Transcript of gr-tut3 ========== _0:06_: Well, as you can see there's a large _0:11_: blank here for the suggestions are and. _0:13_: I didn't send an e-mail around _0:16_: saying I'd suggestions but. _0:18_: And so we'll have to bust a bit and I _0:21_: I do have a a matter of different plan _0:24_: from the previous divisions commercials. _0:26_: So I think that I hope you have just _0:30_: write have written anything down. _0:31_: You have some sort of questions that _0:34_: we're going to talk through and could _0:36_: potentially go look through one of the _0:39_: exercises if there's a very obvious _0:41_: one that lots of people think is a _0:44_: really good one to work through but. _0:46_: A lot of the. _0:49_: Exercises are sort of turning the handle _0:53_: rather than massive epiphanies, so. _0:58_: If there are any. Well, _1:00_: I think a higher value way of using _1:02_: this error is if there are questions _1:05_: that you that that that you can. _1:07_: Either burning you up, _1:08_: or which we think are of interest, _1:10_: or which are puzzling you. _1:12_: I got the best way you can be _1:14_: renewed the error I don't really _1:16_: want to give in and just sort of _1:18_: trundle through an exercise, but so. _1:27_: Is it all just easy? _1:31_: Are there particular? _1:33_: Exercises that are difficult, _1:35_: other particular sections that are difficult. _1:37_: I mean that might be that _1:40_: might be a good way of of of. _1:42_: Sort of getting going with with _1:45_: questions that other other sections _1:48_: from part one to part 4. That. _1:52_: You've stumbled over that you're _1:53_: not sure how how to get from the _1:55_: beginning of that section to the end _1:57_: of that section or or that chapter, _1:58_: or is it all of us have a blur _2:01_: or are there some of them? _2:03_: Are there some that are easy? _2:04_: I mean that's quite usually there's _2:06_: some sections where you just think, _2:08_: oh, that's the section that _2:09_: had made it all clear for me. _2:16_: Nope. Other sections which? _2:20_: Where? _2:26_: Even different parts. _2:29_: And Part 2 Part 3, _2:30_: part 4 which ones those? _2:33_: Where did you find I? _2:34_: I have a notion of which ones are _2:36_: probably the harder ones of those 3. _2:38_: But do you agree with me what which _2:42_: of those three things hard is? _2:45_: How would you rank them in _2:46_: terms of which was your easiest? _2:47_: Which ones are hardest in _2:48_: terms of Part 2 Street, _2:50_: I'm getting the part one, _2:51_: we're just sort of getting going thing, _2:53_: but Part 2, three and four and four, _2:55_: which of those are easy and easy and hard? _2:58_: Anyone. _3:02_: That, that's that fan is annoying. _3:04_: I'm not gonna be. Stop it. _3:06_: Say again. Part 2. Yeah 4/4. _3:14_: Right that's that's not what I guessed _3:16_: right but that that that that's very _3:18_: useful to know because I suppose the _3:20_: difficulty there is we can, we can, _3:22_: we can't sort of just turn the hand _3:24_: over them with the maths so that _3:26_: there's that there is back to physics. _3:28_: Would would others agree with that? _3:30_: Is that any that. _3:31_: That's right that should have _3:33_: a fairly quick noting there. _3:35_: OK well that's that. _3:36_: That's interesting because that's _3:37_: not what I would have guessed, _3:39_: but I can see why because that _3:41_: is in a sense where the where _3:43_: the maths hits the road. _3:45_: So, right. _3:45_: Well, that's, that's good then _3:47_: that gives us our place to start. _3:50_: If by the way, I'll look back to the page, _3:52_: if any of you are online and _3:54_: want to add questions there, _3:55_: then I don't entirely honestly _3:57_: then we can do that. _3:60_: So. _4:09_: One of these must be there. _4:22_: Well, just by the way, the. _4:25_: The videos on the Microsoft Stream _4:29_: from last from two years ago. _4:32_: Are any of you looking at those? _4:34_: Are they useful at all? _4:38_: But I think it's also _4:39_: seems rather tentative. _4:40_: Nor do I'm not gonna ask _4:41_: you questions about it. _4:42_: Is it just ask people for information _4:44_: so so that that that's not. _4:48_: Sorry. And I think so, yes. _4:55_: I I think part if part if elected _4:58_: for part 4 aren't there, then they _5:01_: certainly should be I and I haven't. _5:04_: Um, so I might not be able to. _5:15_: And. _5:22_: Yeah, right. _5:36_: One second. _5:54_: Well, no, but that's what we're looking for. _6:01_: Oh, come on, I'm Clinton. _6:06_: Thank you. _6:10_: Ternals uh. _6:15_: OK, the lecture 11 from last year isn't. _6:19_: So I I need to actually, _6:21_: in fact, why don't I just? _6:27_: Um. _6:32_: Because if I don't, then I will forget. _6:39_: This is 2020. Ohh, there's such _6:42_: a lot of room, wasn't it? Umm. _7:09_: Noted. OK. _7:13_: That wasn't what we do. Oh yes, _7:14_: that's what we're doing, was this. _7:19_: So give me that bigger. _7:27_: So that was done. So the stuff on _7:30_: the incremental tension and and that _7:32_: was all about trying to get out. _7:37_: Explain for the argument potentially _7:39_: comes from and this notion of _7:41_: dust so and the point of that the _7:43_: reason why we're interested in in _7:45_: in the age of mentor is because _7:48_: that's what is the source of the. _7:54_: Changes in in the curvature of space-time. _7:56_: That's, that's that, that, that that's _7:58_: the right hand side in 19's equation. _8:01_: That's the bit which says _8:03_: Mattel specialty curve. _8:04_: So the geodesics