: makes sense. _40:24_: And that's also an important point, because in the two _40:26_: different frames, _40:28_: the two sets of observers have different physical explanations _40:32_: of what's going on in terms of what the sequence of events is, _40:35_: what the physical things are happening are. But they have to _40:39_: come to the same overall conclusions in terms of things, _40:42_: of what things happen at the same place at the same time. _40:49_: With that introduction from me, it's probably a good idea, as I _40:52_: say, to go through that step by step in the notes. Just let _40:55_: Reese resettle. _40:56_: We've we've ploughed the ground _40:60_: again. Other puzzles that are immediate there. _41:06_: The other clever, you know, insightful, thoughtful, or just _41:10_: frankly baffles questions. _41:12_: All good. _41:14_: OK, _41:17_: that there is a lot of stick in there. _41:19_: The last thing I would mention I'm not going to go through in _41:23_: much detail is Belle's Spaceship Paradox. _41:27_: And John Bell was a _41:32_: you looked at sound for most of his working life. He was very _41:36_: thoughtful about quantum mechanics and other other things _41:39_: that you know of varieties. And the let's talk about John Bell _41:44_: this, he raised this this puddle in the CERN canteen one time _41:49_: and and said ohh, what would happen? And there's a big _41:52_: argument and the majority of the question go whatever be the _41:55_: wrong answer. So this confused them. _41:57_: They're only particle physicists, but yeah, _42:02_: two spaceships _42:04_: to to to rocket launchpads A kilometre apart. _42:09_: The spaceships take off and head off along the X axis with the _42:14_: same _42:16_: navigation programme _42:18_: for the accelerate in the same way. _42:20_: The engines are designed to prove the right mouth thrust and _42:23_: they celebrate in the same in the same way, so they follow _42:27_: the same trajectory, just displaced A kilometre _42:32_: don't range, _42:35_: so the speed up after a rough for short time, roughly after _42:39_: after quite a short time the good it doesn't take a lot of _42:43_: time going at sea accelerating about G to get up to fraction of _42:47_: the speed of light. It remarkably short time it takes _42:51_: to work it out, but _42:53_: to the moving at Russell's speed at some point, _42:57_: so that's fine. _42:60_: Before they take off, we attach a bit of string to the two of _43:02_: them. _43:03_: One kilometre long piece of string, _43:05_: the kickoff and the the move along move rustic speed. So _43:09_: after after a point, this string is moving at a realistic speed, _43:16_: so at length contracts, _43:18_: so it can no longer stretch between the two rockets. _43:22_: So it's not _43:24_: fairly obvious, _43:26_: but _43:28_: in the rocket stream _43:31_: they're stationary _43:34_: and the length of string is kilometre long and they're _43:38_: kilometre apart, so the _43:41_: there's no problem _43:42_: so the string wouldn't snap. But there's another case where you _43:46_: could have the same answer in both frames. Either string snaps _43:50_: and it destroyed and the bits of string of fragments of string _43:54_: flying around or it doesn't snap. So which is it? _43:59_: And as I say this caused an argument in the quarantine and _44:02_: some people go the wrong conclusion _44:05_: and we _44:07_: the the place where I have up where I went wrong in telling _44:11_: you that story _44:13_: the place where _44:16_: you should have got ah you know you've jumped a step there _44:20_: the police were I smuggled in something when I said _44:25_: in the rocket stream _44:28_: in seeing in the rocket stream there's still a kilometre apart _44:32_: assumes that there is a rocket stream. But is there. _44:44_: This is the main coffee diagram of of the two trajectories of of _44:48_: the new rockets R1 and R2 _44:50_: and P1 _44:52_: is _44:53_: this one year anniversary party _44:55_: of the people in the rocket the the one year out from earth and _45:00_: they have a a celebration _45:04_: and the _45:08_: but in their frame at that point _45:11_: they are moving in in that direction with tangent to the _45:14_: the the water lane. So in their frame and that frame also _45:18_: instant configuration with respect to, _45:20_: that's their cheaper time and that's their extreme axis. _45:24_: But _45:25_: look at what they're telling us. It's telling us that the event _45:30_: which is simultaneous in their frame _45:34_: with the one your party P1 is not P2 _45:39_: but P3. _45:43_: In other words, it's an event which happens later on, _45:47_: after the one year party in the other rockets frame, _45:51_: at which point the rocket will have, you know, moved, moved _45:54_: further, _45:56_: so _45:57_: P1 and P2. _45:59_: Those two events are simultaneous in the _46:04_: launchpad stream _46:06_: and so because of these trajectories are the same, they _46:09_: will be a kilometre apart. _46:12_: But the _46:17_: the _46:18_: events P1 and P3 are further apart than that, _46:22_: more than a kilometre apart to the street and the string if _46:26_: it's to be simultaneously at this at this _46:30_: when your party and it and it P3 is having to stretch more than a _46:34_: kilometre. So it breaks. _46:37_: And I give a couple of other things explanation of how you _46:41_: can think about this in the notes. But the point is there _46:44_: isn't A-frame _46:46_: where P1 and P2 which are simultaneous in the Earth frame _46:50_: where they are simultaneous in the rocket stream. _46:55_: So there isn't our frame, _46:57_: which is the rocket string. _46:59_: There are those all the way along that that that water lane. _47:04_: There's a sequence of of frames _47:07_: which, because the Rockets are accelerating, are not all In _47:10_: Sync configuration with respect to each other. And there's an _47:12_: infinite number of frames there you're talking about. And so _47:15_: there isn't one where the _47:17_: rocket, _47:18_: which is the rocket frame, where the two vans are still 1 _47:22_: kilometre apart, _47:26_: that also needs going through, again settling down in your in _47:29_: your head. And I think that's one of the cases where the _47:33_: utility Makovsky diagram in getting things straight in your _47:36_: head on paper is, is cleanest because this is a really _47:39_: perplexing thing. And of course, time lets you write, draw it, _47:43_: draw it down. _47:46_: That's the end of Chapter 5. We'll go on to Chapter 6 _47:49_: kinematics next time, which I think is tomorrow. I will get _47:53_: the notes up this afternoon and I will see you then.