Transcript of gr-sup2 ========== _0:09_: OK folks. Hello. Welcome back. _0:14_: The division two. _0:18_: Thank you for those who have put _0:21_: notes on the ballot and there were a _0:24_: couple of other ones who added one, _0:27_: quite detailed ones by e-mail, _0:28_: but both a couple of general ones _0:30_: which have transferred to this. _0:32_: So I think we have a number of things _0:34_: we can usefully talk about. And. _0:38_: I think there's a fairly natural order here. _0:42_: Which is I think I can talk about _0:44_: this the time reduced first. Umm. _0:47_: I think this question is what one of detail. _0:52_: I'll make you that a moment these, _0:54_: this one and this one. _0:57_: Think are linked. _0:58_: I think I'll do, _0:58_: I'll talk about that that that _1:01_: that this question 12 that is. _1:06_: Are fairly say you're turning _1:09_: the handle exercise that part of _1:12_: the the the the peel off of that _1:15_: exercise is more familiarity with _1:17_: with export manipulation and so on, _1:20_: and the idea of of components _1:23_: in different frames. _1:24_: It's there aren't some any magnificent _1:27_: epiphanies available there but _1:28_: it is really useful practice. _1:30_: However it is fiddly and it's easy _1:33_: to get lost and the notes that _1:35_: there's of answer slash notes that _1:38_: I have in the in all exercise _1:41_: compendium goes you says and as _1:43_: you will see the answer so it is it _1:45_: is a little bit vague so I think _1:47_: I do need to someone else pointed _1:50_: out as well I think I do need to _1:52_: go and expand that some point so. _1:54_: Of or postpone. _1:55_: I'll defer discussion of that question _1:58_: there until I get around to making _2:01_: an expanded version of that note. _2:03_: And so I think therefore the useful _2:07_: thing would be to talk about that, _2:10_: that I had trouble that for that first. _2:14_: This business of the time like so. _2:16_: So where we were there was the _2:20_: timelike geodesics of the local _2:23_: space-time and the picture. _2:26_: Here's this, _2:27_: yes. _2:29_: If we. We've said to here the. _2:39_: Alright. Well. The picture I _2:44_: want you to have in mind there. _2:47_: Is. This one here, a particular _2:51_: restaurant in a national frame. _2:53_: They are used in Minkowski space. Now uh. _3:00_: So, so where did that? _3:02_: So that that sort of makes sense, _3:04_: but I hope but this question of timely _3:07_: geodesics and mikovsky space sort of _3:09_: leaps out from nowhere to some extent. _3:12_: So where did that come from? _3:15_: If you think back to your _3:17_: study of special relativity, _3:18_: you will don't look at some point. _3:20_: Remember the twins paradox? _3:21_: That's where the where someone _3:23_: stayed on Earth and someone else _3:25_: heads off to another planet, _3:26_: comes and comes back and they end up. _3:30_: Younger than the person who stayed _3:32_: put you you remember that that that _3:34_: that's familiar is it, Wallace? _3:37_: At that point it's just that, that, _3:39_: that, that, that, that diagram there. _3:41_: So so all it is is, is the idea of, _3:45_: I mean configure diagram. And and. _3:48_: The idea that of of a world line _3:51_: which is going along the, the, _3:53_: the, the, the, the the time axis. _3:56_: And if you think of the. _3:59_: The twins paradox. Then if you draw the. _4:04_: That story if you're like ohh _4:07_: mccottry diagram. Then someone stays. _4:10_: On earth. _4:11_: So they they they they they they _4:13_: just stay at X = 0 for the entire. _4:16_: Duration. _4:18_: And someone else heads off. _4:20_: That's some. _4:21_: That's some speech to to turn _4:23_: around point and then comes back. _4:27_: And the traveller? _4:28_: And the stay at home person _4:31_: end up being different ages. _4:35_: Why? And there's a page where you can _4:38_: you you you you can. Talk about this. _4:42_: One of them is the to to to note. That. _4:50_: Because of the way that the interval _4:51_: works out, you know the, the, the, _4:54_: the the the invariant intervals. _4:55_: DS squared equals D T ^2 -, _4:59_: d X squared, the interval along the. _5:04_: Time like but not time axis. _5:08_: The these paths here is shorter. _5:12_: It's smaller than the interval _5:15_: along the the direct route. _5:17_: In other words, the less proper time elapses. _5:20_: Therefore in other words, _5:21_: that person ends up younger. _5:23_: That that that's one way of of _5:25_: just slicing right through the _5:27_: twins paradox and seeing why the _5:29_: the person moving is younger. _5:31_: We're just talking special relativity here. _5:34_: But you can make it a very so So what? _5:36_: What's happened here is that _5:38_: this is straight line here. _5:40_: It's longer than the dog leg lane, _5:42_: which is the opposite way _5:43_: around to Euclidean space, _5:44_: including Euclidean space. _5:45_: The straight line between points is the _5:48_: shortest distance between two points. _5:49_: And if you look at that a bit more, _5:52_: and and and and and think through it, _5:53_: you can deduce fairly straightforwardly _5:56_: that in Minkowski space, _5:58_: A straight line is the longest distance _6:00_: between two points is a straight line of _6:03_: makowsky space is the longest distance. _6:05_: In other words, _6:06_: it's the extremal