Transcript of gr-l01 ========== _0:12_: Hello, I'm Norman Gray. _0:14_: You are a 345 as we have, _0:19_: just fairly conclusively established. _0:21_: We're all back to face face teaching. _0:26_: It's been a very hard couple of _0:28_: years and I think some of you will. _0:32_: None of your third years, is that right? _0:33_: You're all 4th and 5th years, right? _0:35_: So all of you will have had at _0:37_: least one year at university. _0:38_: But your second and third years _0:39_: maybe have been a bit complicated, _0:41_: but well done cleaning, _0:42_: making it through one of some of you _0:45_: I think I may have taught special _0:47_: relativity to in a 2 two years ago. _0:50_: Yep, I'm seeing some nods. _0:51_: Good, good, good. Then you are you. _0:54_: Some of the diagrams you will _0:57_: will be familiar to you. _0:59_: OK, without too much further ado, no. _1:03_: Before I talk about that, _1:04_: what I do want to do is mention _1:07_: as Professor Contar did, _1:09_: the Moodle and a couple of things _1:11_: that are on the relevant page. _1:14_: So. _1:17_: Um. _1:20_: We go there, we go there. Umm. _1:29_: Well. _1:37_: No, I haven't forgotten my. _1:45_: OK, right, as long as equal 5. _1:52_: Cycle B. I want to draw _1:54_: attention to a couple of things. _1:56_: First is there's a folder of lecture _1:59_: notes there which contains lecture notes. _2:03_: A whole little more say _2:04_: about these in a moment, _2:06_: so I'll come back to that. _2:08_: Very shortly, I also want to _2:10_: draw attention, however. 2. _2:16_: Padlet. And padlet is. _2:22_: And. Either to show them to describe _2:26_: it's various bits of fragments of _2:29_: questions if you have a question. _2:31_: Double click on that. Ask a question, _2:34_: but a bit of extra content context. _2:37_: It's all anonymous, _2:37_: have no idea who's asking questions. _2:39_: I will sporadically, _2:40_: but not entirely reliably check that _2:43_: and trying to update the questions _2:45_: that are there or just ask me. _2:48_: This is for the shy. _2:51_: For those who think they are _2:52_: slightly complicated question, _2:53_: they want to to to to work _2:54_: out what how to get down. _2:55_: So all all modes of asking _2:58_: questions are good. Umm. _3:01_: There's also a Microsoft Stream channel. _3:06_: Unlike previous years, _3:07_: that's not a core part of the whole thing, _3:10_: but there will be one or two things there. _3:12_: I do have from previous years _3:13_: some brief 5 minute videos, _3:15_: actually overviews of the various _3:16_: parts and I will post them there. _3:19_: I think they're quite useful just _3:20_: to to be useful to let you getting _3:22_: over all impression of what the _3:24_: the uppermost part is going to be. _3:26_: So we'll post them there, _3:28_: but that's not a big going to _3:29_: be a big deal for this year. _3:32_: OK, let's move away from the middle. _3:37_: Get back to the slides. _3:39_: I don't have a a bigger big thing _3:43_: with slides because the other _3:46_: pages ohh I'd mentioned ESR. _3:48_: I I think you should be able to see that _3:51_: my A2 like special election notes you might. _3:55_: Enjoy those. _3:57_: If you if you want to see those you _3:59_: can't see them then let me know. _4:01_: And. _4:03_: Now before I go on to the onto this, _4:05_: I'll say that the the way I've developed _4:09_: of teaching this and added at the SSR is. _4:14_: Feeling specific? _4:14_: I do put the lecture notes, _4:17_: which are, you know, _4:18_: quite a lot of text up beforehand. _4:21_: And what I want you to do is _4:23_: look at those beforehand, _4:24_: not read them through and _4:26_: digest and fully understand, _4:27_: but look at them to get an impression of _4:29_: what's coming up so that in the lecture, _4:31_: the lecture isn't a surprise. _4:33_: The the notes there are spoiler _4:34_: alerts all the way through. _4:35_: You know they're there for you _4:36_: to to to not be surprised when I _4:38_: start talking something, right? _4:39_: The lecture is the main event. _4:42_: That's me explaining it to you. _4:45_: I will. I will often allow details. _4:48_: I'll say the details are on the notes, _4:50_: or read about this year, _4:51_: or read this in this book. _4:52_: I'll aim to communicate the _4:54_: main idea in the lecture, _4:56_: but I won't always fill in the details, _4:59_: because after the lecture you'll be _5:01_: able to go back to the notes and go. _5:03_: Now I see the point. _5:05_: So it will be possible I think. _5:08_: To just go to the lectures _5:10_: and ignore the notes and pass. _5:12_: It would be possible, I think, _5:13_: to ignore the lectures and just read _5:15_: the notes and pass, but it'd be harder. _5:18_: In both cases. _5:19_: The expectation of the plan is that _5:21_: the two can complement each other. _5:23_: There's not a huge amount of _5:25_: stuff to learn in this course, _5:28_: it's just what there is is quite hard. _5:30_: And it's a case of understanding _5:32_: rather than accumulation of knowledge. _5:34_: And that means a couple of things. _5:36_: It means you do have to think. _5:37_: It means you do have to assemble _5:39_: an understanding from the various _5:41_: resources that are available. _5:42_: And it also means you can't cram it all _5:45_: at the end. That really doesn't work. _5:47_: I mean, with some courses, you know, _5:49_: OK, don't do that, but you know, _5:51_: you can sort of get away with it. _5:52_: And this one I will Part 2 bills on part one, _5:56_: Part 3 builds on Part 2, Part 4, _5:58_: absolutely build on part three. _5:59_: So if you're not, you know, _6:01_: getting some sort of idea of what's going