is space _8:07_: just matter to move. _8:08_: Einstein's equation is matched _8:10_: or specific curve matter? _8:12_: It's not matter we're _8:13_: talking about really is it? _8:14_: It's not matter that just beat _8:16_: out curve it's energy momentum _8:17_: that's beta curve are you question? _8:19_: No, you're discussing it incrementum _8:22_: that tell specific curve so the so _8:25_: the step one there was turning. _8:28_: Our some of our intuitions about matter. _8:31_: Which is the most compact form of elementum. _8:35_: How do we turn that into a geometrical form? _8:37_: So the goal with that section one is just. _8:39_: It's just geometric geometries, _8:42_: the notion of. _8:44_: Lumps of matter. Next question. _8:49_: Details based on the curve? Yes. _8:51_: Does that mean that light curve space? _8:53_: Yes. _8:53_: But then why do we think that light _8:56_: moves along the curved space? _8:58_: Because the light curves, _8:60_: because this is the both houses _9:03_: of that of that. _9:05_: Slogan that energy momentum _9:07_: tells space how to curve. _9:10_: And then test particles within that _9:13_: space are what explore the space _9:16_: and are curved by it so as a star. _9:21_: Curves the space-time around _9:23_: around it and a planet. _9:26_: I test particle in that it which is _9:29_: a a lump of energy momentum does _9:32_: follow duties and and and and and and curve, _9:34_: but at the same time that planet is _9:36_: curving the space-time around it. _9:38_: So that the, the, the the the the, the, the. _9:43_: So these are recursiveness of that slogan. _9:45_: It is part of the the on the thing _9:49_: to meditate on in a sense. _9:51_: But it is true that that enough light, _9:54_: if there was enough energy in a small _9:56_: enough like energy in a small space, _9:59_: then it would curve space-time. _10:00_: And that's sort of The Big Bang. _10:02_: So the the the The Big Bang is where _10:04_: there is enough energy and sufficiently _10:06_: small space that you have the whole _10:08_: universe being curved around it. _10:09_: And another thing is that if you _10:12_: think of gravitational waves. _10:13_: So these are these oscillatory _10:16_: solutions in in in space-time. _10:19_: Those themselves. _10:20_: Are. _10:24_: They have energy momentum in them, _10:27_: so they themselves curve space-time. _10:30_: And that's why I'm saying equations are _10:32_: hard to solve because they're nonlinear. _10:36_: With, um, things like. _10:39_: Maxwells equations if you take a _10:41_: solution of Maxwell equations, so. _10:44_: Like about a library and add another _10:46_: solution to Maxwell equations, _10:47_: another library. _10:48_: Then the sum of the two is a _10:50_: solution of Maxwell's equations. _10:53_: In other words, _10:54_: like light can pass through plate, _10:56_: racing pass through each other, _10:57_: so so you add 2 lightweights together, _10:59_: you also get questions. _11:01_: That's not true for. _11:03_: A nonlinear differential equation _11:05_: like Einstein's equations. _11:07_: So the solutions to Einstein's _11:09_: equations are not additive. _11:11_: You can add two solutions to _11:12_: identify equations together and get _11:13_: a solution of intense equations. _11:15_: That's why it's hard to solve. _11:17_: So because there's a whole. _11:20_: Chunk of mathematical methods, _11:22_: which is all about decomposing differential _11:24_: equations into ones you can solve, _11:25_: and that the idea that the _11:27_: additive is part of that. _11:28_: So the, the, the, the, _11:30_: the the recursiveness of of of _11:32_: that slogan is in a sense talking _11:34_: to several different things. _11:35_: It's it's talking to the idea that _11:38_: it's masters is both the the thing _11:41_: which explores the space-time and the _11:42_: thing which creates the coverage. _11:46_: So the point of that first section is to. _11:50_: Geometrized the idea of. Mass. _11:54_: And to remind us perhaps, _11:56_: that mass is not the only source of energy. _11:57_: Momentum if the important, _11:59_: most important source, _11:60_: instrumentum in our near experience. _12:03_: But it's not the only source. _12:07_: And then the second part and we've _12:08_: got to come back to push back, _12:10_: we can explore a little more at the moment. _12:13_: The next part is that the guessing bit. _12:16_: And just to to to reiterate the the. _12:22_: It is guesswork. _12:25_: Einstein was was inspired by the _12:29_: the formal Poissons equation which _12:32_: relates the Newtonian gravitational _12:34_: potential to the distribution of matter. _12:37_: And to the you know there's a second _12:39_: derivative equals a mash thing and _12:40_: and that sort of giving a hint, _12:42_: but the form of integration is a guess. _12:46_: And the experimental corroboration of. _12:50_: Generativity. Is the experiment, _12:53_: experiment experimental _12:54_: corroboration of that guess? _12:57_: And the idea that you might have _13:00_: the cosmological constant in there. _13:03_: Was there? In response to an _13:05_: apparent falsification of that guess, _13:08_: it appeared that the that the _13:12_: the expanding solution expanding _13:14_: cosmological solution that was found _13:16_: fairly early on was clearly wrong. _13:18_: Therefore that guess was clearly wrong. _13:20_: Therefore the cosmological _13:22_: constant was added. _13:24_: As in response, but it turned out no, _13:27_: that's not wrong. _13:27_: This is actually the case that _13:29_: the universe is expanding, _13:30_: therefore there's no need to _13:31_: add the cosmological constant. _13:32_: Therefore the original guess _13:33_: would perfectly good. _13:34_: And still later it turned out that _13:37_: with things like gravity or blah blah _13:39_: blah dark dark energy it might be. _13:41_: In fact is another case for _13:43_: adding that term in. _13:46_: To the to, to, to the equation. _13:48_: So there's still a certain professionality. _13:52_: Of a slight professionality, _13:54_: but certain professionality to the, _13:57_: UM, the form of Einstein's equations. _13:60_: And the fact of the equivalence principle _14:02_: that says there's no coverage coupling, _14:05_: that says there's no extra things _14:07_: added to special activity, _14:09_: that no extra added to physics _14:11_: because of curvature. There's no. _14:13_: There's no term in the general _14:15_: artistic version of of a micro screens, _14:16_: for example. _14:17_: Which is involves the curvature, _14:20_: local curvature. _14:21_: That's a physical statement which _14:22_: says no you don't you you won't _14:25_: have to add anything to intent _14:26_: equations to deal with coverage. _14:28_: You won't have any other _14:30_: other physical with coverage. _14:31_: So Part 2 there is the guest bit. _14:36_: And part three is OK, so so. _14:40_: End of 4.2 is basically _14:43_: the end of this course. _14:46_: The G1 is getting up to the point where _14:49_: we've said we're declared inside equations, _14:52_: and so 4.3 just a bonus. _14:55_: No, it's a bonus. _14:56_: That's that's I come up _14:57_: with the objective say. _15:03_: Perform some dynamical calculations. _15:04_: If you look through the past papers, _15:07_: you'll see that the dynamical calculations _15:09_: equation are things like can you _15:12_: draw very simple deductions about. _15:16_: I can't. There's one that _15:18_: I've asked in the past about. _15:23_: The duties equation in our coverage _15:25_: space-time that if you start off moving _15:27_: radially you carry on moving really. _15:29_: Yeah really radially and stuff like that. _15:32_: So that's that's what I mean by _15:33_: simple the number calculation if you _15:35_: look back at past papers that I'm _15:36_: not asking you to solve as orbits _15:40_: in in instructional space times, _15:43_: it's basic simple things. _15:46_: Because they simply don't have time in _15:49_: question and that and and that is what _15:52_: Georgia two is all about in in a sense. _15:55_: So Patch 4.3 there. _15:58_: Is really just a. Going, you know. _16:04_: We're treating one particular solution _16:06_: just because I can't resist not doing so. _16:08_: I don't want to leave you with no solutions. _16:12_: Um, so that's how those three _16:14_: things sort of slot together. _16:17_: In terms of the of the sequence _16:18_: of ideas and and the separation _16:20_: between the different ideas. _16:22_: Um. We could dig into, _16:25_: I mean of those three parts of the other _16:27_: bits that I should we should be useful _16:29_: to talk more about in detail which, _16:31_: which which are the parts of those? _16:33_: What good to dig into? Could I? _16:37_: I I I think you're right. _16:39_: This is actually, from some points of view, _16:43_: the hardest. The hardest the course. _16:45_: Because it is actually about trying _16:47_: to use these mathematical tools. _16:50_: In practice. _16:51_: And trying to connect them to the physics _16:53_: that you've spent the rest of your degree. _16:56_: Absorbing. _16:57_: So now tools. _16:59_: Nice tools, we finally get to use them. _17:02_: So which of those sections would _17:03_: be useful to dig into you think? _17:08_: 4.14 point 24.3. _17:15_: Yeah. Excuses. _17:33_: Or should I just blather _17:34_: on about going through? _17:37_: The second objective section, OK. _17:43_: Explaining the Congo semi colon rule. _17:44_: OK, that right that the _17:47_: comical semi colon rule is. _17:50_: I think what which section is that? _17:57_: Specifically. _18:01_: Comical second rule is come on. _18:09_: Yeah, is is is this section here on. _18:13_: The equivalence principle. _18:14_: So the common goal semi colon _18:16_: rule is one of the ways of _18:18_: saying the equivalence principle. _18:19_: So the current principle is in. _18:23_: One version of it is the thing that _18:26_: we sort of covered in part one. _18:29_: We talked about the we way back in _18:31_: part one and the idea that you can't _18:34_: tell if you are if you're free fall, _18:37_: you can't tell whether you're _18:39_: way out in space. _18:41_: Away from all gravitating matter _18:42_: or you're in lift shaft for around. _18:45_: You can't. They aren't different. _18:46_: It's not you can't, it's not you. _18:48_: It's hard to tell. _18:49_: The difference is you cannot _18:50_: tell the difference. _18:51_: So I as a physical statement, _18:53_: those two things are the same. _18:55_: It's what we said in part one _18:57_: of what we're repeating here. _18:60_: Another version of that. _19:03_: Is a stronger version of that. _19:07_: Is, but because that's consistent with. _19:12_: And freefall being a sort of special case. _19:15_: It might be. _19:16_: There's all sorts of things _19:17_: that aren't there in freefall _19:19_: in the local inertial frame, _19:20_: because being in the local frame, _19:22_: you're in freefall. _19:23_: You have the coordinates that are _19:25_: attached to that are nice and simple, _19:26_: and you can do calculations _19:28_: in that frame very easily. _19:29_: So the local and frame is