distance. _6:08_: And if you recall what I _6:09_: said about you desiccate, _6:10_: one of the problems of eugenics is they are _6:12_: the extremal distance between two points. _6:14_: That's the the the definition of of _6:16_: one of the definitions of geodesic. _6:19_: In creating space, _6:21_: that's demonstrably true, _6:22_: because the if you if you could _6:24_: get string and you pull it tot, _6:26_: you have discovered the shortest _6:27_: distance between those two points, _6:28_: and that's a straight line. _6:29_: If the surface of a sphere _6:31_: you pull the string taught, _6:33_: then it forms a great circle, _6:35_: the shortest distance in miskovsky space. _6:38_: The straight line is the longest distance. _6:40_: It's a geodesic, so the. _6:43_: The person going along the. _6:46_: The the state home person here. _6:49_: They were the. _6:53_: The party who just moving the whole world _6:56_: line moves along their own time axis. _6:59_: They're the one following the geodesic. _7:01_: So the point of this is we _7:02_: have identified where geodesics _7:04_: are in Minkowski space. _7:07_: And this this, this person here the the _7:10_: the traveller is also following their _7:12_: duties because they are following the the _7:14_: the time axis in their own inertial frame. _7:17_: So time like straight lines you're like _7:20_: are there UD pics in Minkowski space? _7:24_: And what the equivalence principle _7:26_: tells us is that. _7:30_: OK, we understand how how motion _7:31_: works in Minkowski space. _7:33_: Especially every we move along _7:34_: the our geodesic is moving along _7:36_: the time axis of inertial frame. _7:38_: And the Council says the same is true in GR. _7:43_: So in GR the description of how _7:46_: you move is you move along the. _7:50_: Time like geodesics of our _7:52_: local natural thing, _7:53_: because the local national frame, _7:55_: the freefall frame is the one which _7:58_: is the same as Minkowski space. So. _8:01_: That's why the that's why it's significant. _8:06_: This diagram that, _8:07_: that, that, that, that, _8:08_: that's the point of the of the of this _8:11_: argument here because we understand _8:12_: how physics works in special activity. _8:15_: We can immediately transplant that _8:16_: to an understanding of how physics _8:18_: works in the local national frame. _8:20_: And because we can always identify _8:21_: the local natural frame at any point, _8:23_: we just jump up and down. _8:24_: We understand how motion works. _8:27_: We're definitely around a particular _8:29_: point and thus by integrating that _8:31_: up we understand how motion works in, _8:33_: you know, _8:34_: in a non impossible displacement _8:36_: in our curved space-time. _8:40_: So this is the first application, _8:42_: if you like, of the college principle in GR. _8:47_: Yeah, importing a bit of _8:49_: physics from special relativity. _8:51_: And I think that there's quite next we _8:54_: think about that because that it is. _8:57_: No good harks back to me you're familiar _8:59_: with, I think from special activity. _9:02_: That I at this point over over complicating _9:06_: this or talking too much or this. _9:09_: Have different properties and. _9:12_: Without spaces. _9:14_: How can you say that? _9:17_: The longest distance, yeah, _9:19_: because of the minus sign in this sphere. _9:23_: Yes, but the sphere on the _9:25_: surface of a sphere, so. _9:30_: The L ^2 is equal to. _9:35_: The. Theta squared plus. _9:39_: Sine squared Theta, D, Phi squared. _9:45_: With the plus sign. _9:47_: So if you if you think that _9:51_: the deviations variations, _9:52_: then any what that is telling you _9:55_: that what plus is telling you is that _9:58_: any deviations from a straight line _10:00_: are going to increase this length. _10:03_: Whereas in this case a deviation is _10:06_: straight line. Delta X ^2 being nonzero, _10:08_: or delta Sigma squared, whatever. _10:10_: I'm going to decrease that, that, _10:11_: that, that, that thing. So it's this. _10:13_: So the answer is the same. _10:15_: And the signature, _10:16_: so the the the signature of. _10:20_: This officer is here. _10:21_: It's the same with the signature so-called _10:23_: of Euclidean space which is all, all, _10:25_: all, all, all positive and the and the _10:28_: signature of special activity special _10:30_: activities metric is plus, minus, _10:31_: minus, minus which is the same as the _10:33_: signature of all of generativity metrics. _10:35_: So all of the solutions to instance _10:37_: equations are plus, minus, minus, _10:39_: minus or minus depending depending _10:40_: on your on your convention, _10:42_: but they they are a ± 2 and _10:46_: all the solutions of. _10:48_: Of general activities, _10:49_: all the solutions of our _10:50_: sense equations generativity. _10:52_: All the solutions describe the universe _10:54_: at the same signature because they are _10:56_: all mappable to the Minkowski space one. _10:59_: And so yeah, so other signatures as possible, _11:02_: mathematicians care about those. _11:04_: As physicists, we only care about _11:06_: signatures which are plus or minus. _11:10_: That's. Question there. Twins. _11:16_: Move along the geodesics. _11:18_: Who does? _11:19_: Both of the twins have, yes, _11:21_: so they are both moving along _11:22_: a geodesic in their frame. _11:24_: So do you