on, _6:02_: it will get progressively harder. _6:04_: OK. Um, I do record the lectures. _6:11_: Audio, however, that depended on _6:13_: batteries and me not accidentally _6:16_: turning the recording off and stuff, _6:19_: so don't depend on that. _6:20_: I'll put those up once I sort of _6:22_: got a couple of technicalities, _6:24_: but so there's sort of revision aid rather _6:26_: than as something should depend on. _6:31_: I'm getting going with the the _6:32_: actual relativity in a moment, _6:34_: but the last thing I want to mention _6:36_: in a sort of pedagogical sense is _6:38_: that although it's a couple of of _6:40_: pedagogical fashions that are date, _6:42_: I do like aims and objectives. _6:44_: And the distinction to them _6:46_: between them is clear to me. _6:47_: In the aims are the point of the _6:50_: course that the things that the _6:52_: why you're doing this course, _6:54_: the why I'm teaching it, _6:55_: that the things you'll remember _6:56_: after the course is finished, _6:58_: you know, in years to come. _7:01_: And there are things they appreciate, _7:03_: understand. _7:03_: Quite general, quite general things, _7:06_: quite high level things. _7:08_: The problem, that's the point _7:09_: that that's why you're doing this. _7:11_: The aims. _7:12_: The problem with that is it's easy to say, _7:14_: Oh yes, I understand that, _7:15_: I appreciate that, but. _7:17_: You can't do an examiner appreciation so. _7:21_: There are also companions to them, _7:23_: not a one to one relation, _7:24_: but companions objectives and _7:26_: these are the party tricks. _7:29_: These are the things that you can do. _7:31_: They are explained, quote, explain. _7:34_: There are things you can do, _7:36_: which are the sort of things _7:37_: you can do in an exam. _7:39_: So when I write the exam, _7:41_: I have the list of objectives in front of me. _7:44_: And the exam is basically conformant _7:46_: or consistent with those objectives. _7:48_: Not a sort of precise 1 to one thing. _7:50_: But if somebody is clearly not _7:52_: covered in the objectives, _7:53_: then I think that's for the. _7:55_: If it's broadly mentioned, you know, _7:57_: alluded to in the objectives _7:59_: then that's in there possible? _8:01_: So that's the distinction for me. _8:02_: So these are not exciting things. _8:05_: But the things you can do and the _8:09_: exercises that are attached to the. _8:11_: To debate, _8:12_: parts are keyed fairly precisely _8:15_: to these objectives. _8:16_: OK. _8:18_: I think I'm about to start _8:20_: doing relativity now, _8:21_: so the administrative and _8:22_: pedagogical things are over. _8:24_: Are there any questions about _8:26_: where we've got to so far? _8:29_: Any questions, anxieties, worries, _8:32_: neuroses or other dislocations, _8:35_: psychological dislocations of that type? _8:38_: None of the English people admit to OK. _8:40_: That's not necessarily good. _8:41_: I mean, _8:41_: I like questions because the _8:43_: questions are good because they _8:45_: help me understand if I'm going _8:47_: too quickly or too slowly. _8:48_: So I like questions. _8:50_: OK. _8:53_: Still nothing. OK, well, _8:54_: that's that's that's good, _8:55_: that's good, that's good. Right. _9:01_: You've seen this before. _9:03_: This is Newton's second law. _9:06_: Force is proportional. It is. _9:08_: It is proportional to the _9:09_: rate of change of momentum. _9:11_: You're familiar with that. _9:13_: That's not wrong. _9:14_: We're going through, _9:15_: we're more safe with that, _9:16_: but that's that's good. _9:22_: And I did have the other one, _9:23_: I thought another version of that. _9:27_: The key thing about this, _9:28_: from our point of view is _9:30_: that it's a vector relation. _9:32_: It says that the force vector is. _9:37_: Proportional to the rate of _9:40_: change of momentum vector. _9:42_: And you say, well, yeah, _9:43_: that's is that that's not really _9:45_: a big it is a big deal because _9:47_: that is a geometrical statement. _9:49_: It's seeing these are these things _9:51_: have a magnitude and A and a direction, _9:53_: and that side is proportional to that. _9:55_: We're not talking about coordinates, _9:57_: we're not talking numbers here. _9:58_: Just saying that's proportional _9:60_: to that is a physical statement. _10:02_: You can imagine a world, but that's not true. _10:05_: A science fiction world, but that's not true. _10:07_: But in this universe, that's true, _10:08_: and that's a very big deal. _10:10_: And it's true whatever the coordinates. _10:13_: Either you pick. I mean it in most cases. _10:18_: And in most cases you you'll solve that, _10:21_: but using Cartesian coordinates. _10:22_: It might be that the case where you _10:24_: solve that using what you know, _10:26_: fair polar coordinates and the masks will _10:28_: look very different in each of those cases. _10:30_: Possibly very hellish _10:31_: complicated one of those cases, _10:33_: but the basic geometrical _10:35_: statement is the same in both, _10:37_: and that's what we're holding on to _10:39_: all the way through this course. _10:43_: And indeed, that is the principle that _10:46_: that that statement is the principle _10:49_: of general covariance, which the grand _10:51_: we have seen what I've just said. _10:53_: Which is that all physical laws have _10:55_: been under coordinate transformations. _10:57_: All physical laws must look _10:59_: the same or be the same. _11:01_: Independent of the coordinates _11:03_: you choose to describe them. _11:05_: So it doesn't matter if I choose Cartesian _11:07_: coordinates to do to to to work out the _11:10_: consequences of F equals of people, _11:12_: city or or use spherical pullers, _11:14_: they're just numbers. _11:15_: That's just maths the physics, _11:17_: the physical laws must be independent _11:20_: of the choice of coordinates. _11:23_: You think and and that doesn't _11:24_: sound like a big deal. _11:25_: I mean you think, well, _11:27_: that's sort of could be us in next mean. _11:28_: That's certainly been my experience in in _11:30_: in the physics I've learned in the past, _11:33_: but it's not just a nice thing. _11:35_: It turns out to be massively important, _11:36_: and Einstein beast the whole of GR on that. _11:39_: That's that's where all of GR comes from, _11:42_: that statement. _11:43_: The principal general comedians the _11:45_: demand that that be true is where _11:47_: everything else is right falls out from. _11:52_: Um and? And the principle relativity here, _11:58_: I don't see anything that I _12:01_: forgot to say here. Yeah. _12:03_: And we can combine that with _12:05_: the principle of relativity. _12:07_: Which is that? That's not I mean, _12:10_: I illustrated that with F equals PDT using. _12:17_: Basically accommodation _12:18_: first and second laws. _12:21_: It's not just through mathematics or _12:23_: mechanics, it's true of all physics. _12:26_: So Einstein here is saying, you know, _12:28_: we the principle companion applies to _12:30_: all of physics, not just mechanics. _12:31_: What we'll we'll talk about mechanics _12:33_: most of the time because it's _12:35_: straightforward talk about and we _12:36_: can all understand that very well, _12:38_: but it's placed all of physics. And. _12:43_: What? Although the although the _12:46_: equivalently but not obviously identical. _12:49_: It is a. It is equivalent, _12:51_: but is that the link between them _12:53_: is not trivial. _12:54_: No experiment performed wholly _12:56_: within one local national frame. _12:58_: Can detect its motion related to _13:00_: any other local natural frame now? _13:02_: The words local, _13:04_: inertial and frame are all important there. _13:07_: A-frame just a coordinate frame. _13:09_: It's. It's a coordinate system. _13:10_: It's an XY and Z. Very little he's been. _13:18_: A Freeman just coordinate system. _13:19_: It's a choice of where your origin is and _13:22_: where and where the axes are. Inertial. _13:26_: From your recollection of special relativity, _13:28_: you will remember inertial natural _13:31_: means it's something we're using. _13:33_: Laws work basically. _13:34_: We'll have more to say over that is _13:36_: what we'll we'll we'll complicate _13:37_: that a little bit in a moment, _13:39_: but that's basically the same idea _13:41_: or most of the same idea as you've _13:43_: as you know from special activity. _13:45_: And local is important because _13:47_: what we're talking here is the _13:49_: local part of our national frame. _13:52_: We're not talking with things _13:54_: that are far away from us. _13:56_: Because and and that will _13:58_: become more important later on. _14:00_: Um, but the word local is is meant to _14:03_: mean within a short distance of here. _14:06_: Now, what does short mean? Short? _14:08_: Depending on the experiment you're doing, _14:10_: the moment you're doing should be a meter. _14:11_: It could be a party, _14:12_: but the point is, it's not infinite. _14:15_: And it also means short in terms of time. _14:18_: Could be 10 seconds or it could be years. _14:21_: Or millennia, whatever. _14:22_: So the the point is it's it's it's a bounded, _14:26_: the bounded box within which your your, _14:28_: which is your local and national, _14:29_: the local part of your national frame. _14:30_: And this says in that you can't tell. _14:35_: You can't tell which of the of the of _14:37_: the base bottleneck frames you're in. _14:39_: That is equivalent to the thing you _14:41_: might remember from special activity. _14:42_: You can't tell if you're moving, _14:43_: but this is more profound than that. _14:46_: And so. Yeah. _14:50_: Are there any questions at that point? _14:53_: OK. _14:55_: And. _14:55_: Another thing that is unexpected here, _14:59_: which could drop in here but and _15:01_: and leave slightly hanging. _15:05_: No, I'm ugly saying, _15:06_: but we're going to pick up again, _15:07_: which seems to be unrelated to _15:09_: the the two statements I've just _15:12_: made is the strange business of _15:14_: inertial and gravitational masses. _15:17_: No. You don't love gravitation? _15:20_: That's F = g Big M / r ^2. _15:24_: Says that the gravitational force _15:27_: between two objects is proportional. _15:30_: To the gravitational mass of those objects. _15:32_: With the gravitational mass, that's the _15:34_: amount of gravitational charge you like. _15:36_: That's the amount of stuff that _15:38_: the gravitational field acts on. _15:40_: OK, so that's a it's how. _15:43_: How? How? Coupled based? _15:44_: How coupled with gravitational _15:46_: fields is this mass? _15:47_: And the government voted proportional to. _15:51_: Accelerate the acceleration. _15:54_: Is proportional. _15:56_: Invest proportional to the mass, _15:58_: the inertial mass. _15:59_: So if you push something. _16:01_: Then how much accelerates is a _16:04_: function of how inertial it is. _16:07_: A heavier thing, I think with _16:09_: more inertial mass will resist the _16:11_: pushing more and accelerate less. _16:13_: And that inertial mass is is that is _16:16_: the amount of resistance to being pushed. _16:20_: Nothing to do with gravity. _16:21_: It is resistance but and and and and and and. _16:26_: Galileo's contribution to _16:27_: physics was essentially walking, _16:28_: discovering that and and Newton _16:31_: something tied that down _16:34_: mathematically in in existing laws. _16:36_: But girly with this business of _16:37_: dropping things over the Leaning _16:39_: Tower of Pisa should this is what _16:41_: you thought you were showing. _16:43_: But what he did show was that _16:45_: this gravitational mass. _16:46_: And this inertial mass? _16:49_: Are precisely proportional. _16:51_: So if you double the amount of _16:54_: resistance a mass has to being pushed. _16:56_: The only way of doing that? _16:58_: Also double s the coupling _17:01_: to the gravitational force. _17:03_: Even though these two things have nothing _17:04_: to do with each other as far as we can see. _17:06_: You know what? _17:07_: What would you've never thought of that of, _17:08_: of, of them being distinct. _17:09_: But in fact they have nothing _17:10_: to do with each other. _17:11_: And yet they are precise, _17:13_: proportional to each other. _17:15_: And This is why you've never heard _17:16_: of these terms of gravitational mass, _17:17_: international mass? _17:18_: Because I saw some of you looking so worried. _17:20_: Am I supposed to know the national mass? _17:22_: No, you're not. _17:23_: Because it's never been used before? _17:24_: Because I introduced here in _17:26_: order to remind you that that _17:28_: distinction doesn't matter. _17:29_: But there's no explanation for that. _17:32_: In your in Newton physics. _17:35_: That's a completely unexplained _17:37_: thing in Newtonian physics. _17:40_: And and and I'm not sure _17:41_: it's a question of history, _17:42_: of of science and how how much _17:43_: Newton was worried about that. _17:45_: I think there were some things Newton _17:47_: didn't like the law of universal gravitation. _17:49_: He thought, he thought, _17:50_: that can't be right. _17:51_: But the there are physiological _17:52_: problems with this, _17:53_: with it from his point of view. _17:54_: But I don't know if he would _17:55_: worry about that. _17:56_: But he should be because there's _17:57_: no explanation for that. _17:58_: Insurance physics. _18:02_: So that's a puzzle. _18:06_: How do we approach that? _18:08_: Imagine you are in a box. _18:13_: A special out in floating around somewhere, _18:17_: very far away from gravitating masses, _18:19_: and in there there's a _18:21_: bit of mechanical stuff. _18:23_: Watch, but of biological stuff. _18:26_: Some, some, some, _18:26_: some other bits of energy, _18:28_: photons and so on, _18:29_: and they're all wondering they're _18:31_: floating around because you're floating _18:33_: well away from the application masses, _18:34_: you're just floating around. _18:37_: And if you push him off the wall, _18:38_: you move in a straight line at a constant _18:40_: speed until you hit the other wall. _18:41_: Nuisance laws work. _18:42_: In other words, they really, genuinely work. _18:44_: I mean you. _18:45_: You push him off and you stay. _18:47_: You move in a straight line at a _18:48_: constant speed until until you hit something. _18:50_: And so on. _18:51_: So Newton's laws work there. _18:56_: Good. No, I trust no one surprised at that. _18:60_: OK, so you, you've got that picture. _19:02_: I'm not telling you anything new here, _19:03_: I'm just reminding you that that that's _19:05_: in fact you do think of that picture _19:07_: and see that makes that makes sense. _19:12_: No, we put our. _19:13_: A rocket under this very big rocket, _19:16_: but a small rock. And we push the the box. _19:21_: We don't push things in the box, _19:22_: we just push the box. So what happens? _19:25_: The people stay where they are, _19:27_: or people in the and and and the _19:28_: and the clock and the and the and _19:29_: the pot plant stay where they are, _19:30_: but the box moves around them. _19:33_: And they don't. _19:35_: And what we're doing this _19:37_: is there's no air in this. _19:38_: Rain is complicating, _19:39_: but the box will just move around _19:41_: them until they hit the ground, _19:43_: until they hit the bottom surface, _19:46_: and then they'll stay there _19:47_: and they will accelerate at _19:49_: the same rate as the box does. _19:51_: That's not surprise. _19:55_: So, and if the rocket is designed _19:59_: to with the right force, _20:02_: then it can make accelerate _20:04_: this at 9.81 meters per second _20:06_: squared and even inside will go. _20:08_: I can't tell good Earth or not. _20:11_: And they will be uncertain. _20:15_: I like to think what experiment you might do. _20:18_: They would let you tell _20:20_: the difference between. _20:21_: Being in that situation and being on Earth. _20:25_: You would think of what an _20:26_: experiment that would be. _20:27_: How would you tell you where you were in _20:29_: the situation rather than being on Earth? _20:35_: The only one I can think of is having _20:38_: someone in this side of the box, _20:39_: that side of the box with _20:43_: very precise inclinometers. _20:44_: And if you're on Earth, _20:46_: then they would sleep. _20:47_: They point toward the center of the. _20:50_: And if they were in this situation, _20:51_: they point they both point down, _20:52_: but that wouldn't be a local frame. _20:55_: Because the word local here, _20:57_: and we're talking to the local frame, _20:59_: the word local means it's small enough _21:01_: that you can't do that sort of stuff. _21:03_: That your experiments aren't precise _21:05_: enough to be able to detect that so _21:08_: within the bounds of your of your _21:10_: experiments that you can do 2nd _21:13_: order effects like that you can't, _21:16_: you wouldn't. _21:17_: I defy you to think of any real term _21:20_: that is with that and be on Earth. _21:23_: And I'm confident you will not be able _21:24_: to come up with something because _21:26_: it is a foundational principle, _21:27_: GRR, that you can't. _21:29_: That there is no difference. _21:31_: Between that and being on Earth. _21:34_: That the. _21:36_: Experience of being accelerated at _21:39_: constant speed and the experience of being. _21:42_: In our. _21:46_: And St. Gravitational field _21:50_: are equivalent. Not just close. _21:54_: Not just the same, but equivalent. _21:57_: They are the same thing. _21:58_: It's what it seems. _22:02_: OK. And that but and that's and _22:04_: that's a physical statement, _22:05_: that's that. You know, _22:06_: this is called the equivalence principle. _22:09_: Uniform gravity. _22:10_: I will put these slaves and there's _22:12_: nothing in these slides isn't in the notes. _22:14_: These slides are just here. _22:14_: So I've got something to Peter _22:16_: in front of and talk about, _22:17_: and I will put the size up in the lecture _22:19_: notes folder just because why not? _22:21_: But but I encourage you to write things down. _22:24_: Writing things down is engaging and _22:25_: putting things in in the notes. _22:27_: The equivalence principle. _22:28_: Uniform gravitational fields that you _22:30_: know that's just straight up and down, _22:31_: not ones which are pointing inwards, _22:34_: are equivalent to frame that accelerate _22:35_: uniformly relative to an. 2 frames. _22:38_: There's that word national happening again. _22:40_: What an inertial frame. _22:43_: An inertial frame. _22:44_: Is. _22:47_: One like that. _22:49_: It's one where Newton's laws work. _22:53_: Now there are more elaborate _22:55_: way of of putting that, _22:56_: but in that frame Newton's laws work. _22:59_: If he goes a precisely right if _23:00_: if you leave something, it stays. _23:02_: If you push something, _23:03_: it moves at constant speed. _23:06_: So that's a national, _23:07_: you're prototype national frame? _23:10_: And uniform graphical fields are equivalent _23:13_: to 1 accelerate relative to that. _23:18_: So. Um, where we going with this next thing? _23:24_: So open parenthesis. And the _23:29_: critical general comedians is? _23:33_: It restricts the IT it _23:34_: it had critical content, _23:36_: it has physical and mathematical content. _23:38_: It's not just a beautiful philosophy _23:39_: that we can think about and and _23:41_: and debating the pub it had. _23:43_: It had consequences. _23:44_: It restricts the category of mathematical _23:47_: statements that we prepared to countenance. _23:50_: As Portal scriptions of nature. _23:53_: So if you have a theory which violates _23:55_: the equivalence principle, is wrong. _23:58_: You would have to think about it. _23:60_: It's just wrong because the _24:02_: ethical principle is true. _24:04_: Then it supervenes on on those, _24:08_: so it restricts what you can see. _24:11_: The relativity principle. _24:13_: I just mentioned picks out _24:15_: inertial frames as special those _24:18_: frames like the floating box. _24:21_: In which nuisance laws work. _24:23_: Are special. _24:23_: They have a special status and _24:25_: we'll carry on talking with them _24:26_: again and again and again they are. _24:30_: There's things we could say about _24:32_: them that we can't see other things. _24:34_: And the equivalence principle further _24:36_: constrains the set of the set of, of, _24:39_: of, of, of these special frames by. _24:43_: Seeing that those frames. _24:49_: The acceleration business _24:50_: for the constraints what the? _24:52_: What the frames can be. _24:54_: I I forget what the for the quite what I put. _24:57_: Why I've written that thing _24:60_: probably in the notes. Right. _25:03_: And so I'm going through the course, _25:07_: talk about things, certain things, _25:08_: physical statements. _25:09_: And what I mean by that. _25:10_: A physical statement is a statement _25:13_: that picks out one possible universe. _25:15_: From all the ones you can think _25:17_: of mathematically or in some sort _25:19_: of dream state and says this, _25:20_: one of all the ones that _25:23_: mathematical insistent is ours. _25:25_: Other universes are possible. _25:27_: They're not this one. _25:28_: And so physical statement is a _25:30_: statement that says the universe _25:32_: could be otherwise logically, _25:34_: but it's not. _25:35_: And when I see physical statement _25:36_: that that is that notion that _25:38_: I mean there are plenty of _25:39_: mathematical statements in this. _25:40_: Mathematical statements just follow _25:42_: from what came before and they can't be _25:44_: otherwise other or else logic is wrong. _25:47_: Physical statements could be otherwise. _25:48_: There's a contingency to physical statements, _25:50_: and there is, _25:51_: and that's where the physics is. _25:53_: So that's a good point. _25:54_: At which you mention. _25:55_: Some of you will be doing maths, _25:57_: some of you will do maths, yes. _26:00_: And you do months. _26:02_: Well, if you OK so. _26:05_: This is of course in interactivity. _26:08_: It's course we therefore also _26:12_: covers different geometry. _26:14_: It's nonetheless a physics _26:15_: course or astrophysics course. _26:16_: It's not a maths course, _26:17_: and so we will be sloppy. _26:20_: From the point of view of map additions, _26:22_: there's gracefully so. _26:23_: But we don't care because we're physicists. _26:26_: And everything's done. _26:28_: Being a physicist means the _26:30_: entire universe is analytic, _26:31_: that that's being a physicist means. _26:34_: It nonetheless, nonetheless it is, _26:36_: I will mention just parenthetically to _26:38_: that and they take mathematical course, _26:40_: so, so yeah, hold on to your, _26:42_: hold on to your seats. _26:45_: So close parentheses. _26:48_: Would you do badly OK? _26:52_: Let's go back to this floating box. _26:54_: So we're going to know is 3 thought _26:57_: experiments, 3 things that we. _27:01_: Fancy scenarios which we can analyze _27:04_: using what we've learned so far and _27:06_: and discover the physical content. _27:08_: The consequent consequence of those _27:11_: statements, we're back to the _27:13_: to the box floating in in space. _27:18_: On the observer floating in the middle. _27:22_: And we. Flash a light bulb _27:24_: on one side of the box. _27:27_: And we have a detector of some type, _27:29_: perhaps another observer on the _27:30_: other side of the box and given _27:32_: that this this light is pointing _27:33_: across the across the box, _27:35_: then you know of course it's going _27:37_: to go straight across the box and _27:39_: be detected the same distance down. _27:43_: No surprises there. _27:44_: There are no surprises there, OK? _27:48_: No, let's do the same thing. _27:51_: But in our box which is falling. _27:56_: No, that box is in free fall. _27:59_: And I think the first time I've _28:02_: mentioned this that phrase. Um. _28:08_: I think there is a link to _28:10_: to to to to that there's a. _28:11_: There's a site link of I think _28:13_: I've forgotten from the note. _28:14_: Check the notes for the for, _28:15_: for the for the little bridging movement, _28:18_: but the. The point here is that we _28:23_: are also taking things just that's it, _28:25_: things which are. Things which are _28:30_: moving only under gravity like. _28:33_: Which is floating out in a way _28:37_: more gravity gravitating bodies. _28:38_: They are inertial frames. _28:39_: They are in freefall, so-called. _28:42_: Moving all into gravity and they we _28:45_: are saying are also inertial frames. _28:48_: I'm sure there's bridging a little _28:50_: bridging statement I I've skipped _28:51_: here that doesn't matter. So this. _28:54_: Falling box and imagine it's a _28:56_: lift and the keeble's been cut. _28:58_: It's falling, in other words, _28:60_: moving purely under under control of gravity. _29:03_: And so it's a national frame. _29:06_: And the equivalence principle says that. _29:08_: Situation there with the box _29:10_: falling down the lift shaft. _29:11_: And the people in it is equivalent. _29:14_: To being in the national frame well _29:17_: with gravitating sources, therefore. _29:20_: And. _29:20_: This person can't tell the difference _29:22_: between these two these situations. _29:24_: They can't tell which inertial _29:26_: frame they're in. _29:27_: Therefore, _29:28_: when the they they see this that this light. _29:32_: Yeah, shown across the across the box. _29:36_: What they see is this. _29:39_: And for the late Good Cross and. _29:41_: Hits. _29:41_: I'll I'll little bit don't know _29:44_: from the top as you'd expect, _29:46_: but the point of view someone _29:48_: standing by this watching. _29:49_: In horror as this lift shaft plummets _29:51_: to the ground and hoping that this _29:54_: person is paying attention to their _29:56_: relative lectures and taking notes, _29:58_: hopefully very quickly. _30:01_: The light takes a finite time to cross. _30:05_: Not much, but a finite time to cross and so. _30:10_: The would it reach his, _30:12_: the other side will be slightly lower. _30:15_: By the time the light gets there. _30:18_: It must get there because _30:20_: the observer in the box, _30:22_: if it didn't get to that point, _30:23_: that expected point, _30:24_: which is the same distance down from the top, _30:26_: then this person would tell _30:27_: that something was missed. _30:28_: So it does get to that point, _30:30_: but from this person's point of view. _30:33_: That point is lower than the starting point. _30:36_: In other words, _30:37_: the light has followed a curved path. _30:39_: And the only things we've used in _30:42_: that argument are the equivalence _30:44_: principle and the relativity principle. _30:46_: In other words, light is bent by gravity. _30:50_: It has no mass. _30:52_: But it's still bent by gravity. _30:55_: And that's strange. _30:57_: But that conclusion I mean it. _30:59_: Which is true. _30:60_: But that conclusion comes only _31:01_: from the physics we've learned _31:03_: in the last half hour. _31:07_: Um. I don't change to work through this, _31:10_: but I I I have a few of _31:12_: these quick questions here. _31:13_: No, no, I will ask this question. _31:15_: The question just this point in _31:18_: following lift. I am a spring gun. _31:19_: Just do not have a light. _31:21_: Not a relativistic spring gun, _31:22_: just an ordinary, you know. _31:25_: Charles Toy across the lift _31:27_: intend to hit the bulb. _31:29_: I want to hit hit the bulb _31:31_: which is BEM above the bulb. _31:33_: Should aim directly at the bulb or should it? _31:35_: In below the bulb, who would see a? _31:38_: Who would say be? Who say see? _31:42_: Correct. OK. _31:43_: And you've all been paying _31:46_: excellent attention more than. _31:47_: This isn't just a a story about light, _31:51_: it's a story about things falling and _31:53_: you're not being able to tell the. _31:56_: The difference? _31:56_: So there's no difference between _31:58_: this and do the same thing with _31:60_: the weightless cabinet in orbit. _32:01_: OK, next. _32:07_: So you have a mass. _32:09_: Here. And you? Drop it. _32:15_: And it drops it acquire kinetic energy from _32:18_: transitional energy into kinetic energy. _32:20_: So it ends up down here. With. _32:25_: Oh oh, C is 1. OK, just like in _32:31_: special relativity as we use natural units. _32:33_: I'll have a little more to that in a moment. _32:35_: I better hurry up or I won't get to that bit. _32:37_: See series one. So E is equal to M, right? _32:40_: So the energy of this of of this _32:43_: thing is equals MC squared C is 1. _32:45_: Andrew there. When it gets down to here, _32:47_: the energy has gone up by the amount of _32:50_: potential energy loss in going down here. _32:52_: OK, then we take this slightly _32:56_: more energetic mass. _32:58_: And convert it into pure energy. _33:01_: That's kinematically impossible, _33:03_: but energetically perfectly reasonable. _33:06_: OK, and we send that photon. _33:09_: You know the energy equivalent _33:11_: of that mass back upwards. _33:14_: Until it gets to the top here _33:16_: itself with energy and the energy _33:18_: E primed at the top there no. _33:20_: Either we have invented a way of extracting _33:23_: free energy from the universe, or. _33:26_: E is the same as M. _33:30_: So that this must be a closed cycle. _33:34_: Or else we've got perpetual motion machine. _33:37_: In other words, what that means is _33:40_: that photon which starts off with _33:43_: energy E here which is bigger than M. _33:46_: Ends up with energy E prime, _33:47_: which is the same as it was a photon _33:50_: climbing through a gravitational field. _33:52_: Loses energy. Simply because of _33:55_: that could have special activity, _33:57_: an argument and and because yes. _34:00_: So that's another thought experiment that _34:02_: that that that that that says interesting _34:05_: physical things have to happen just _34:07_: because of what we've learned so far. _34:09_: And. Let's do that twice. _34:12_: So this is a setup called Shields photons _34:16_: because some someone Shields you use _34:18_: this example to as part of someone _34:21_: as an argument in the 50s I think. _34:23_: So this is a Minkowski diagram _34:25_: Z the vertical direction. _34:28_: That way and time in this in this direction. _34:33_: And we do the same thing as _34:35_: in the previous slide. _34:36_: We fired a photo on upwards in the Z _34:40_: direction. And it loses energy. OK. _34:46_: In which it goes from frequency F energy HF. _34:50_: To frequency F prime and F probably _34:55_: because F prime at the top. _34:58_: OK, just like what we said before. _35:01_: Then we could hit. _35:04_: Wait, and periods? _35:05_: We'll do it again. _35:08_: OK, so we do the exact same thing again. _35:10_: We send a photon of frequency F up _35:14_: and it arrives the top frequency. _35:17_: Different frequency, _35:18_: a lower frequency because that's lost energy _35:20_: by climbing through relational field. _35:24_: So that time difference. Is north over F. _35:29_: Obviously because we we _35:30_: had that many periods. _35:32_: This time difference is also _35:35_: the same number of periods. _35:37_: But the frequency is different _35:39_: so the period is different. _35:40_: So the time interval between B&B primed. _35:43_: Is not the time interval between A&E Prime? _35:50_: You're entirely surprised at that. _35:52_: What that is telling you is that isn't. _35:54_: This isn't a parallelogram. _35:58_: The. You know the the the _36:02_: distance is in both cases, _36:03_: but the the time distance on opposite of _36:06_: the parallel or parallel is not the same. _36:08_: In other words, this is not a Euclidean. _36:12_: Um. Uh. Surface is like are _36:16_: you clicking in space now? _36:18_: It's because of space. _36:18_: We didn't expect it to be anyway, _36:20_: but the the the point here _36:23_: is that just with that. _36:25_: Gravity redshift argument. _36:26_: You can just, _36:27_: you can just discover the first hint _36:28_: of that and there be lots more of that. _36:34_: OK, I think that's. Well, things to see. Um. _36:41_: I have a quick question with that which you _36:43_: can look at in the in the in the notes. _36:47_: Yep. And another one to we don't have _36:50_: time to do have have votes and and and. _36:53_: This is also an important diagram _36:55_: which you will see again in about. _36:59_: 10 lectures. _37:02_: And it's two objects. And. _37:08_: Well up above above the earth. _37:11_: And they both fall down towards _37:13_: the center of the Earth, _37:15_: direct towards center of the Earth. _37:17_: But this frame is big enough, _37:19_: there's not a local inertial frame, _37:21_: so the two balls will fall directly _37:24_: toward the center of the Earth. _37:26_: Now they're both. And if we fall, _37:30_: they're both in inertial frames locally. _37:34_: But if you ask what's the separation _37:37_: between them, what's that sigh. _37:40_: Distance? _37:41_: Then you discover that the second derivative _37:43_: of Phi with respect to T is non 0. _37:48_: So the second derivative of the _37:51_: position is known as zero. _37:53_: But they're not accelerating. _37:56_: So the the the the they are in freefall, _37:58_: so they are non accelerated frames _37:59_: and yet the second derivative _38:01_: position is is known as zero. _38:03_: What's happening here? _38:04_: What's happening here is the secondary _38:06_: position is just a coordinate number. _38:08_: It's just a number. _38:09_: Which is not the same as being pushed _38:12_: in the back and being accelerated. _38:14_: So I will aim to keep the sanctions separate. _38:17_: When I mean acceleration, _38:18_: I mean pushing the back. _38:19_: I mean something that you can detect _38:22_: locally and unequivocally and absolutely. _38:24_: And there is a difference between. _38:28_: That, and I think we've been different. _38:29_: And the reason