clearly somewhat. _19:32_: But it's it's only special because for _19:35_: calculational reasons or special otherwise. _19:38_: And the strongest principle is saying no, _19:39_: it's not special at all. _19:42_: In a way. _19:43_: Um. _19:43_: Any physical law that can be _19:45_: expressed intention rotation in Sr, _19:48_: so a geometrical statement of physical law. _19:52_: The key to the idea of geometry has _19:55_: exactly the same form in a locally _19:58_: inertial frame of a curved space-time. _20:01_: And what that means is, _20:03_: do I see who was cook? _20:08_: And with no extra covered your terms _20:10_: repeating on the right hand side. _20:13_: In other words, _20:15_: it's not that there is some _20:17_: extra that if that's a that's a _20:21_: conservation law in special activity. _20:23_: Of other than that is with, _20:25_: with, with single partial with _20:27_: partial differentiation. _20:28_: The tensor form of that is this. _20:30_: That's. That's a tensor. _20:32_: The tense of the tensor form the _20:34_: geometrical form of a special _20:37_: activity conservation law. _20:38_: And the strongly equivalent _20:40_: principle says that's true in GR. _20:44_: In the story, _20:45_: it's not that plus some curvature _20:46_: terms which happened to be zero _20:48_: in the local inertial frame. _20:50_: It's there are no extra terms. _20:53_: So there there are no tidal terms, _20:55_: there are no. Just do extra energy in. _21:01_: In the cover. _21:03_: Which is implicit in the coverture. _21:04_: It's just there's nothing else _21:06_: on the right hand side there. _21:08_: Because the basic equivalence principle _21:10_: is consistent with there being _21:12_: other terms there who just simply _21:14_: zero in their local national frame. _21:18_: The strong conference says no, _21:19_: they're not there at all. _21:21_: And what that is is, is is that for others, _21:23_: for for that, for example, _21:25_: for the geometrized, for the, for the, _21:29_: for, for macro equations, which are _21:31_: already in basically geometrical form, _21:34_: there's no extra terms and Maxwells _21:36_: equations which are to do with geometry. _21:39_: So late isn't doesn't propagate _21:40_: differently in a curved space-time. _21:42_: It propagates in a, A, the, the, the. _21:44_: There's a very little ways _21:46_: of saying there's no extra. _21:47_: There are no, there's no coverage coupling, _21:48_: there's no curvature terms, _21:51_: and that is the physical. _21:54_: The slogan for that is. _21:59_: That. _22:01_: Good, because there that the the the _22:03_: the the the point being made there _22:06_: is that you can go from the special _22:08_: artistic version team you knew comma _22:11_: new equals zero to the generalistic _22:14_: version tbu semi colon U = 0. _22:18_: And that sort of physical law so that _22:20_: is so this version this this quote, _22:23_: comma go semi colon rule is an A way _22:25_: of thinking about that that that that, _22:28_: that physical statement. _22:29_: That there's no extra coverage _22:32_: coupling now and and and the point _22:34_: of the of the of the Congo Senegal _22:37_: rule is that there's sort of two. _22:39_: Only one of which is being referred to here, _22:41_: because if you if you remember _22:43_: all the stuff about going to The _22:45_: Walking in the local natural frame. _22:47_: The reason why we work in the _22:49_: local national frame is because _22:51_: the calculations are easy. _22:52_: But if we have our if in that easy frame, _22:57_: we have a an equation. Which involves. _23:02_: Angled involved just components or _23:05_: single derivatives of components. _23:07_: Not set double second derivatives _23:09_: but single Dr Components then. _23:11_: That is the. _23:17_: And although we calculated that in _23:19_: the local natural frame these special _23:21_: coordinates those that that all _23:23_: the bits that would survive of are. _23:26_: More complicated expression which _23:28_: would be which are transformed _23:30_: into these coordinates. _23:32_: In other words that is equal to the. _23:36_: Covariant version of that. _23:41_: Expression of trying to for example, _23:45_: for example, for example. _23:49_: Let's go back to. Here. _23:58_: And I was looking at this. _24:03_: Earlier I think we may have. _24:07_: Talked about this briefly last time. Umm. _24:13_: For example. I'll be better, bigger. _24:24_: That's one of the exercises later on, _24:26_: little later on in in part three, _24:29_: and it's about. Completing. _24:33_: You're calculating this expression here, _24:36_: so that going slightly beyond that. _24:40_: OK. It's about it. It's it sure that. _24:45_: And the point is that there the. _24:49_: We do the calculation in _24:51_: the local inertial frame. _24:55_: The point being that the _24:56_: rest of this expression, _24:58_: this complicated expression here, _24:59_: is all zero because the _25:02_: Christoffel symbols are zero _25:04_: in that in those coordinates. _25:05_: So in this frame. _25:08_: This potentially very complicated _25:10_: expression turns, you know, _25:11_: collapses to just that. _25:15_: And. _25:18_: We can. We can. In these coordinates we _25:22_: can compare that with the expression _25:25_: like 3349 for the Riemann tensor. _25:29_: And discover that that question there. _25:32_: Is all the bits of the of that _25:37_: expression in 349 that survive. _25:39_: In other words, this. In other words, _25:41_: this is equal to this in these coordinates. _25:45_: But. This equation here. Is