mean all the inertial _11:27_: frame for special relativity? _11:31_: Moving along the yes so because _11:34_: sorry I said international so in in _11:37_: each of these cases and the well so _11:42_: something like like so that's the. _11:47_: 2 frame access. _11:50_: That from there to there is also a judic, _11:52_: but there's a space like geodesic. _11:56_: So all all of the straight lines _11:59_: in Minkowski space are geodesics. _12:01_: And because of the way _12:02_: this speculative works, _12:03_: it turns out that there is _12:06_: always a A-frame in which. _12:09_: With Travelocity is not _12:10_: moving in the offering. _12:14_: So. My parrot. _12:21_: Heard you could contact. Control tab. OK. _12:29_: So that was what this this was about. _12:37_: How do you go from application _12:39_: application in Windows? Ohh. _12:47_: Is that that's not very pages. Ohh right. _12:53_: Did that illuminate? OK. And the? _12:59_: I think this is a good question to to _13:03_: look at the what's essentially the. _13:06_: Um comma goes to semi colon rule in the _13:10_: case of breach of 420? I think it was. _13:15_: That's what we're looking at here. _13:18_: Was and I will, I'll, _13:21_: I'll just talk through this _13:23_: rather than writing about it much. _13:25_: That'll be this bigger search up. _13:33_: And here. _13:39_: So. _13:41_: The way we get to this. _13:43_: Expression here. Is. _13:45_: We're doing this calculation _13:47_: in our local inertial frame, _13:49_: so we're doing this calculation _13:51_: in the frame, which is. _13:56_: Locally flat. And in which the _13:59_: first derivative of the so. _14:01_: So the metric is the Minkowski _14:03_: metric diagonal plus, _14:04_: minus and the first derivatives of _14:07_: the metric are all zero and when we. _14:13_: Do that we discover Green 419 _14:15_: that the reason that the the _14:17_: the derivatives of Riemann. _14:19_: Tanger are that there will third _14:23_: derivatives of of the metric. _14:25_: And based on the, the, the, the, _14:27_: the expression we had for the room _14:29_: intensive in terms of the metric _14:31_: in the local natural frame earlier _14:33_: on a couple of questions back. _14:35_: And from that equation 419. _14:39_: We can then play some games late into the _14:43_: night and discover that there's a this _14:46_: particular combination of permutations. _14:48_: Of the remaining of the component of _14:51_: the Ranger has it is an identity. _14:54_: The various things cancelled _14:56_: out so that that that they are. _14:59_: The Alpha, beta are the same in these cases, _15:01_: and the new new Lambda cyclically, _15:04_: cyclically, _15:05_: cyclically permute in the other three terms. _15:07_: OK, so that is a deduction. _15:11_: Fiddly but not you deeply intellectually _15:14_: challenging deduction from the _15:17_: expression we had earlier on in the, _15:20_: you know, _15:20_: a few things back from for the _15:22_: Riemann tensor in terms of _15:24_: derivatives for the metric. _15:26_: But this is only true in the local natural. _15:30_: We calculate this in the _15:31_: local inertial frame. _15:32_: So it's true in the local inertial frame. _15:35_: Are we stuck in the local national frame? _15:37_: No, we are not. _15:39_: Because this involves just _15:42_: single derivatives. _15:43_: And in the local natural frame, _15:44_: one of the the key property of _15:47_: the local nature frame is that _15:49_: all the gammas or the chronicle _15:51_: deltas or or the chronicler? _15:53_: All the Christoffel symbols, _15:55_: all the gammas, are all zero. _15:58_: Because the local national team is _15:60_: flat and the the coordinates the _16:02_: basis vectors in the local frame. _16:04_: You don't rotate if you like as you _16:07_: move around the around the space. _16:09_: So that means that. _16:10_: If we were to calculate what _16:13_: the covariant derivative. _16:15_: Of the remains of these components _16:18_: of the human tensor. _16:20_: Was. _16:21_: Then we would and I got something like this. _16:27_: We wrote that down and asked, OK, _16:29_: what is that? How do we evaluate that? _16:30_: Well, in the local national frame, _16:33_: all the chronicle deltas _16:35_: are see chronica deltas. _16:38_: All christophel symbols are zero. So. _16:43_: The expression for what this this, _16:45_: this covenant drive is, you know, _16:47_: semi colon U would be common _16:51_: mu plus chronicle delta. _16:55_: Christoffel symbol. _16:55_: You know the gamma blah blah blah, _16:58_: so it would turn into this. _17:00_: Should be calculated that in the _17:03_: local natural frame we get this _17:05_: and we get the answer to 0. _17:07_: So a question there. _17:13_: Each term in terms of the partial. _17:17_: And then you will find out. _17:21_: So, so in getting to getting to _17:26_: from 419 to 420. What you do is it _17:30_: just write that whole thing out and _17:32_: and and lots of things just cancel _17:35_: so so you there just by algebra. _17:37_: A bit. It's just fiddly and you _17:40_: get all the things wrong. _17:42_: The question is how it was the, _17:44_: the, the, the, the, the smart move, _17:45_: the the the speaker of the move _17:47_: that gets you from there to there. _17:49_: And it is that when you evaluate _17:51_: that in the local natural frame, _17:53_: it turns into that. _17:54_: In other