Transcript of a2-l04 ========== _0:10_: Login lecture 4 and we are off to the races. Now before I get _0:15_: going I'm going to mention a couple of things on the rural _0:19_: page. One is the folder of lecture notes which I trust you _0:24_: have found already. I really hope you have found that already _0:28_: and that contains _0:33_: the. _0:35_: There's a four format notes _0:39_: there as PDFs. _0:41_: It contains _0:43_: a note formatted nicely different way, which may or may _0:47_: not be more suitable for reading on a tablet I have. I have no _0:51_: idea how that works for you, entirely up to you what you use, _0:55_: They're the same content. I will retrospectively post the slides _0:59_: there as well. OK, one behind, but it doesn't matter. I will _1:03_: say again that there's no difference between the content _1:06_: in these various formats. The tablet format and the printable _1:10_: format are the same, just would have margins basically. The _1:14_: slides are just things on the notes I want to highlight here. _1:18_: So there's nothing extra in the slides. Just _1:23_: I, I, I do tend to end up with a lot of stuff here. And I I think _1:26_: it sometimes happens that you end up going, Oh my God, there's _1:29_: so much stuff and like going there, there's less there than _1:32_: than their peers with a lot of the duplicated. OK. So we'll _1:35_: make that really clear. _1:37_: Also in the _1:41_: little Peach, _1:44_: if we can get to the right page, _1:47_: get down to here, _1:50_: back down to here. Um, _1:53_: there are some some on _1:58_: I thought it _1:60_: the _2:03_: right and _2:04_: there are some things that are are from previous years _2:09_: a I'm going to post but I haven't done so yet. My _2:13_: recordings of the lectures, _2:17_: the the the E360 does sort of work, but I can't work out how _2:20_: to get it connect that to the Moodle. It's terribly hard. So _2:24_: that might be easier to do it myself. We'll see what happens _2:27_: there, but I will be posting the the the the, the audio _2:30_: recordings. I just haven't done that yet, _2:33_: but I do want to drink for your attention. And ohh, this is _2:37_: quite important writing Greek letters because legibility is _2:40_: important there. Have a look at that document it it contains _2:43_: bitter experience. Well, bitter on previous students for I do _2:47_: want to draw attention to this padlet. _2:52_: I I really am. Honestly, _2:58_: yes. _3:06_: OK well I I think because I've rejected all cookies this is not _3:10_: working at work anyway. The the, the, the padlet. Perhaps I have _3:15_: to stand by. You know for the pilot is a good thing. It's _3:20_: basically a collection of sticky notes on on a on a web page. Ask _3:25_: questions you entirely anonymous. I can't say who you _3:29_: are. Ask questions and I will you know when I remember, go _3:33_: back there and look at what questions have been asked and _3:37_: and and and and add some comments. Answer them or or add _3:42_: further context. Different years. Sometimes _3:45_: no one looks at this deserted wasteland, Sometimes it's _3:48_: notable all over the place. Different classes seem to get _3:51_: the hang of it in different ways. I don't know. Whatever. _3:54_: We'll see what happens this year. I think it's generally a _3:57_: good thing, but it provides a nice way of asking questions to _3:60_: me. If you're shy and don't want to come and ask questions at the _4:03_: end. Also everyone else gets to see the questions. Questions are _4:06_: good questions because if you realise someone else has that _4:09_: puddle as well, it's they may have expressed it really well in _4:13_: a way that you had to go down to, or you may be reassuring you _4:16_: just go. Oh, someone else didn't understand that either. So quite _4:19_: all questions are good questions. _4:22_: I think actually pretty much all questions are good questions. _4:25_: Yes, _4:26_: So we're turning to this _4:29_: just to recap the end of last lecture, last chapter I talked _4:34_: of the light clock, this idealised way of talking about _4:39_: the passage of time and the passage of time in space. _4:44_: I don't. I talked to the difference the light clock when _4:48_: you were standing by it ping ping _4:51_: and the light clock when you were watching it go past you on _4:55_: for example a train where the the the point where the light is _4:58_: reflected, it's little bit further down the track from the _5:02_: point it was admitted and from your point of view on the _5:05_: station platform and the point where it's received again is _5:09_: further down again. And because in this context _5:13_: you are seeing the light moved at the same speed in the two _5:16_: different frames, _5:18_: that allows you to come to some surprising conclusions. But the _5:21_: time interval between emitted and being received in this frame _5:25_: and then the other frame. _5:28_: If you were doing this in a non relativistic context, say with a _5:32_: a looking at the direction of a tennis ball being thrown back _5:37_: and forth across a _5:38_: train carriage, _5:39_: and the thing that would allow you to go from one frame to _5:42_: another is the fact that time goes the same in both cases and _5:45_: to the tennis ball clock, the tennis