an equation _25:52_: between components of a tensor. _25:55_: Which is true in in this. Frame. _25:59_: Yeah, in this frame, in these coordinates. _26:02_: But if it's if those components _26:04_: are equal in that frame. _26:06_: Then they will be equal in any frame. _26:10_: If you transform the left hand side from _26:12_: the local frame into something else, _26:14_: and transform the right hand side from _26:15_: the local industry into something else, _26:16_: then you'll still have an equality there. _26:20_: You have a massive more terms right hand _26:22_: side, but you still have inequality. _26:24_: In other words, this is a tensor. _26:27_: Equation. _26:29_: It works because the sub the sub _26:32_: scripts of the knob line different and. _26:37_: He's well, yes. Because because this. _26:41_: Hear that? That's natural. _26:42_: I apology VK minus nabla G _26:45_: nable I VK But if they were? _26:49_: OK, that would work because _26:50_: then we would have a second. _26:54_: Ohh alright but. _26:57_: But that we wouldn't see, _26:59_: but that that's true in principle, _27:02_: but we would what we see here would be. _27:08_: A zero here because I think if you _27:11_: swap I and G then you choose sign, _27:15_: so you end up with you know. _27:20_: One more thing. _27:23_: We have a Ji comma hi. _27:28_: Yeah, as well. So then we have 3. _27:34_: Over M I'm not sure. _27:37_: I'm not sure what you mean. _27:39_: And we also have it. _27:44_: J5 comma I. _27:48_: I'd have to go back and look at the, _27:50_: the, the, the, the, the, the, _27:51_: the intermediate steps there, _27:52_: but I think that we would end up with _27:57_: this if that I would end up with zero _27:60_: on the right hand side. Because again, _28:02_: I have to go back through the steps, _28:05_: but because because that's, that's a so, so. _28:11_: I V minus Napoli Napoli V will be equal to 0. _28:15_: But the point here is that here _28:17_: you have a changer. Here you have. _28:22_: Yes, so it's a tensor. _28:25_: If the component of the tensor are equal, _28:28_: then the tensor are equal _28:30_: as geometrical objects. _28:31_: And that's sort of a common _28:33_: goal semi colon rule. _28:34_: In the sense that you can go from. _28:38_: And that's not perhaps a terrifically _28:41_: good expression, but but. _28:44_: Is there another version of that _28:46_: which illustrates that better? _28:48_: Which I? _28:54_: Umm. _29:02_: Let's see. _29:06_: And I'm not going to find _29:08_: one immediately, but. _29:22_: Four and. _29:26_: I'm not gonna be mediately but but the. _29:32_: I mean, I, I think I, I, _29:33_: I'm risking over complicating _29:34_: this by speaking too much. _29:36_: So, so the, the, the, the, the, _29:38_: the key point I want to to stress _29:40_: is that there's two things. _29:42_: There's two cases where you go _29:44_: from a a commentary semi colon. _29:46_: One is in the general context of _29:48_: this thing about doing calculations _29:50_: of the local national frame _29:51_: and then deducing that you've _29:53_: actually got a tensor equation. _29:55_: Therefore you can turn the _29:56_: comment into semi colon. _29:57_: And that a mathematical trick. _29:60_: And the other is this version is _30:03_: rephrasing of the equivalence principle. _30:06_: To see that there's no coverage coupling. _30:09_: There's no that that, that. _30:13_: A. A differential equation _30:15_: such as that conservation _30:18_: equation in special activity, _30:20_: which will just involve our. _30:22_: A comma, I think. _30:23_: I think, I think partial derivative _30:25_: can be turned into a physical law, _30:28_: that physical law and specialty can _30:29_: be into physical law generativity by _30:30_: turning the common into semi colon. _30:32_: And that's a physical statement, _30:33_: not a mathematical trick. _30:36_: So that the the distinction _30:38_: between those two things is the _30:39_: important thing. And so, so. _30:45_: So where are we? So that. Objective. _30:54_: Is basically see what I just said? _30:59_: It it it's, you know. _31:02_: In your room or to explain. _31:05_: That. In a way which indicates _31:08_: that you do understand it. _31:10_: Is it cool there? _31:11_: So I think the objectives in part _31:13_: four are not actually terribly hard. _31:15_: I think the idea is that the _31:18_: aims in that's one of the ones _31:20_: with the aims are quite hard, _31:21_: but the objectives aren't massively _31:23_: hard for what I want you to be _31:25_: able to to be testable on. But. _31:30_: I think these two things _31:33_: are both explained things. _31:35_: The not do a calculation, _31:36_: but you show me you understand. _31:39_: Good questions which are? _31:41_: A pinch to mark, _31:43_: but I think quite good as ways of I _31:45_: think the quite useful objectives in _31:47_: terms of if you can do this then I _31:49_: sort of believe you understand it. _31:51_: Which is the getting back to _31:52_: the point of exams and the fate _31:54_: of rubbish we're going. _31:55_: But obsessing things would be so much _31:57_: simpler if we just say they are the aims. _31:59_: Do you have you tried the aims? _32:01_: Yeah, yeah, yeah, I would agree. _32:03_: That would be simpler. _32:08_: 4.2 OK, don't look at that. Umm. _32:18_: So 4.2. Ecomo. _32:28_: Alright, did you have all the answers? Right. _32:33_: OK. I think there actually was one of the. _32:38_: Yeah, the things mentioned in the thing _32:41_: last week where I think I say I can't, _32:43_: why I didn't talk about that last time, _32:44_: I think whatever, right? _32:53_: And. I remember this do the whole sensors. _32:59_: OK. And one graduate. _33:01_: I'm good, I think rather than right. _33:03_: And we'll talk through the, _33:04_: the, the, the, the, the, _33:06_: the, the, the, the note. _33:09_: 4.2. _33:17_: OK. And? No, it's quite brief, _33:22_: isn't it? Yeah, as as a note, _33:24_: perhaps should expand that. _33:27_: Well, OK, let's let my body should break, _33:29_: break, break this down to some extent. _33:31_: Just what's in place to? _33:41_: And. _33:52_: OK. _33:57_: So the. _34:03_: I think I think that this _34:06_: ansatz is suggested in. _34:09_: In the question so. _34:12_: A row plus B. P. You cross you. _34:20_: Plus CRO. Plus DP. _34:25_: G and equation 4.4. _34:30_: Let's go back to. This. _34:36_: Oh, oh duh. _34:44_: Um, so for dust. Um. _34:56_: So. _35:07_: What was said there? Is. _35:14_: If we are to be consistent with. _35:16_: Equation 4.4 as equation will be 4. And. _35:24_: Why is it? Why is it obvious that _35:26_: we must have equals one is equals 0? _35:35_: Right. _35:39_: If we are in the movie reference _35:42_: frame, then G is. Diagonal. _35:49_: What are we choosing? _35:51_: What's minus plus here? _35:56_: Just one. OK. So. If. _36:05_: In the case of dust. _36:08_: This is a question. _36:09_: Here is if we going with _36:12_: this and that's here. Then. _36:17_: If that's consistent with this in the _36:19_: case of dust, dust is the remember _36:21_: the case where the pressure is 0, _36:24_: so it is a an ideal fluid, _36:27_: which being made of dust, _36:28_: you know, which is stationary, _36:30_: is not banging against the dust particles, _36:31_: not bang against each other or banging _36:33_: against the walls of notional container. _36:34_: There's no pressure. So, so, so. _36:38_: P in this and that is 0. _36:41_: That turns into a row U cross U + C _36:46_: row G and if this is to be the case. _36:50_: If if she wasn't zero in _36:53_: this general expression, _36:54_: then would we would see an AAG in this term, _36:58_: which we don't. _36:59_: So this is so this general term. _37:03_: The general expression here has to. _37:06_: If it's to be consistent with that, _37:08_: then all the way that can be is _37:11_: if A is 1 and C is 0, so we can't. _37:15_: So we deduce that that term can't be _37:19_: present in the Azure momentum transfer. _37:24_: So. And. _37:27_: Dust implies see a equals one, _37:31_: C = 0. _37:34_: Um and. _37:38_: OK I I can pop you expand on on _37:40_: on the note for that a little bit _37:42_: to to make it less need a little _37:45_: less unpacking now in the moment _37:47_: I'll move reference frame the the. _37:49_: There you is equal to 1000. _37:55_: And G is. You could just that. _38:00_: So there T00 is a row. Plus BP. _38:10_: You 0U0? Plus. And. DP. _38:19_: G00 which is minus one which is. _38:26_: Row plus BP. _38:31_: You one thing one is 1. Minus. _38:36_: Probably DP. Yep. So. OK. _38:43_: Does that make sense? So so. So. So. _38:45_: So again we're we're we're picking _38:47_: A-frame that with with this simple _38:49_: so this is the frame in which the _38:51_: the dust is not moving so all these _38:53_: motions of dust are just sitting there. _38:55_: Not moving with respect to us and not _38:57_: moving with respect to each other. _38:59_: OK. So all for so for each of _39:02_: them or the moats of dust. The. _39:06_: Velocity 4 vector is nice and simple. _39:09_: Just one the the the time component _39:11_: is 1 and the IT is that familiar from _39:15_: your recollection of special activity. _39:17_: I'm seeing some some nodes and _39:18_: some sort of slightly, _39:20_: I'll look that up later take notes _39:22_: and G is a simple in this frame. _39:25_: So just plugging these numbers _39:27_: in with T00 is that. And tig. _39:33_: It will be raw plus BP. _39:38_: Well. You I the the the UI are _39:42_: are are are 0 * 0. And plus DP _39:48_: and and and the. Metric here. _39:55_: Is 0 operational. And all of the terms on _40:01_: the diagonal are ones, so that will be. _40:05_: One if I and J are equal and 0 otherwise. _40:10_: So you see again. _40:15_: Because in the end if you think _40:17_: of the special artistic 4 vector _40:21_: for velocity. And that is. _40:27_: So. And. If you could X well. _40:34_: New by TD tour. Which is. _40:41_: Well. If, if, if our particle. _40:49_: Is is is seeing in one place then the. _40:54_: So the the the four velocity. _40:56_: Is the rustic velocity. _40:58_: It's the the the special _41:00_: components of it are well. _41:02_: Be X0 by D Tau, DX1 by D Tau. And so on. _41:10_: But X0 the, the for for a displacement, _41:14_: the the, the sort of. _41:18_: X cosky diagram. I I displacement. _41:23_: We'll have a. _41:25_: The X and A. _41:28_: The T so if a displacement has no _41:30_: displacement in the spatial direction _41:33_: then that will be purely timelike _41:35_: but purely parallel to the time axis. _41:38_: So X0 will be DT by D. _41:43_: Tall. _41:46_: 000, which is equal to 1000 _41:50_: because the proper time is the _41:52_: same as the coordinate time. _41:54_: In the case where the displacement _41:56_: is purely in the time direction with _41:58_: displacement isn't moving, so some. _42:00_: That's a very potted recollection _42:03_: of what the special University _42:05_: for for velocity is. But. _42:10_: Team work for at the moment that _42:13_: that that velocity is. Simple. _42:15_: In the moment helical moving reference frame, _42:18_: because a particle is _42:20_: then moving only in