words, _17:55_: you can go in the other direction. _17:57_: You can see if we found it found _17:59_: this in the local freedom, _18:01_: then it would also be the case that. _18:04_: We'd have this. _18:05_: Because these two things were evaluate to _18:08_: the same thing in the local national tree. _18:11_: But this that isn't. _18:13_: But this is a potential equation. _18:16_: And so if it's true in the _18:18_: in the local natural frame, _18:19_: if those can put your components _18:21_: have that relationship in _18:22_: the local national frame, _18:24_: then when you switch back to a _18:27_: non inertial frame this tensor _18:29_: equation would still be true. _18:31_: And this is one of the if you like 2 _18:34_: separate comma go semi colon rules. _18:36_: That you are exposed to this one, _18:38_: which is the mathematical trick, _18:40_: which is the. _18:44_: The IT involves you getting out _18:46_: of the local inertial frame. _18:48_: And the other one is the _18:50_: equivalence principle. _18:51_: Who says that if you have a law expressed in? _18:56_: In in geometrical form in special activity. _19:00_: We are the great derivative is. _19:01_: It's just with the at the _19:04_: ordinary derivatives comma. _19:05_: Then there's nothing more complicated _19:08_: in in in in going to the. _19:12_: Equals generativity. _19:13_: So there's an another _19:14_: remark about the stuff, _19:16_: but the the the time logistics. _19:23_: And our physical law expression geometrical _19:25_: form is special activity turns into the _19:28_: corresponding one in general activity, _19:30_: the comma in social activity, _19:31_: to assist the semi colon in generativity, _19:34_: and that's also called the _19:36_: Komodo semi colon rule. _19:38_: But the two things happening there, _19:40_: this one is a mathematical trick. _19:42_: The other one is a is a is a _19:44_: physical statement that you're _19:45_: allowed to do that and nothing else. _19:47_: There are no extra terms that _19:49_: appear because of curvature in the. _19:55_: Does that make sense? _20:02_: No, no, no, no. I mean that _20:03_: would take you an afternoon to go _20:05_: through all the yeah, so, so I. _20:12_: It's. In some quarters it is _20:16_: a useful exercise to say, _20:18_: be able to derive such and such. _20:20_: It's sort of useful if it obliges people _20:22_: to think through the notes and so on. _20:24_: But most of the things in general _20:27_: relativity and everything that's _20:28_: terribly useful as an exercise anyway, _20:29_: has or as an assessment thing. _20:30_: But in most cases in general activity _20:32_: is just infeasible in an exam because _20:34_: it just takes too long writing exam. _20:36_: Writing exam questions for GR is murder _20:39_: because anything is either trivial or an _20:41_: afternoon's work to to to to calculate. _20:44_: Things in the middle which are, _20:46_: you know, not just insulting to you, _20:49_: but which are still doable in a _20:52_: readable amount of time without _20:53_: panic is is is tricky. _20:55_: Share my pain. _20:59_: I hate it. I actually mind marking less. _21:03_: OK. That was. Um. So that's sort of. _21:11_: This put another thing it's useful to. _21:20_: Right, I'll come back to that and _21:23_: useful to look at this at 3:29. _21:27_: If you do anything. _21:28_: After with you. And which was? _21:40_: I think this would also be a _21:42_: useful thing to talk through. _21:49_: Yeah. And so it it. We have bigger. _22:01_: So. It appears like say you can't _22:05_: do the christophel symbols for the _22:06_: surface of unit sphere and just asking _22:08_: you to at that point turn the handle _22:10_: and calculate the component of the _22:12_: cover Centre in these coordinates. _22:14_: So extra 3.4 we got. _22:19_: I'll even write them down on this _22:21_: bit of pig paper. These would be 4. _22:25_: Just for reference in a moment. _22:32_: And. _22:39_: We we had a G Theta, Theta equals 1G Phi, _22:46_: Phi equals. Sine squared Theta. _22:50_: And other zero and the because that's _22:55_: diagonal the components of the. And. _23:05_: And. _23:08_: Inverse matrix. The matrix with _23:10_: indices raised is just one over those, _23:13_: because the, the, the, the, the, _23:15_: the those two matrices are diagonal. _23:19_: And the Castro symbols we worked out. Where? _23:25_: And. _23:29_: In one may have worked out in _23:31_: that where if they're nice neat _23:33_: table of them, yes there is. _23:38_: That. Gamma. A Theta Phi Phi _23:44_: is equal to minus sine Theta. _23:48_: Cos Theta, gamma Phi. _23:52_: Theta, Phi which equal to. _23:55_: High Phi Theta equals. _23:59_: Same teacher and was there another one? Yeah. _24:06_: I think the other ones were all zero. Yeah. _24:14_: You know. Thank you. _24:17_: So that's that that I'm just copying _24:19_: and actually 324 at that point. _24:22_: So calculate the components of the _24:24_: coverage sensor for these coordinates. _24:27_: So what's R? In this case, _24:30_: and what we we know is that our. _24:35_: The answer is. I think it's. _24:44_: 349. _24:47_: Yes, I've got that place, 340. _24:58_: It is this. _25:02_: So it's a fairly messy thing _25:04_: which involves the Christoffel _25:06_: symbols and derivatives of them. _25:10_: So if we ask. If we. _25:15_: Pick out one of the components of _25:20_: the human tensor. I think what I. _25:28_: What I'm picking here is R. Utah Phi. _25:34_: Peterffy. Then that's going to be. _25:39_: G Theta Alpha, R alpha. Phi. _25:45_: Teacher's fine. So that. _25:49_: So for instance this, _25:51_: and we obtain that by doing that contraction. _25:54_: And looking back at it agreed to 49. _25:58_: That's going to be G. These are alpha. _26:02_: Ah, right, that's great. _26:04_: Tedious gamma alpha. _26:07_: Chucky and five. Fiji. _26:12_: Comma, Theta minus. _26:15_: Comma I was alpha. And GQ. _26:25_: Phi, Theta comma. Phi plus. Alpha. _26:35_: How can we you? Why don't I _26:38_: just copy that down rather than _26:42_: trying to to to to do it? Sigma. _26:47_: It. Sigma. 