ball would pick up some _5:47_: speed from the from the train that's moving in and so there _5:50_: will be discriminated at the speed but an agreement about the _5:52_: time. _5:54_: The point where we move into special relativity territory in _5:57_: this case is the point where we see the thing that we can use to _6:00_: make that bridge. The thing that we can hold on to in these two _6:03_: cases, the 2 frames, is that the speed of the light is the same. _6:07_: That's the point where the magic happens. If you like to drive _6:10_: that home, _6:12_: OK, _6:16_: yeah, _6:18_: I, I, I, we we allow that allowed us to deduce our _6:22_: relationship between the time on the watch of the person of the _6:26_: observers in the moving carriage and the times on the watches, _6:30_: plural, of the observers in the on the station platform. With _6:34_: that expression, we will see again and again _6:39_: and I made some final remarks about the what I call the clock _6:43_: hypothesis or the the clock clock hypothesis which just the _6:48_: time that watches _6:50_: clocks _6:51_: measure, the measure, the passage of time in in, in a way _6:54_: that we are going to assume was unsurprising _6:57_: and an image that I think is very useful. _7:00_: This idea that I think I mentioned, I think in the first _7:03_: lecture of this thing called, you know Taffrail log, the thing _7:06_: that if you're a yacht person, you've put you, you tow behind _7:09_: you and it measures how much water you've has been pulled _7:12_: through and that's how far you've gone. And a clock. Think _7:15_: of a clock like that. _7:17_: A watch measures how much time it's been pulled through, _7:20_: and that's that's what that was recording. It sounds fussy to _7:23_: have to meet these fine distinctions, but it turns out _7:26_: with relativity, if you don't make careful distinctions you _7:29_: end up confusing yourself. _7:31_: But that's the recap of last time. _7:37_: Are there any questions about that? Things have been puzzling _7:39_: over _7:40_: since last thing. _7:44_: OK, _7:48_: space-time and jump to this. We start with more exotic _7:53_: terminology _7:54_: and I and I plan to cover this in I think 2 lectures, _8:02_: objectives, aims and so on, right? _8:06_: No, we'll come to that in a moment. _8:13_: There's a couple of things which are It's important to preface _8:18_: this. I'll show you to report to preface this section with _8:25_: in in the last chapter last when we're talking about lens and so _8:28_: on. _8:31_: What we were doing was _8:35_: noting that different sets of observers in different reference _8:38_: frames, the ones attached the platform, ones attached to the _8:42_: station platform, _8:44_: both make _8:45_: observations of the positions and times of events. _8:50_: And what we what we are aiming to do is relate the observations _8:53_: of the people in the in the carriage to their observations _8:57_: in the in the station platform. And that's not just because we _9:00_: don't we want to know those numbers what that tells us is _9:04_: the physics of the relationship between them. So that's _9:07_: applicable in all sorts of other contexts all the way through the _9:11_: rest of physics, just that's what's important. But we're _9:14_: we're boiling that down to this question of of of of changing _9:18_: coordinates with these two different frames which have _9:21_: they're different. And it's not because we've chosen different _9:25_: units for the on the, on the, on the, on the platform and on the _9:29_: the train that's trivial. It's not because children are _9:32_: different origin _9:34_: because the whole business of special of standard _9:37_: configuration resolves that problem. I mean that does make _9:40_: that we sort of define that problem away. It's not because _9:44_: we have to worry about the time of flight of the light that is _9:47_: that is a I think that is a potential issue but the fact _9:51_: that all our observations are taking place entirely locally _9:54_: means we don't we we that that's to avoid having to even worry _9:57_: about that even think about that. It's not terribly hard, _10:01_: but you can just avoid thinking about buying the observation. _10:05_: What it turns out _10:07_: is that the difference between these different frames is _10:11_: because of the finite speed of light and because we the 2nd _10:15_: axiom says that the _10:17_: speed related the same in all reference frames. That's where _10:21_: the the change happens, as I, as I was seeing _10:25_: remarks about the end of the last chapter. _10:32_: And the point is that the _10:35_: coordinates of the events _10:37_: are measured by the moving observer are systematically _10:41_: related to the coordinates of the same events as measured by _10:44_: other observers. The notion of randomly different, they are _10:48_: systematically differ _10:51_: and the way we are going to. So we're going to explore more _10:55_: about the the relationship between those events and _11:01_: with the approach that there are a couple of different approaches _11:04_: one could take to ratify the approach. I take it very _11:06_: affirmative. Geometrical 1. _11:09_: Now you grow up with an intuitive understanding of _11:12_: Euclidean geometry. _11:15_: In school