time. _42:22_: That's another way of thinking about it. _42:23_: So the the the speed of this particle, _42:27_: it's just sitting there. _42:28_: The speed of me in this in this _42:30_: room is I'm moving through time, _42:32_: but not through space. _42:34_: So the rate of the rate of _42:36_: change of my time coordinate is _42:37_: the same as the proper time. _42:39_: And I'm not moving through _42:42_: space at all in this frame. _42:45_: So plugging those in to this _42:49_: I recover this and this one. _42:52_: And equation 4.7. _43:04_: And is the one which I know we've _43:06_: talked about a bit before about the. _43:12_: The argument of a perfect _43:15_: fluid being diagonal because _43:18_: there's no preferred direction. _43:22_: Because there's no shear. _43:24_: So if that's to be if that this _43:28_: question here is to match that, _43:31_: then this D will have to be 1. _43:35_: And. Uh. Throughout 4.7. _43:43_: And. Put with other thing _43:46_: is sage. 4.4. About 4.4. _43:53_: Um. _44:04_: Yes, which is that that _44:07_: implies that T00 is equal to. _44:12_: Rule plus. B P -, P. _44:18_: But 4.4. In the stream. _44:23_: Is that T is is equal to rho. _44:27_: And U0U0 but equal to row so _44:31_: this has to be equal to row. _44:34_: Which implies B is equal to 1. _44:44_: So the. The, the, _44:49_: the the logic there is that. _44:52_: When we were talking about but this, _44:56_: but here we were asking what _44:58_: essentially we're asking. _44:60_: So this is modelling, _45:01_: this is mathematical modelling. _45:02_: You're asking what mathematical _45:05_: structures can I use to? _45:07_: Pick up the important features _45:09_: of the these physical. _45:13_: Objects, and so the mathematical _45:17_: modeling of dust. What do we _45:20_: have available? We have the. _45:24_: Momentum of the particles. _45:27_: Which is, you know each each _45:29_: individual particle has a momentum. _45:31_: It's a momentum 4 vector. _45:32_: It's a special artistic you know _45:35_: the mass times the the you you get. _45:41_: M * U the If the special artistic _45:44_: form momentum we have the flux that we _45:47_: worked out asking how how these this _45:50_: assembly of particles so so that's the _45:53_: momentum of each individual particle. _45:55_: We also worked with the flux of vector, _45:57_: which describes how the assembly _45:59_: of particles is moving. _46:01_: How many particles are there going _46:03_: through a unit area in the speaker _46:05_: directions and the time directions, _46:06_: and and and so on and. _46:08_: So those two things we have to play with. _46:10_: Richika punch. What? _46:11_: What happens if we multiply if we take _46:15_: the tensor product of those two vectors? _46:17_: Call it the this this this tensor tea. _46:22_: Because we can compose, _46:24_: we can construct our second rank _46:26_: tensor from first rank tensors by _46:28_: somebody taking the other product. _46:30_: And we know we have to have a _46:32_: second rank tensor here somewhere, _46:34_: because if you remember if we look at this. _46:38_: And. Box of dust. _46:41_: Then as the dust is moving and _46:45_: then it's got dust in it. _46:47_: And add the box is moving if we _46:49_: are instead in a different frame. _46:51_: In which the box is moving. _46:54_: Then it will be length contracted. _46:56_: By a factor of gamma? By of of gamma, yes. _46:60_: So the density of the number density _47:02_: of of the things of the box will _47:04_: go up in our frame similar to _47:06_: the boxes got smaller. _47:07_: But when those particles are moving past us? _47:13_: All the same speed because they're just, _47:14_: they're not moving relative to _47:16_: each other at the moving past us. _47:19_: There's another factor of gamma which is _47:21_: the you know the gamma in gamma v ^2. _47:26_: So the. _47:28_: Number density goes up by a factor _47:31_: of gamma because of the contraction. _47:33_: And the energy density goes up _47:35_: by a factor of gamma squared. _47:36_: Because of that times the the _47:39_: change in the rush momentum of each _47:43_: individual particles so that gamma _47:46_: squared so when you change frame. _47:49_: From the moment alcovy frame, _47:51_: moment alcove with the dust _47:52_: to something else, _47:53_: you're getting a gamma squared. _47:54_: That's telling you that is not _47:57_: lowering transformation of a vector. _47:59_: It changes that that that gamma _48:01_: squared is telling there's _48:02_: rank 2 tensor somewhere here. _48:03_: That's hinting. _48:04_: You've gotta find a Rank 2 tensor _48:06_: to play with this here. _48:10_: What you said is that number density _48:12_: goes up when two objects several _48:14_: have speed relative to each other. _48:16_: Meaning that if I were able _48:17_: to move at the speed of light, _48:18_: would they see everything be a black hole? _48:21_: Can't really. _48:24_: And so that's just not a question. _48:27_: I mean, I think that. _48:31_: If you were. If something's _48:34_: moving past you very rapidly. _48:36_: Then yes, it becomes smaller and and _48:38_: and more and more compact and so and _48:41_: so the space-time around it would be. _48:45_: The measure would be different _48:48_: because you're moving. _48:50_: In the same way that that, that, _48:51_: that the length contraction, _48:52_: time dilation, the measurements of _48:54_: things happened differently because _48:56_: you're because you're moving. _48:57_: So that's the so the the the metric _49:01_: tensor will be the same but the _49:04_: coordinates