5-5 minutes. And. _26:56_: Alpha. Sigma Phi. Sigma Theta. _27:05_: Bye teacher. And. So all I'm _27:10_: doing here is looking back at. _27:15_: This expression here and so _27:18_: filling in the filling, filling, _27:20_: filling, filling in the slots. _27:23_: And he got a bit of a mess, _27:25_: but I can do the alpha summation immediately. _27:29_: Very quickly, because we know. _27:33_: Yeah. That. G Theta Phi. _27:38_: Is 0. And GT to Theta is 1, _27:42_: so the alpha will be the the some _27:45_: some there where alpha is equal to _27:47_: Phi is 0 and what we're left with is. _27:51_: Just that expression there with alpha _27:54_: turned into Theta. And so we have. _27:60_: Theta, Phi, Phi comma, Theta minus. _28:04_: The Phi, Theta, comma Phi plus. _28:09_: Sigma Theta. Sigma Phi Phi minus Theta, _28:16_: Sigma Phi, Sigma Phi Theta. _28:22_: And then we look back at. _28:25_: This expression here. _28:26_: And we discover. _28:28_: That. _28:30_: And. _28:33_: That's. With this. _28:39_: Did I see that was going to _28:40_: be for that with that teacher? _28:41_: That was theater 55. And. _28:48_: 2424. _28:52_: And. If you don't, yeah. _28:57_: Three to five Phi. At the. _29:01_: Teacher is going to be. _29:05_: Minus Cos squared Theta. And. _29:10_: Plus sign. Squared Theta. _29:17_: Um Theta 5 Theta. Is is 0. _29:23_: Because it's one of the others 30 here. _29:27_: Theta. Something Theta _29:29_: is a is is going to be 0. _29:34_: For any for any any Sigma because _29:35_: it's one of the others are zero. _29:39_: Phi, Theta something. _29:41_: Phi is going to be nonzero only if _29:46_: when when Sigma is equal to. Phi. So. _29:50_: That the only time that was five from _29:53_: the sum and that would be minus gamma. _29:56_: Peter, Phi, Phi. Phi. _30:02_: Five future. Which is. _30:07_: Equals to future Phi. _30:09_: Phi is minus sine Theta Cos Theta times. _30:18_: Costita over. Sine Theta. _30:23_: That gives us our. Cost square Theta. _30:27_: My mosquito it so that cancels with _30:30_: that we end up with sine squared Theta. _30:39_: I'm not impressed about that. _30:40_: I didn't make a assign error there. _30:44_: Immediately impressed, actually. _30:46_: But the point even if I had made _30:48_: a sign error, then the point of _30:50_: this looks rather more forbidding. _30:52_: The thing that needs practicing here, _30:55_: the thing that something like that size is _30:58_: the sort of thing that I think would be, _31:01_: is the sort of thing that would _31:02_: fit in an example question. _31:03_: That's not a promise, _31:04_: but that's the sort of side of thing that _31:07_: that that fits in exactly that question. _31:09_: And it's easy if you've practiced _31:11_: them and you've you've failed to. _31:13_: So you've managed to avoid just _31:15_: losing track and getting exposure _31:17_: of of signs all over the place. _31:19_: The practice comes in in in saying I, _31:22_: you know, I spot that zero, _31:24_: so I don't have to do anything else. _31:25_: That's zero for all the all the _31:27_: possibilities of stigma, this one is. _31:30_: Is nonzero only when taking equal to five. _31:32_: So which to identifying which _31:34_: terms survive the sum and and and _31:37_: just keep track of the algebra. _31:38_: So an easy question is if you mark _31:41_: if you have practiced it because _31:44_: it's easy to understand, _31:45_: it's easy to understand than to do _31:47_: on the on the hoof, as it were. _31:50_: And that's really one of the _31:53_: components of the Riemann tensor. _31:56_: There are what? Two by two? By two by two? _31:59_: There are 8 components of the of of the. _32:04_: Or even tension. So what are the other ones? _32:07_: Well, that was our Theta Phi. _32:12_: Peter fight. _32:14_: But we know from the symmetries of the. _32:20_: Remind tensor that were noted down _32:23_: that is equal to minus R, Theta, _32:28_: Phi, Phi Theta. _32:31_: We were just swapped over the last two. _32:36_: Terms is equal to minus. _32:39_: Are Phi, Theta, Theta, _32:41_: Phi whose walked over the 1st 2? _32:44_: And it is equal to the swapping over _32:47_: the first pair and the second pair. _32:49_: That's just gets back gets back _32:50_: what we what we started off with. _32:53_: And what I'm looking at there are. _32:56_: And. _32:60_: And the two. _33:06_: These things here so the the. _33:10_: The symmetries of the human _33:12_: tensor which are obtainable _33:13_: from this expression in the. _33:17_: Local national frame. _33:20_: Which is and and and that _33:23_: expression there is not. _33:25_: We can't turn the this comment _33:27_: into semicolons in that case. _33:29_: Can anyone see why not? _33:32_: In in this case here, why can we not _33:34_: just turn those comments into semicolons? _33:41_: It's Robert geodesics. _33:43_: It's because the second derivatives. _33:45_: Because the it's that that _33:48_: is not an obvious remark, _33:51_: but these are second _33:52_: derivatives of the metric, _33:54_: so if you think of what the the. _33:58_: colon MK would be it would involve _34:01_: terms which were, you know, _34:03_: comma M comma K plus derivatives _34:05_: of the Christoffel symbols as _34:07_: well and they don't necessary. _34:10_: And the derivatives of the christophel _34:12_: symbols don't necessarily disappear. _34:13_: So that by itself can't turn doesn't is is _34:17_: not the local national frame version of our. _34:21_: Covariant derivative but this one here. _34:25_: We can use that to calculate _34:27_: these symmetries here. _34:28_: Because these symmetries don't _34:29_: involve any current drives at all, _34:31_: their attention equations again. _34:33_: Even though we covered them in _34:35_: the local national frame and so. _34:37_: Be intentional equations. _34:38_: We can say they they are they _34:40_: are true in an arbitrary frame, _34:43_: but that's a long way of pointing _34:45_: out that that that these particular _34:47_: index swaps give us information about _34:49_: what the other things are here. _34:51_: If we look at something like our Theta. _34:55_: Theta, Phi, Phi. _34:58_: We can. _34:58_: Know that that change is sign if we swap the _35:02_: last two indexes so that is equal to minus R, _35:06_: Theta, Theta, Phi, Phi. _35:09_: And which means there's of course _35:11_: there has to be a has to be 0. _35:13_: So although there are two terms, _35:15_: 2 instructions, _35:16_: 2 components of the Riemann tensor in. _35:21_: On the surface of the sphere _35:23_: in two dimensions. And. _35:25_: Most of them are equal. _35:28_: Some of them are zero. _35:30_: The long way I have to actually calculate. _35:33_: Is a sample one a sample code 01 _35:37_: which we do by doing this relatively _35:40_: simple once you get used to it some. _35:45_: Did that make sense? So anyway, _35:49_: in, in, in, let's go back to. _35:56_: So with that, yeah, because this is _35:59_: the excitement of mine in a way. _36:01_: There's slightly less to that _36:02_: exercise than meets the eye. _36:03_: It looks so forbidding, but it is, _36:06_: it's actually quite useful exercise _36:08_: to go through step by step. _36:09_: And you've seen me do this here, _36:11_: go off and do that again. _36:12_: You've in a sense we, _36:14_: we from a standing start and and and and _36:16_: and and and rehearse that sort of thing. _36:19_: It's just a matter inject gymnastics. _36:23_: Put the hand together as I thought. _36:26_: OK, question there. _36:30_: The matrix. The matrix, sorry. _36:32_: And. Well, we have, we have, _36:34_: we, we, we, we did there. _36:37_: But we we sum that over alpha that was that. _36:41_: That was that with alpha equals Theta and _36:45_: that equals Phi but the G Theta Theta. _36:48_: Is 1 and G Theta Phi is 0 so the _36:52_: second term with alpha equals Phi. _36:55_: Is you know this is something _36:57_: they multiplied by zero, _36:58_: so it just disappeared. _36:59_: So that's how we got from there _37:01_: to to to to to there that that _37:02_: that is the answer to that sum. _37:06_: Again, something that's really _37:07_: obvious once you've done once, _37:08_: you've got through it by hand once. _37:13_: With the water of that exercise, _37:17_: I think was a was a sort of _37:21_: codicil to that exercise. Umm. _37:29_: Why would you not use equation 355? Um. _37:39_: Hi yes so this expression here _37:42_: looks like it's a a a shortcut _37:45_: to what we calculated there, _37:47_: but that applies only in _37:50_: the local national frame. _37:51_: So the the the which is _37:53_: office of the tangent to the. _37:54_: But in this context that applies _37:56_: only to the tangent to the _37:58_: the the surface of the sphere. _37:59_: But what what we what we want is the _38:02_: agreement tensors in the in the on _38:04_: the on the surface of the sphere. _38:06_: So you had waiting yesterday, but. _38:09_: Uh. So more than with the pilot _38:13_: as I speak, speak faster. _38:15_: Ohh, thank you. Right, good. _38:17_: That was a an e-mail question. _38:20_: So this I think that that that's _38:22_: a natural move on to what the _38:25_: richest tension scaler and. _38:26_: Mean and I think that this naturally _38:29_: follows on because ohh yes. _38:31_: So the rest of that of that question 329 and. _38:37_: Had eyes calculate what the? _38:42_: Richie Tensor was in these these on _38:45_: on the in the in the space and these _38:49_: coordinates are are Theta Theta equals one. _38:53_: Are Phi, Phi equals sine squared Theta and? _39:04_: Now I'm afraid I'm going to _39:05_: give you a slightly hand _39:06_: waving answer to this question. _39:08_: What does what do these things mean? _39:11_: Because physically. _39:17_: I I think I I should be able to find _39:20_: a deeper way of of saying this, _39:22_: but if you look at those. _39:25_: Those expressions then what you. Can see is. _39:34_: And. _39:38_: Yes, that's not a very pretty. _39:45_: That's going to be. Five. _39:49_: And so that is going to be. _39:53_: Teacher. What is? _40:00_: And that's going to be. So