you learn a little you you learn that word and you used _11:18_: to systematise your your, your intuitions about it. But you you _11:22_: understand Pythagoras theorem intuitively. You you know that _11:25_: if if, if, _11:27_: if I _11:29_: stick or something. _11:33_: OK, you know that if that's about a metre long, _11:38_: but it's still a metre long when I turn it gasp, and that's not _11:42_: surprising you. That's your intuitive understanding of _11:45_: Euclidean geometry. OK, _11:48_: now when you you, you, you, you, you, you hold down astronomy one _11:52_: and you will recall you were talking about spherical _11:55_: trigonometry. The distances in the sky, on on on the sky. _11:59_: The geometry of the sky is not Euclidian _12:02_: because the trigonometry you learn is different, because _12:05_: things like the internal mechanism, angles of triangles _12:08_: add up differently. _12:10_: It's different and the same true on the surface of the Earth. If _12:13_: you're talking about, you know, large, very large scale scale _12:16_: maps, it's different. So you're you're used to the idea of _12:20_: one step, you're understanding me accompanied by changing _12:24_: geometry. And that's what we're going to discover is true in _12:27_: ratifies will Just the geometry in question here, the geometry _12:31_: of four dimensions in space and time. And the startling thing is _12:35_: those end up being closely related to each other, in the _12:38_: sense that going from one to the other isn't a isn't going from _12:42_: talking about talking about watches, but our rotation in _12:46_: that, in that geometry. Slightly mind-blowing, I thought, but you _12:50_: will get used to it, I assert. _12:53_: So this chapter is taking a first look at these ideas. _12:56_: That's the direction we're heading. _12:59_: But first _13:02_: we'll have to talk about units. _13:05_: Now the speed of light is quite, quite fast. It's it's 303% to 8 _13:11_: metres per second _13:14_: shift. _13:16_: So _13:20_: the the the things that we are, we're used to _13:24_: crawl compared to this beautiful light. So it makes sense to _13:26_: have. You might want me to try to discover that we have to have _13:29_: a little think about units, _13:32_: but you're astronomers, you know that we were talking about _13:35_: distances within the solar system or within the Galaxy. We _13:39_: have to talk about light, years and so on as a measure of _13:42_: distance because we have to big distances _13:46_: and you're familiar with that and that makes sense. And the _13:49_: standard thing that confused about is that a light year is a _13:52_: measure of time. It's not, of course. You know I like you as a _13:55_: measure of distance. Is the distance that light travels in a _13:58_: year. _13:59_: We don't have and. Light second is also a distance _14:04_: as the list that travels in a second. It's the and. And the _14:07_: radius of the sun is I think 2 light seconds. _14:10_: It takes 2 seconds for for for like to go back to that distance _14:13_: and you could talk about the light nanosecond _14:17_: the just the late travels in a nanosecond is but that that that _14:20_: far there's not very far but it's it's it's a real time. So _14:23_: we could if if we're talking about things moving at the speed _14:26_: of light, talk about things moving at late nanoseconds per _14:29_: second or something like that or late seconds per second, that _14:33_: would make sense. There's nothing wrong with that. It's _14:36_: not conventional _14:38_: or can instead do is talk about the light metre _14:42_: and then just a unit of time. _14:45_: The same idea, it just so the other way up. _14:48_: So a light metre is the time it takes for light to travel a _14:52_: metre _14:53_: and that's not, I mean that's a very small unit of time. It's I _14:56_: think 3.3 nanoseconds, _14:58_: but it's a but what that means is that light travels at one _15:02_: metre _15:04_: per light metre, _15:05_: which is very convenient. So clearly the light is a useful _15:09_: unit of time when talking about _15:12_: about things moving. It's late, so we're just at this point, I'm _15:15_: just talking about picking our units, right? _15:19_: So what we could do then is, um, _15:26_: you recall how to convert units between different, _15:32_: of course you different units. _15:35_: Have we got, _15:36_: I think this is the one that's being recorded on. _15:39_: OK. And _15:46_: can we see that? OK, _15:51_: OK, good. _15:53_: So see, we had a quantity of _15:59_: 10 Joules. _16:00_: OK. That's 10 _16:03_: kilogrammes, which is squared per second squared. _16:07_: You know, this is first year stuff. OK, _16:12_: now you you know how to convert between different units, _16:17_: you multiply or divide by the conversion factor as _16:19_: appropriate. It's also confusing. I mean, I'm supposed _16:21_: to multiplying and dividing, but there's nothing nothing exotic _16:24_: about it. _16:28_: So what we could if we do is if one _16:33_: light metre _16:34_: is and I've I've gotta try work hard here trying to which we _16:38_: update these things are is these things tend to the eight _16:44_: second _16:45_: OK so there's a very small amount of time then one _16:51_: per second is these things tend to the _16:54_: 88 