in a different frame. _49:06_: Would be differ. _49:08_: Because everyone's frame is is _49:11_: equally valid that that's that's _49:13_: the relativity in the relativity, _49:15_: but in in in the generativity. _49:17_: It doesn't matter who where you are, _49:20_: who you are, how fast you're moving, _49:22_: your coordinates are as good _49:24_: as anyone else's. _49:25_: And GR is all about the mathematical _49:28_: consequence of that statement. _49:30_: You call your quarter perfectly _49:31_: good at everyone else's, _49:32_: so when you move at some very _49:34_: high speed and extra mass. _49:36_: Your coordinates would be _49:38_: different my coordinates. _49:39_: The metric will be the same the metric. _49:43_: The distance element of the _49:44_: space station moving through will _49:46_: be the same as geometrically, _49:48_: but the numbers you get out _49:49_: will be different. _49:50_: And what we learned from special _49:52_: activity is that if you have a. _49:57_: Collision of. _49:58_: This is this is it. _50:02_: One of the things that that provokes _50:04_: system of alarm when I'm teaching _50:07_: this in second year is if you have a. _50:10_: A collision between two relativistic things. _50:13_: Yeah, _50:13_: you have a a lump of a putty _50:15_: and a A some sort of barbarian _50:17_: commanded relativistic speed and _50:19_: hit the party and the and the two _50:21_: inelastic collision and and and _50:23_: and the thing goes off afterwards. _50:25_: And you, _50:25_: you balance Ross 54 momentum _50:27_: and you work out how much the _50:29_: different components are and and _50:31_: and the mass and blah blah blah. _50:33_: And there's more mass afterwards _50:35_: than there was before. _50:36_: And folk go what? _50:37_: How can we be more mass afterwards _50:38_: than before? _50:39_: Does that mean there's more? _50:43_: No, it doesn't mean. Yes, that's it. _50:47_: That's it. Once you have that _50:50_: in in that box if you like. _50:53_: There's no because this box _50:55_: has now had this other. _50:58_: Rustic party will come in and smash _50:60_: into the party and and go off. _51:01_: There's an awful lot of kinetic energy there. _51:04_: And that adds to the to the _51:07_: incrementum in that box. _51:08_: So there's no more gravitating _51:09_: stuff in that box. _51:10_: Not there's more mass in the box, _51:12_: but because or there's more mass _51:13_: in one frame and another frame, _51:15_: there's not more mass. _51:16_: They're just in the environment. _51:19_: Yeah, etcetera. _51:20_: I see running again, _51:22_: but but the the the point is _51:24_: that if the argumentum that is _51:26_: different is the source of the mass. _51:28_: Is why we have a tensor here, _51:30_: a geometrical object, _51:32_: and how we were able to, you know, see. _51:38_: These things to play with her, _51:39_: to make changes out of that. _51:40_: And so that question, _51:41_: that exercise is seeing, OK, _51:43_: stepping back a little bit, _51:44_: if you really were starting this _51:46_: from scratch by saying what _51:47_: do we have to play with, _51:48_: how could make changes. _51:50_: Then all you have to play with are you, _51:53_: you cross U&G, _51:53_: so how do you add those together in a _51:56_: way that's consistent with what you _51:58_: have sort of seen in in this case? _52:02_: So. _52:04_: So yes, that's a. Well, _52:08_: people are putting on their coach now, _52:09_: so I'll be quick question, _52:10_: but why do we need to, _52:11_: why do we need to add on top of the _52:14_: velocity vector here the the metric? _52:16_: Because physical condition can tell _52:17_: us why we necessarily need attention. _52:19_: Because we have this, yeah. _52:22_: We we don't have to I mean this _52:25_: isn't telling us we have to but _52:26_: but the the the point of the of _52:29_: the exercise is saying if you. _52:33_: If you start from nothing and see _52:36_: how would we go about about making _52:39_: attention from what we've got. _52:41_: Then what do we what do we have? _52:42_: We have, we have you and we have G. _52:45_: And and we can put them together in _52:48_: what seems to be the most generic form. _52:50_: Given some constraints. _52:52_: What we otherwise know the expression. _52:55_: And here this has to be consistent with this. _53:00_: What does that do? _53:01_: What does that tell us about the, _53:03_: the, the, the, the, the, the, _53:05_: the in principle things, _53:07_: the, the, the, the, _53:09_: the values of EBC&D in in that express? _53:14_: So that's that's using your imagination. _53:18_: What does the other part of the argument _53:22_: constrain that imagination to to to be? _53:25_: And I'd better stop there. And. _53:27_: No. Could you get somebody _53:28_: else to go but because, well, _53:30_: you do have any lectures to go to, _53:31_: but you probably have somewhere _53:33_: else to go this evening. _53:35_: And that I think that is I think _53:36_: and 5:00 o'clock this Friday is the _53:38_: end of teaching for this semester. _53:39_: OK, one moment too soon. _53:41_: And I hope you'll have a good time _53:43_: over the Christmas break and New Year. _53:46_: And I look forward. _53:48_: Well, I won't see you next semester, _53:51_: but I will I think keep the officers _53:55_: going every, every, every couple of weeks. _53:58_: I'll try to make an announcement _53:59_: of that in the next semester. _54:02_: But unless you are the officers. _54:05_: And have a good time. I won't. _54:08_: I'm sure we'll bump into you. _54:10_: Enjoy and.