if you _40:04_: look at stephco pullers. Then. The. _40:10_: The fact that as you. A change. _40:15_: Fi are you change change Theta _40:18_: I as you change the the, _40:20_: the sort of longitudinal coordinate, the? _40:26_: The the volume element, _40:28_: the area element at that point is, _40:32_: is is is proportional to the changes _40:35_: in proportion to the site to the the. _40:38_: Change you make in the. Theta. _40:43_: But as you changed Phi as _40:45_: you open up that that angle, _40:48_: the area element in that little block. _40:54_: There. _40:57_: Changes in proportion to the sign _40:59_: of the of the of this angle here. _41:02_: So it what the the the Richard _41:05_: Chance is picking up is the the _41:08_: sort of sensitivity if you like. _41:10_: I'll just be that I've just was just _41:13_: popping about the the the sensitivity of _41:16_: the this area element to the changes in _41:20_: depending on where in the coordinates _41:23_: in the coordinate surface you are. _41:26_: And. So it's. _41:30_: Another slightly hand waving. _41:31_: We think the Richie tensor is giving you _41:33_: information about the way the coordinates _41:35_: change as you move around the space. _41:37_: In a way that's useful for the sort of _41:39_: sort of calculation the Riemann tensor is. _41:41_: Is giving you more information about exactly _41:44_: how you how how a vector will change. _41:47_: All its components will change _41:48_: as you go in in a, _41:50_: in a circuit or in a space, _41:53_: but the remains the richest center _41:55_: is of summarizing that to the _41:56_: sort of need to know bits. _41:58_: For doing calculations on the _41:59_: surface of the sphere in a way. _42:02_: So there's more in the in the Riemann tensor _42:04_: than you sort of need for calculations. _42:06_: And the richer tensor is the _42:09_: encoding the the the behaviour of the _42:11_: coordinates I think as you move around. _42:13_: And the and in this case so, so. _42:16_: So the way we we we we calculate that is. _42:21_: Well, _42:21_: the way we we we we calculated by _42:24_: seeing our Theta Theta is equal to and. _42:28_: She's. _42:32_: Um. Could you be G? _42:38_: Alpha. How do I write this? And. _42:51_: Yeah, so _42:55_: RGL is equal to G. Key R. _43:01_: IG KL so R, Theta Theta is going to be G. _43:08_: And. Let's see Alpha, beta R. _43:13_: And. Of. Theta, beta. Teacher. _43:22_: And that will only be. _43:28_: Which would be equal to G _43:30_: Theta Theta, R Theta, Theta, _43:33_: Theta Theta plus G Phi Phi. Are. _43:40_: Phi, Theta, Phi Theta. _43:44_: And since that is equal to 0, _43:47_: that is equal to. Umm. _43:53_: 1 / 1 over sine squared Theta times. _43:57_: Sine squared Theta equals one. _43:58_: That's how we calculated these. _44:01_: And the curvature tensor, _44:02_: I'm not gonna go through it. _44:04_: Similarly, ends up covering scaler. _44:07_: Similarly, ends up being a number. _44:10_: Which is constant over the whole _44:12_: surface of the of the space. _44:15_: So the, the the the the sphere _44:17_: has a constant curvature, _44:19_: the same curvature at at at every point. _44:21_: The, the, the, the behaviour, _44:22_: the sensitivity of ordinary element _44:24_: to the the position on the sphere _44:27_: varies as you move over the over the _44:30_: sphere in ways that this picks up, _44:32_: but the but overall curviness of the _44:34_: sphere is is the same everywhere. _44:39_: The, the, The you know the the _44:41_: units of of of that culture are _44:44_: not particularly interesting but. _44:46_: I would like if you have a _44:48_: space that changes curvature _44:49_: then I would not be concerned. _44:51_: Yes so I speak so sorry. _44:55_: For example so see this with with _44:57_: the the surface of an ellipsoid. _44:60_: Then there are would have a probably a _45:03_: variation on depending on the on the on the, _45:05_: on the feature, the feature parameter. _45:07_: Yeah so there would be it would _45:09_: be a coordinate dependence to _45:10_: the to the curvature. _45:15_: OK, so talk faster, talk faster. And. _45:23_: Right, I think. _45:26_: I'm not going to mention that _45:28_: just now because that section 413 _45:30_: is the is the alternative way of _45:33_: getting the engagement retention, _45:34_: which I I isn't that dangerous band thing, _45:36_: but which I sort of wanted to give up, _45:37_: give bodies worth to. _45:39_: I can put pick that up in an office hour _45:43_: if anyone wants to grab me next Thursday. _45:47_: Like I said, this point here _45:51_: is picked up by the stuff in. _45:55_: Part 4. Which? Is. Um. 4. _46:10_: Yeah, the stuff in 413 is quite nice, _46:13_: but I I, I it's, I want to think _46:16_: of it and it's quite nice. _46:18_: I don't want to put a dangerous _46:19_: bend thing there to discourage _46:20_: you from have a look at it, _46:21_: but I sort of don't want you _46:23_: to spend too much time on it. _46:26_: What I said back here was. _46:30_: There's this thing here. _46:31_: So the question. Was. With the. _46:39_: Right the the the the audio component. _46:41_: But when I'm saying so, _46:43_: I think this is essentially it'd _46:45_: be useful to to. Pretty good. _46:49_: To. Go go to these arguments. _46:54_: I again. And no. _47:07_: And. Trying to recall the steps _47:12_: just before this, so the. _47:19_: The the. It