metres so one per second squared is the same as 10 to the _17:01_: eight _17:02_: like metres. It's nice too. _17:06_: So 10 joules equal 10 _17:09_: kilogrammes metres squared, _17:12_: 3 * 10 to the eight _17:16_: let me just squared, which is, um some number. I think I see _17:20_: it's _17:25_: 1.1 times _17:30_: grammes. You just squared, _17:33_: black metre squared. OK now I haven't done anything at all _17:38_: exotic there. I've just changed to to from seconds to a _17:42_: different a different unit of time _17:46_: I think. _17:48_: OK, _17:50_: it's the next bit that's confusing. _17:54_: I'm going to look I I could carry on talking light metres, _17:58_: but I'm going to stop talking about light metres and just talk _17:60_: about metres, _18:01_: OK? So when I say a metre _18:05_: and it's going to be so ambiguous that I'm talking about _18:07_: metre in distance or a metre of time, I'd like metre of time. _18:11_: OK. And you think that's a really stupid thing to do _18:14_: because that creates an ambiguity. It turns out it _18:17_: doesn't. It's a bit confusing, but it turns out it doesn't. But _18:21_: look what happens if we do that. We'll see. That's 1.1 * 10 to _18:24_: the 16 kilogrammes of metre squared _18:27_: per metre squared. Ohh. So we can cancel. _18:35_: I think that's just a mistake, isn't it? _18:39_: And it does look a bit like a mistake. But if you hold on to _18:43_: that, that's what's happening here, _18:46_: and this step here just a notational convenience. You _18:49_: won't go far wrong, _18:52_: OK, _18:53_: but this does look weird. You just have to stay here for a _18:56_: bit. But all that's happening _18:58_: is East _19:00_: a decision about the time units were choosing and BA bit of _19:05_: notational trickery because what happens is the other advantage _19:09_: of that _19:11_: is that we if we ask what is the speed of light? _19:13_: Well, this beautiful light we all know is 1. _19:17_: It is 3 * 10 to the eight, _19:21_: he says. _19:22_: 2nd _19:26_: for a second but if we convert from seconds to lay metres, _19:32_: discovered that one metre per _19:36_: but like me here _19:37_: and we make the second step of just writing metres for light _19:41_: metres. _19:42_: She got that one. _19:44_: So speeds, _19:45_: for example, if you're late, are all dimensionless. Are you? Are _19:49_: all unitless. They're all unitless. _19:52_: They're not dimensionless because the speed is still _19:57_: a length divided by time. _20:00_: But because their children same units were, both the units _20:03_: cancel. Even the dimensions don't. I mean with speed. So you _20:06_: can think of that either as a _20:09_: as writing down the units where you've magically made units _20:12_: disappear, or else just think of speeds as being all quoted in _20:16_: fractions of the speed of light. _20:19_: So any speed slower than that will be some number which is _20:23_: less than one _20:24_: in units of metre per light metre or less than one. _20:29_: And _20:30_: that is all there is to the idea of what we're calling natural _20:34_: units. These are called natural units. This choice, _20:39_: I'm not going to say more about that There there, there's more. _20:41_: There are more words in the relevant section of the notes _20:44_: which which go through this in a couple of different ways. _20:48_: So I'm not gonna say that because if I see more about you, _20:51_: you think there's much that you think there's a really big deal _20:54_: happening here. There's not a big deal happening here. It is a _20:57_: bit confusing, but it's just a change of units to convenient _21:01_: set of units. And there's a couple of exercises which I _21:04_: encourage you to do _21:05_: just to settle this idea in your head. But hold on to the thought _21:08_: is that there's not any particular topic here, right? _21:13_: You will get used to these but it will take you go through a _21:17_: couple of couple of exercises and it is just the standard _21:20_: confusion of trajectory of of switching between units and I _21:24_: noticed as well. Another thing you'll see is if I say something _21:28_: like 1 _21:29_: inch _21:31_: equal to 25.4 _21:34_: millimetres and I think in in the notes I got the decimal _21:38_: point in the wrong place there because inches are weird weird _21:41_: colonial unit. And _21:43_: the other thing that you wouldn't see often is some it _21:48_: writes some writing. One is equal to 2025.4 _21:52_: millimetres _21:54_: per inch, and that's a weird way of writing about it, or without _21:58_: a weird way of writing it, _21:60_: but you're often not wrong. OK, _22:04_: so if you see something like that and and see something like _22:11_: one is equal to 3 * 10 to the eight _22:15_: which is per second _22:17_: to do _22:20_: see C is equal to 1 is equal to that, then that looks that. _22:24_: Again, that just looks looks wrong, _22:27_: but that is all that that equality there is, _22:33_: is that _22:34_: it seems The thing is that it's just that's the conversion _22:37_: factor between metres and seconds, and saying that, that's _22:41_: the conversion factor between inches and millimetres. So it _22:44_: looks wrong, but it's just a rotational, rotational fluke. _22:47_: OK, _22:49_: go in. Think about that for