why this happens, _38:30_: the reason why you can tell _38:32_: why if you if you were, _38:33_: if you were these two people in _38:34_: radio contact with each other, _38:35_: you could say, _38:36_: ohh, _38:36_: we appear not to be in our _38:38_: uniform gravitational fields. _38:40_: It's because this frame is big _38:43_: enough that it's not local. _38:46_: So there's a a tidal effect _38:47_: so-called happening here, _38:48_: which lets you detect that _38:50_: you're not in in this case. _38:55_: There is an international frame _38:57_: which covers both of these. _38:59_: OK, this is an important slide _39:01_: because it's meant to be reassuring. _39:05_: So that's the end of of _39:08_: the thought experiment. _39:09_: A few a few final remarks. _39:12_: That's the definition of differentiation. _39:16_: That you are very familiar with _39:17_: because you learn about it in school. _39:19_: OK. The derivative of F with respect _39:23_: to X is the limit of F X + H -, _39:26_: F of X / H as you take that H to 0. _39:30_: You really understand that? _39:34_: The will be doing the same a lot, _39:37_: the part, but a large part of the _39:39_: mathematical structure of this course _39:41_: is learning to do the same thing. _39:43_: In a curved. Space-time. _39:46_: And the the 10 lectures gets you to the _39:48_: point where the where you learned that _39:51_: mathematical practice like I'll just do _39:53_: that G2 which comes next semester is OK, _39:55_: what follows from that in in physical terms. _39:58_: So this first semester is _39:59_: basically learning how to do that. _40:01_: Intercourse. _40:02_: Beasting and it's difficult because. _40:06_: In this case, it's easy to know to to _40:09_: think about what F of F at a different _40:11_: point is and how subtract it it. _40:13_: It needs to know what what _40:15_: dividing by H means. _40:17_: In a country's time, _40:18_: both of those things are tricky, _40:20_: so we have to learn how _40:21_: to say the same thing. _40:23_: But it's the same thing happening. _40:25_: It's fundamentally the same thing happening. _40:27_: It's just it will look hellish because the. _40:30_: The coverage does that does that for you. _40:36_: In terms of overall structure, _40:37_: this Part 1 lecture one almost done it is. _40:43_: Setting the scene, _40:43_: saying why there's a problem, _40:45_: say what the problem is. _40:47_: Part 2 is introducing tensors, _40:50_: mathematical structures that you _40:51_: may have somewhat across before, _40:53_: but have always seemed a little _40:54_: bit exotic and and you don't _40:56_: necessarily have much to do with them. _40:58_: They are vital to an understanding _41:00_: of of of generativity. _41:02_: Part three is letting her to _41:05_: differentiate those sensors. _41:06_: Which is each of them, _41:07_: one might think, _41:08_: but but still not not easy. _41:10_: And then part 4 is actually using _41:13_: all this mathematical apparatus to _41:15_: do what Einstein did and describe _41:17_: a theory of gravity in these terms. _41:20_: So today's been physics. _41:23_: The next 7 lectures were basically maths. _41:28_: The last few lectures are back to physics. _41:31_: So say bye to physics for seven lectures. _41:34_: We will look forward to coming back, _41:36_: but that's what's happening. _41:38_: At the last thing I'll mention natural units. _41:41_: I've lost more than natural units. _41:43_: Natural units are the units you _41:45_: choose when you say C is equal to 1. _41:47_: So you measure distance in meters _41:49_: and time in meters. _41:50_: A meter of time is the is the _41:52_: light meter is the time it takes _41:54_: for light to travel a meter, _41:56_: and in those units in light meters, _41:59_: light meters in meters per light meter. _42:01_: The speed of light is a nice, _42:02_: easy to remember one and I have noticed it. _42:06_: You will find stuff on the _42:07_: way by have a natural unit. _42:08_: How terribly confusing. _42:09_: Oh my God, _42:09_: awful. _42:10_: But it's just a matter of picking the right _42:12_: units in which back to the sea disappear. _42:15_: And I think that's the _42:16_: last thing I've got to say. _42:17_: And I know you have to rush _42:18_: off to gallop around the that _42:20_: you'll be so fit this semester, _42:22_: galloping around the the, the, _42:23_: the the campus trying to find things. _42:26_: I have a section in the notes about reading. _42:29_: There are many good books on _42:31_: relativity on general relativity. _42:34_: This course is highly compatible with shoots. _42:38_: Highly compatible said. _42:38_: I'll even refer to it as the _42:40_: same overall set of ideas, _42:41_: or so. _42:42_: It's it's a good book to use cattle _42:44_: is very it's it's slightly different, _42:46_: and that's a good thing, _42:47_: because a nice contrast, _42:48_: it might treat you better. _42:49_: Rindler is very old fashioned as far _42:51_: as the treatment of generativity goes, _42:54_: but the first half of that book is _42:57_: really all about explanation of _42:59_: this of of subtleties and the things _43:02_: you three different subversions _43:03_: of the government principle. _43:05_: Rindler knows all about the physics of GR. _43:08_: Little thought Wheeler. _43:09_: Great big doorstop. _43:10_: Wonderful book which irritates _43:11_: the hell of some people, _43:12_: but good and and some of these _43:15_: books are on available at that URL, _43:18_: which in the notes which you can _43:20_: find at the library, there are. _43:22_: There are electronic versions of some _43:24_: of these books and you might be able to. _43:28_: The wild. OK, I think that's the end.