seems comes _47:24_: from from rearranging this. _47:27_: This version of the components in this _47:31_: frame here so the the rule comes about _47:34_: from the consideration of the the dust. _47:37_: So the energy momentum in in in in _47:40_: this box just sitting there moving _47:42_: forwards into time is dependent on just _47:45_: how much mass there is just so much _47:48_: material there is in the case of dust. _47:50_: Dust remember is this idealized thing _47:54_: which is isn't moving it there's _47:55_: a frame in which it isn't moving. _47:58_: So there's no. _48:00_: Contribution to the argumentum _48:02_: from from that. _48:04_: There's no, there's no internal, _48:05_: there's no internal shears. _48:07_: It's it's a perfect fluid. _48:08_: There's no internal stresses and strains _48:11_: and because it's not moving inside the box, _48:14_: it's not banging off the edge _48:16_: off the side of the box. _48:18_: So it's not creating any _48:19_: pressure in the in in the box. _48:20_: So the only source of energy momentum in _48:23_: that notional box is due to the the mass, _48:26_: the mass density move forward _48:28_: into the future. _48:29_: And that's so that's where the the the. _48:34_: The the, the, _48:35_: the the T00 component comes from. _48:37_: I have to just take the the the energy _48:40_: density in the in the box which is is in _48:43_: this country the same as mass density. _48:45_: And. This. _48:48_: This question of of why the? _48:51_: Engagement engagement tensor is _48:54_: proportional to the unit unit matrix. _48:59_: Is um. _49:01_: And I have to find a better _49:03_: way of explaining this. _49:04_: Because one of these things _49:05_: that when you so feels right, _49:07_: but I'm not sure how I would _49:08_: go through the steps to to _49:10_: to to to to expand on it. _49:13_: If it isn't a preferred _49:15_: direction, then the that. _49:20_: The the that tensor had to _49:22_: be rotationally symmetric. _49:24_: And if there's no shear, _49:26_: then there can't be any. _49:29_: Contributions to environmental _49:31_: push depend on as something moving _49:35_: in this direction across a plane _49:39_: in a different physical plane. _49:42_: Um, and so you can so I _49:45_: see this the the expansion, _49:47_: I should tie down a little more. _49:51_: Fundamentally, it's the rotational symmetry. _49:53_: Rotational symmetry of of that situation _49:56_: means it has to end up being diagonal. _49:59_: And and and so if you've got that far, _50:02_: then the rest is just sort of algebra _50:04_: in the sense that you can expand _50:07_: that or you're adding subtract this. _50:13_: Matrix here. _50:14_: Turn that into something which is. _50:18_: Purely temp time like I think at _50:20_: least one question, 22 questions. _50:22_: Also important because you have the matrix. _50:27_: They don't understand. _50:31_: Yeah so so this we've just because _50:35_: we've we've we've taken that out _50:37_: into two matrices added AP here _50:39_: into practical corresponding P _50:41_: there purely so that this is then _50:43_: proportional to the the the metric. _50:45_: So that the so that ends up _50:48_: being paintings the metric. _50:50_: And in the local national freedom, _50:52_: that means that being proportional to T _50:55_: times the velocity 4 vector in that frame. _50:59_: Therefore we can write this as _51:02_: an in that frame. _51:03_: In the local inertial frame, _51:05_: the we just got we can write this. _51:10_: So we can write down this in the local frame. _51:13_: And because the at that point _51:15_: at the local dash frame, _51:17_: the still instantaneously _51:19_: call moving reference frame. _51:22_: And because of tension in that frame, _51:24_: it's a tension equation everywhere. _51:28_: So that's our. That, that, _51:31_: that, that is quite a lot _51:33_: happening in that paragraph. _51:34_: There's a couple of paragraphs, _51:36_: there's certain amount of of of _51:38_: of of physics motivated see to the _51:40_: pants hand waving combined with _51:43_: a certain amount of, you know, _51:46_: like cunning in prisons up right away. _51:48_: Combined with. _51:51_: Identifying the fact that in in this _51:53_: moment helical movement reference frame _51:55_: these vectors take a nice simple form. _51:59_: Combined with turning this into a matrix, _52:00_: therefore turning it into a _52:02_: matrix tensor equation in that _52:05_: local moving reference frame. _52:06_: At which point you realize that _52:08_: once you've written equation, _52:09_: you've written it down quite quite generally. _52:11_: So there are a number of different steps _52:13_: using a using everything from physics to, _52:16_: you know, dumb algebra. _52:18_: In that passage. _52:20_: Which because. _52:24_: With the general through _52:25_: between everything, yes. _52:29_: Pretty specific, no? _52:30_: If a tensor equation, geometry, _52:32_: geometry because it can't be _52:34_: from specific the the components _52:35_: might be worked out in A-frame, _52:37_: but the tension equation, _52:39_: because it's purely topic geometry, _52:40_: it's really dependent has to be reminded. _52:44_: OK, that was more talking _52:47_: than I braced myself for, _52:49_: but I'll see you. _52:51_: The last lecture is on Wednesday, _52:54_: so I shall look forward with _52:56_: the contribution to your