a bit. OK, you're all going, _22:52_: ohh, _22:57_: this is one of these things that is fundamentally not not deep, _23:01_: it's just a bit confusing. You know? There are we will talk _23:04_: about things which are fundamentally deep as well as _23:06_: being confusing. This is just fairly trivial, but looks weird. _23:09_: OK, _23:10_: hold on to that thought and I again, there's more. There are _23:13_: more words about that in the, _23:16_: in, in the notes, but I don't want to spend too much time on _23:19_: it because because I spend more time on it makes it sound _23:21_: terribly complicated and sophisticated, _23:24_: right. _23:26_: And and that's that working, that working done. _23:31_: OK. Well, just quickly, _23:33_: umm, _23:35_: OK. Well, you know that I've told you that this, but _23:39_: I'll do it. Move this back to _23:41_: and _23:45_: as a quick question, which I've I've sort of spoiled by by _23:48_: walking through the answer beforehand. And what would _23:52_: remain nuisance _23:54_: be in natural units? _23:56_: Well, you know that the answer must be kilogrammes because I've _23:59_: got I've worked this out. But you don't have any arithmetic _24:02_: for that _24:04_: because you can just think of the dimensions of _24:11_: of force. _24:13_: Ohh, no, no. What's the nuisance? I I talked to Jules _24:16_: before, _24:18_: so so OK, _24:20_: hands up. Who thinks that they Newton's is going to be that. So _24:24_: you're doing the arithmetic. Just look at the dimensions. Who _24:27_: thinks it could be the first one? Hands up, _24:30_: this is one of the second one. _24:33_: With the third one, _24:35_: we have a little chat to your colleagues _24:40_: which one it is. _25:07_: OK, who after conversation would think it was it was the first _25:13_: one, _25:14_: Second one, _25:16_: third one, _25:17_: right. Good that the majority of you got good second one. That's _25:22_: sorry answer by the way. It's really useful to talk to people _25:26_: about you know if I'm puddled explain why this answer can't be _25:30_: wrong and it can't be right. And often just explaining to your to _25:34_: the cat why they the the the reason for your incomprehension _25:38_: that can somebody resolve problems. It's the second one _25:42_: because _25:43_: the conversion to natural units. _25:46_: A conversion to units. _25:48_: Well, we're not using seconds, but we're using light metres, _25:52_: so the 9 kilogramme metres per second squared _25:58_: would be converted into something kilogramme metres per _26:01_: light metre squared. _26:03_: If we turn the light metres into metres, just renovate them, _26:07_: that's metres per metre squared, so it's per metre. _26:10_: So that to kilogrammes per metre are the dimensions of the units _26:15_: rather of _26:17_: force. In these units, the dimensions are the same. They _26:22_: are mass length T 2 -, 2. _26:26_: But because of the units we've chosen some, some of the of the _26:28_: units cancel. _26:31_: Good. _26:33_: We have the right speech, _26:37_: right? No. How do we _26:40_: visualise _26:43_: motion? _26:44_: This is a perfectly familiar sort of plot to you. _26:51_: Distance _26:53_: and versus time. So as time. Yep, I'm teaching you about _26:56_: truck X here. You understand this. But the way graphs work is _27:00_: as time moves along. Here _27:03_: you you plot the distance at that time. OK so this _27:07_: first so this is imagine a a light bulb flashing. So it's _27:11_: it's flashing as it as it moves about. _27:16_: If the light bulb is just staying where it is, _27:21_: then as time moves on, it's expedition just stays. Flash _27:25_: flash flash flash flash flash flash along E _27:29_: Nothing complicated there. If the light bulb starts _27:32_: accelerating so it's moving faster and faster along a then _27:37_: you'll get a graph like that of the flashes as the thing _27:41_: accelerates along the X axis. _27:44_: OK, _27:45_: I'm really upset about that and The thing is moving well. We'll _27:49_: choose the the units on the on the axes to be metres and light _27:53_: metres. _27:54_: If things are moving at the speed of light then it will move _27:58_: at one metre _27:60_: per light metre 1 metre per metre _28:03_: I along the diagonal. _28:06_: OK _28:08_: and if you had an event, so all these are events. All these dots _28:11_: are events _28:13_: and and and if we join the dots in each of these cases then we _28:17_: get what we call the warplane. The world lane is the set of _28:21_: points in space and time that an object goes through. _28:26_: OK, it's it's a join the dots of the all these and if an event we _28:30_: call event one then that will happen at a place in the time _28:35_: and we can plot that on the diagram. OK, at at at _28:39_: event one has X coordinate X1 and time coordinate T1. Why am I _28:42_: making so much of A fuss about this? There's nothing _28:45_: complicated about that. OK, you have entirely familiar with. _28:52_: That's the same diagram, but flipped so that _28:58_: the X axis is along its horizontal and the same axis is _29:02_: vertical. _29:03_: Why? Because it's traditional, _29:06_: OK. And it does make some sort of sense, but because _29:11_: makes some sort of sense. But the point of this change of _29:14_: slides is that is exactly the slide, the graph you're familiar _29:18_: with but just flipped on its side and it's called the _29:21_: Makovsky diagram. When it's done that way, it's quite useful just _29:25_: to and you'll see these again and again and again and again _29:29_: and again in this course. OK, _29:32_: end up looking a bit confusing, but the point of the game is you _29:35_: build it up slowly while thinking through the problem _29:39_: in a way. The only so the two things that make up a graph of _29:43_: the coffee diagram is this, this tick, this, this weirdness of _29:47_: flipping the axes, and the fact that we always implicitly assume _29:51_: that the that the units are are the same in the horizontal and _29:54_: vertical _29:56_: actually access. Which means _29:59_: that things were to move at the speed of late, always move _30:03_: at 45 degrees, _30:06_: and things were moving less than speed of light _30:08_: move at a gradient which is bigger than 45 degrees. _30:12_: OK, that's OK. So you will _30:18_: so so so they are the world line A. _30:21_: Maybe something just resting at X = 0 _30:25_: it's a successive times B is accelerating and sees movement. _30:29_: People like OK, but there's nothing weird there _30:33_: that hand up or just we have the _30:38_: right. _30:40_: So let's use that, use the McCarthy die or let's build up _30:44_: some other Mickey diagrams, describe what we've seen before. _30:47_: Because the the point of when Gotti diagrams is that, _30:52_: well, I did. I see this last time. I think it may have done _30:57_: the _30:58_: but but but the assessments in relativity with exams and _31:01_: relativity and special relativity, there's basically _31:03_: only one question that can be asked. _31:06_: Now here are some events in this frame. What are the coordinates _31:09_: event in the other frame? That's basically the only question that _31:12_: gets. It is dolled up in a variety of ways, but that's _31:15_: basically the question. _31:18_: And it it in all cases the the thing that's hard or the thing _31:22_: that makes it challenging. The question is there to test is for _31:27_: some situation, it's described in words, you know this thing is _31:32_: moving and this thing happens. How do you turn that into _31:37_: a set of events _31:39_: with coordinates? You can then turn into the other other frame. _31:46_: So the way you do that, the way you go from a description of _31:50_: this, this thing is moving in this way to assess events is you _31:53_: focus on the idea of the machine diagrams. The McCaffrey diagram _31:56_: is a way of you organising your thoughts _31:59_: and organising the translation of something interesting events. _32:04_: So what we're going to do is look at this _32:08_: again, again, this idea of the of the, of the, of the flash _32:10_: bulb, you know going off in the middle of the train carriage. _32:13_: It flashes reflects from the mirrors at both ends, and the _32:16_: light comes together again in the middle. _32:21_: So how do we draw those? How do we draw that in the Minkowski _32:25_: diagram? _32:28_: OK. _32:31_: And what we do first if we get some more paper, _32:42_: so let's talk about the so. So I describe that in words. Light _32:45_: flashes reflect come back. We're going to turn that into set of _32:49_: events _32:50_: through there are four events there. _32:53_: Event one is _32:56_: right _32:58_: flashes. _32:59_: Ohh no step no step 0 _33:05_: frames. The frame S prime will be the frame _33:10_: of the _33:11_: A tree _33:13_: and the frame S will be the platform. _33:18_: OK, if in one if the flash _33:23_: event 2 will be a reflection _33:27_: from around my flat from the front _33:36_: event three will be reflection _33:39_: from the back and don't fall will be the flashes light _33:45_: arriving. _33:48_: OK, that's eligible. But you know, I'm saying, so let's draw _33:53_: those events on the musky diagram. _33:57_: So we'll draw the _34:01_: the ex prime _34:05_: and T frame axes. So these are the other the coordinate axes in _34:09_: the in the train, in the train carriage. _34:13_: Well see the the origin of these of of the of of of this _34:18_: coordinate system is that is that the light is that at the _34:22_: centre of of of the train carriage. OK, so that event one _34:27_: happens at X prime coordinate equals 0. So it happens _34:33_: along that that frame axis, and we'll see it and we'll see that _34:37_: this carriage is 6 metres long, _34:40_: quite short carriage, 6 minutes long. OK, _34:43_: so it's 3 metres from the centre to each end. So we'll say that _34:48_: the first event happens at _34:52_: X1. Primed equals 0 _34:55_: T one prime equals -3 metres. Could we get to choose our _34:59_: origin? _35:01_: So we'll choose them so they're sensible. So that that means _35:05_: that's event one there, _35:09_: right? And that's and that's T1. Prime equals -3 _35:14_: metres _35:16_: to the right. Flashes _35:18_: should the light goes forward and backwards at the speed of _35:22_: light. It being light, _35:25_: what does that look like on the coffee diagram? Remember that _35:28_: things which are this people like in the midfield diagram _35:31_: move at an angle of 45 degrees, so the light moving forwards _35:34_: when we plot it looks like that _35:39_: and the light moving backwards _35:43_: looks like that. _35:45_: OK. _35:47_: And the light moving forwards, we'll move three metres forward _35:50_: and three metres of time _35:52_: and so it'll get to the point where it reaches. _35:57_: I thought frame equals zero, _35:59_: that's the time when it got to event two. _36:02_: So we can put event 2 on this. On this diagram, _36:05_: event two _36:07_: is there _36:09_: and event three is there. _36:12_: So we know that X2 primed is equal to 3 metres. We know that _36:16_: anyway because the any event which happens at the front metre _36:21_: at the front mirror is happening at an X coordinate of plus three _36:25_: metres, _36:27_: OK? And we know that any event that happens at the back mirror _36:30_: is happening at an ex prime coordinate of -3 metres. _36:36_: So we knew that X2 prime is equal to 3 metres, _36:41_: X3 prime equal to -3 metres _36:46_: and we've just worked out that because of our choice of origin, _36:51_: two due primed is equal to zero. _36:54_: D3 primed is equal to 0. _36:58_: OK, _36:60_: now this is just speed. This is the speed times time you learned _37:02_: about this in school _37:05_: and they reflect there and the light goes back in the opposite _37:09_: direction. _37:11_: In other words, it move, it reflects from three and goes _37:15_: at the speed of light in the positive X direction. And _37:19_: if you write the negative expression, and of course they _37:23_: meet again at event four _37:26_: and then 4X4 _37:28_: primed is equal to 0, it's happening at the origin again _37:31_: the spatial origin. _37:34_: And the time _37:35_: of that _37:37_: is 3 metres, because it takes 3 metres of time, 3 metres of time _37:42_: to get from the mirror back to the centre. _37:46_: And so we have completed. _37:48_: And of course your diagram _37:51_: in the _37:54_: prime. The prime frame. _37:56_: No, I'm doing this. Very sketchy. OK. I'm just scribbling _37:59_: live in front of you, _38:02_: the assessments. _38:04_: Well, you'll be required with that question, yeah. So what _38:08_: does the label label or #4? _38:11_: Ohh _38:12_: the that the light gets back-to-back to the centre so _38:16_: that the the reflection we arrives at the back of the _38:20_: light. So the _38:24_: in the future like sizes and the in in the class test and the _38:27_: exams, you'll be required to do things like draw that Drummond _38:30_: coffee diagram. _38:33_: You're right, if what I I get looks like that, you'll get _38:36_: marks off. That's a mess, OK? _38:39_: And they will take marks off if it's a mess because you, you, _38:43_: you, it'll be a mess. The first version you do will be a mess. _38:46_: Then you think, ohh, no end it. You draw a fair copy, right? _38:49_: Because the communication here is, you know, part of the whole _38:53_: part of the scientific communication thing. But it also _38:56_: is about explaining clearly. You're working, explaining _38:59_: clearly. You're the process of thought. And in a sense it's the _39:02_: it's the explanation that you're getting marked on. You're _39:06_: showing that you understand that you have just pick some numbers _39:09_: out of the air, _39:11_: So that's a little right, which I will probably repeat more than _39:14_: once, _39:16_: but they're supposed to be all 45 degrees. They're a bit _39:19_: squashed there, _39:23_: so we have assembled our recovery diagram. Note I was _39:26_: explicit about the what the frames were. I wrote that down. _39:32_: That's important. I was explicit about what the I about the _39:34_: events that I identified and I thought about and thought these _39:37_: are the events that happened, the things that have happened at _39:40_: a place and time. _39:42_: I wrote down what the events were in words _39:45_: and I worked out one way or another. This is meeting time of _39:49_: what the coordinates of those events were, and I plotted them _39:52_: on the Minkowski diagram in a nice neat eventually way. Those _39:55_: are all steps that you have to go through _39:58_: and you'll get faster. _40:00_: So far so good. Now let's just do the same thing in the other _40:05_: frame, the frame of the platform, _40:08_: and see what happens. _40:11_: And this, _40:13_: um, _40:17_: so the frame, that other frame _40:19_: that that would be a bit more thought, so that frame. _40:28_: XT _40:30_: No prime T, prime X&T. _40:35_: And _40:36_: there's one thing we can draw immediately, which is we know _40:39_: that the train carriage is moving through this frame. _40:43_: OK, _40:44_: that's, you know, the basic setup _40:47_: and we know that the world line of the centre of the carriage _40:52_: is moving at speed VV. It's some number _40:56_: which means we can draw the world line of the centre of the _40:59_: carriage, the world of the length of the light flash of the _41:01_: late bull _41:02_: immediately. _41:04_: That's what we go to the middle there _41:07_: at the water line of the train. _41:12_: But if we look at the first diagram, _41:14_: we can see that the. _41:16_: OK, hold on to that, we'll talk about that. _41:19_: Welcome. We will come back to that later. So that's the water _41:22_: line of the, of the, of the, of the train carriage _41:25_: of the centre of the train garage. _41:27_: But we know _41:29_: from looking at the top diagram the events one and four happened _41:32_: on that world lane. _41:35_: Who they happened on that world line. It happened if they _41:38_: happened at at the centre of the train carriage in the 1st frame _41:41_: and they happened at the centre of the train carriage and the _41:44_: 2nd frame. So events one and a four must be on this model line. _41:49_: So events one and four are something like there _41:53_: and _41:55_: there. _41:57_: This is not a scale drawing. Minkowski diagram has never _42:00_: scaled, but there's something like that. _42:04_: But also _42:06_: we know that event one was a light flash. _42:09_: The late moves forwards and backwards _42:12_: and move forward and backwards. At what speed? _42:16_: Beautiful light so that the water line of that of that light _42:19_: flash has moved? At what angle? The majority diagram _42:23_: 145 reads yeah, at speed one or angle 45 degrees. OK, so it _42:29_: looks like that. _42:33_: It looks like that. _42:36_: Good. _42:38_: And when it reflects and comes back and it comes back and they _42:43_: both arrive together at _42:46_: the at event four _42:49_: up there, but we think reflect back they are also moving at the _42:52_: speed of light. So they are also on this diagram moving at 45 _42:56_: degrees _42:57_: that the late flash. _42:59_: So when they arrive back at event four, _43:04_: they are moving at _43:06_: 45 degrees. _43:08_: OK, so so that's good. But we also know _43:15_: that the point where the looking at the top diagram, the point _43:18_: where the light flashes turned round _43:21_: it events to an event three. _43:24_: So from that drawing we can see where it would plot events 2 and _43:28_: event 3, _43:32_: namely _43:36_: and three. _43:39_: From the top diagram we can see that the that that _43:45_: events one in four happen on the on the on the on the T prime _43:49_: axis. _43:51_: So we can draw the location of the T frame axis on this new _43:55_: diagram. _43:57_: It's just that. _43:60_: And we also just can see that events _44:03_: two and three happened on the X prime axis. _44:07_: So we can draw the position of the expert Maxis _44:11_: of this diagram as well. _44:17_: So we've worked out what the most country diagram in the _44:22_: UM station plane frame is. _44:25_: Rather messily, I reiterate rather messily. _44:30_: And it looks a bit different. _44:32_: Why does it look different? It looks different because in both _44:35_: cases the speed of light is the same. In other words, the speed _44:38_: of light the the the the world line of light flash is 45 region _44:41_: both cases. That's the second axiom again. _44:45_: But look at what we've discovered. _44:49_: The _44:51_: time coordinates who events two and three. So the time _44:55_: coordinate of of of event two is about there T is, T is plus _44:59_: something, _45:01_: and the time coordinates in this frame of van three is _45:05_: T equals minus something, and they're not the same. _45:09_: In other words, in the top frame event is 2 and three with _45:12_: simultaneous, _45:14_: unsurprisingly because if you look at the picture of what's _45:18_: happening, the the lights get to both ends at the same time. But _45:22_: in the moving frame, _45:24_: the light gets to the is reflected from the from the back _45:28_: earlier, it reflects from the front. And that's the same thing _45:32_: that we reason through _45:35_: in the last lecture, _45:36_: just did it in words. We're doing the same thing here, but _45:40_: in a diagrammatic form. _45:43_: So that is the relativity of simultaneity, _45:46_: this phrase that I used last time. Two things being sipped _45:49_: being simultaneous is framed dependent. _45:53_: If it's two or three are simultaneous in that frame. _45:55_: They're not simultaneous in another frame _45:59_: because they are physically separate. _46:01_: To vent a repeat which are which are simultaneous at the same _46:05_: place. _46:06_: You know two car crashing _46:08_: are simultaneous in every frame. That's that is absolute. That's _46:11_: unequivocal to eventually simultaneous but specially _46:14_: separate. _46:15_: That it was unity is relative, _46:17_: and as you can see that made coffee diagram looks a bit of a _46:20_: mess. Not just because of, I've just drawn it quickly, but _46:23_: there's a lot of lines happening there. _46:26_: If you look at that, you look at the end result, you go. That's _46:29_: terrible, confusing mess. _46:31_: But it's not if it's the end result of you working it out. _46:36_: So if you look forward in the notes, you'll see lots of _46:39_: McCaffrey diagrams and they all look very intimidating. There's _46:42_: lines and curves and all sorts of other space. That's because _46:45_: they are the end result of a thought process. _46:51_: So I've I've spent quite a lot of time on that. _46:58_: I will pick that up next time _47:03_: because I just want to go quite carefully through the idea, but _47:07_: it's because it's so important. I'll have to speak up a little _47:10_: bit next to to to catch up myself. I aim to get to the end _47:13_: of Chapter 4. Next time I'm going to move on to Chapter 5, _47:17_: which is the main, the main deal. We'll see you next time,