Transcript of gr-l10 ========== _0:08_: Welcome back to lecture 10. _0:10_: And this is a second last lecture, _0:12_: and we are remarkably enough on time, _0:16_: so we needn't scamper too much. _0:20_: Now, where we got your last time was a. _0:23_: They got this far as the I think _0:25_: I got as far as as as the. _0:27_: I was revisiting the equivalence principle _0:30_: and in in this specific form that. _0:36_: I find the right slide. _0:41_: All three following non rotating laboratory _0:43_: laboratories are fully equivalent to the _0:45_: performance for physical experiments, _0:47_: and the stronger version of that saying _0:49_: that any physical law that can be _0:52_: expressed in tensor notation in special _0:54_: activity has exactly the same form. _0:57_: In a current space-time. _0:59_: And this in a sense, _1:01_: is why we've been talking about why _1:04_: we've been missing about geometry. _1:06_: Because the aim is to is to articulate _1:09_: physical laws in geometrical form and _1:11_: what this this tells us is that once _1:14_: we've done that we can immediately _1:16_: import that into a curved space-time. _1:19_: So we know for example well Newton's _1:22_: laws tell us tells us that excuse me _1:25_: that the if you if you exert a force _1:28_: or something then the acceleration _1:31_: of that object is. _1:33_: In the same direction of the force, _1:34_: and proportional to it detail, _1:37_: the constant proportionality is the mass. _1:40_: But the geometrical aspect of _1:41_: that is the important thing. _1:43_: That's a geometrical law. _1:44_: That's that. _1:45_: That is true independently _1:46_: of the reference you pick. _1:49_: Independently of the coordinates you _1:50_: pick is as true in Cartesian coordinates _1:52_: as it is in polar coordinates and _1:55_: spherical coordinates and anything _1:56_: like it's a geometrical law. _1:58_: We could similarly do things and and and in. _2:02_: In special relativity we can see the _2:04_: full momentum is conserved in collisions, _2:07_: so the total P. _2:08_: Before and the total P afterwards are equal. _2:11_: That's a geometrical law. _2:12_: You know the the P before people _2:14_: are after are in the same direction _2:16_: and the same in the same length. _2:18_: And this is telling us there are _2:20_: no further complications. _2:21_: So if we pick our. _2:25_: An actual local and national _2:27_: frame of freefall frame. _2:30_: Then we can do our physics in Sr. _2:34_: And that still works. _2:35_: The key thing that this _2:38_: excludes is coverture coupling. _2:40_: There are no extra terms which _2:42_: appear in your geometrical law. _2:44_: Which are associated _2:45_: with the local curvature. _2:47_: It's not that F equals MA _2:49_: plus a bit depending on R. _2:50_: So this version of the governance _2:52_: specifically rules that out. _2:54_: It explicitly rules out any other _2:56_: additions to your your physical laws, _2:59_: and so that is by itself _3:00_: a physical statement. _3:01_: So I've, _3:02_: I've, _3:02_: I think I've repeatedly _3:04_: distinguished physical statements _3:05_: from mathematical statements. _3:07_: Mathematical statements are things _3:08_: that follow from other things. _3:09_: They cannot be false. _3:11_: Physical statements are _3:12_: things that might be false. _3:13_: You could imagine the universe _3:15_: where that wasn't true. _3:16_: But in this in this universe, _3:18_: the guess that that physical _3:20_: statement is true is a guess. _3:22_: Which turns out to be _3:23_: confirmed by experiment. _3:24_: But it has to be confirmed by experiment _3:25_: because it could be otherwise. _3:27_: But it's not so this is a physical statement. _3:31_: And since you could imagine a _3:32_: universe where that wasn't true, _3:34_: but it's it's importantly, _3:35_: it is true in our case, and that _3:38_: immediately tells us at least one thing. _3:40_: Because in. _3:43_: Special activity. _3:45_: We know how we we understand. _3:49_: How how we move if we just stand here? _3:53_: I just stand here in a reference _3:55_: frame in which I'm I'm stationary. _3:56_: That's sort of the simplest motion you _3:58_: can imagine me not doing anything at all. _4:01_: What is my movement through space-time? _4:03_: I'm moving along my own time axis. _4:06_: OK, I'm just standing here moving _4:08_: along my time axis, taking taking _4:09_: out the 2nd until they you know, _4:12_: so it's not complicated. _4:13_: And if I were to be, _4:15_: if someone would be moving past _4:17_: me and I'm in a reference frame, _4:19_: a national reference frame, _4:21_: specialistic reference frame _4:22_: moving with respect to them, _4:23_: then my motion, my my world line will _4:26_: be along my my my local time axis. _4:29_: In other words, _4:29_: it would be a time like a time _4:32_: like straight line or which is _4:34_: in Minkowski space A geodesic. _4:37_: So the geodesics of Minkowski _4:39_: space are time like are various. _4:42_: But there are some which are time like. _4:47_: Push and and our our physics, _4:51_: our understanding of physics _4:51_: special activity says we move _4:53_: along the timeline, the timeline, _4:54_: geodesics in another reference frame, _4:56_: and we do so in accurate time as well. _4:59_: Sue. This is telling us _5:00_: half of the famous slogan. _5:06_: Ohh. _5:10_: Free falling particles move on timely duties, _5:13_: pics of the local space-time. _5:15_: The slogan being space tells _5:16_: matter how to move and this was _5:19_: a famous couplet of I think. _5:22_: If Wheeler, I think first enunciated, _5:24_: put it this way, _5:26_: space tells matter how to move. _5:29_: This is the point at which we have _5:30_: to some extent, got rid of gravity. _5:33_: This is it's saying that once _5:35_: you have a curved space-time. _5:37_: The move motion within _5:38_: it is simple geodesics. _5:40_: There's no need for gravitational field. _5:42_: You just followed geodesic in _5:44_: your field and you're sorted. _5:46_: Job done. _5:47_: So that's the second-half of the problem. _5:50_: Once you've set the problem _5:52_: up and got and got a a metric, _5:54_: a solution to answer this equation, _5:57_: and you've got a solution to. _6:00_: Start that sentence again once _6:02_: you have an answer to the to the _6:04_: question of what is the is the _6:06_: is the is the metric of of of _6:08_: the of the metric of space-time. _6:10_: You know what, _6:11_: you know what what happens next. _6:13_: So the first half of that question _6:15_: though is how do you work out what _6:17_: the metric is of a space-time _6:19_: in relevant circumstances. _6:21_: So what is the, _6:22_: what is the constraint that _6:23_: we have to put on that? _6:25_: And that's what we talk about now. _6:29_: Umm. _6:34_: So the relevance of this, _6:36_: but the point being made here, _6:38_: can I make that I think I can make that _6:41_: full screen in a second. A little bit. _6:44_: Get some distractions, are we? _6:48_: The relevance of this is that. _6:51_: Although I didn't spend much time on it, _6:53_: I I in passing earlier mentioned _6:56_: that it's well when you look _6:59_: at the energy momentum tensor, _7:01_: the the team you knew. _7:03_: There was a continuity _7:05_: condition which said that the. _7:11_: That the. _7:17_: The derivatives. _7:20_: Of the energy momentum tensor AT20. _7:23_: And that's just a consequence of you _7:26_: know what good what goes into a box. _7:28_: Plus what's in there is what, _7:30_: what is what you, _7:31_: what you end up with is a _7:33_: straightforward continue condition _7:34_: and that we worked out in a limited _7:37_: circumstance in a local inertial frame. _7:40_: But what the? _7:43_: The equivalence principle is telling us is. _7:47_: That. We can put a dot on that. _7:50_: If you like it, you can put a dot on it. _7:54_: And you're seeing that this law _7:56_: that was a special artistic law _7:58_: is no more complicated in GR. _8:00_: So that also becomes a truth in GR. _8:04_: That the covariant derivative _8:05_: of energy mental tensor is 0 is _8:09_: conserved in invariant terms. _8:14_: That's the remark about. _8:17_: Things moving along time right, georgics? _8:19_: And the summary of that, _8:22_: so just to recap that. _8:25_: From college Principal is a variant of _8:28_: the college principle we talked about. _8:30_: You know, in, in, _8:31_: in lecture one of the course basically. _8:33_: If you think about it, it's. _8:37_: There is more. _8:38_: There is more content to that than the, _8:40_: than the versions of of the coolest _8:42_: principle we talked about last time, _8:43_: but only because we have strengthened it _8:46_: a bit by talking out about specifically. _8:49_: Because these times and so on. _8:51_: But it's the same idea. _8:52_: And the and. _8:53_: The and and the the the the previous _8:54_: version that the immediately previous _8:56_: preceding version of the principle _8:57_: I mentioned is just equivalent _8:59_: to what you learned in lecture 1. _9:02_: And physical laws in flat space take _9:05_: the same formula local national frame, _9:07_: and that's also called the comma _9:09_: go semi colon rule for the failure _9:11_: of his reason that here we return _9:13_: this comma straight into semi colon. _9:15_: OK, so we hear that the common _9:17_: goal semi colon rule, _9:18_: that's what we're talking about. _9:19_: It's just a nice way of a nickname for it, _9:21_: if you like, _9:22_: or space tells matter how to move. _9:25_: Once you get your space then things will _9:28_: move along duties and problem solved. _9:33_: So we've solved the second-half _9:34_: of this problem. First, here we. _9:36_: And what this means is that we _9:38_: have to rethink the way that we _9:41_: think about gravity could. _9:42_: We are used to standing still, _9:44_: being the natural order of things, _9:46_: and falling out of trees being the odd thing. _9:50_: We're apes, you know that. _9:51_: You know, we spent millions of years, _9:54_: you're working out how to hold on _9:55_: to all the trees and not fall out. _9:57_: You know, it's a thing. _9:58_: We have a focus on that, _9:60_: but we're thinking about the wrong way. _10:03_: This set should be. _10:05_: Think of that if you imagine. _10:08_: I I saw shadow version of us. _10:11_: Drop dropping through the news as _10:13_: flushed suddenly disappeared and they _10:15_: dropped down to the center of the Earth. _10:17_: We would see that ghost version of us. _10:21_: Disappearing at an increasing speed, _10:24_: there would be quotes strictly in scare _10:26_: quotes, accelerating away from us. _10:29_: And we think, Oh my God, that's terrible. _10:30_: They're accelerating. _10:31_: But the government tells us we're _10:33_: looking at the wrong way around. _10:35_: That if you like, _10:36_: that that version of us plummeting _10:37_: toward the center of the center of the, _10:39_: of the gravitational, _10:40_: local gravitational concentration _10:41_: is the real, is the natural motion. _10:43_: That's the real thing. _10:44_: And we are the ones being accelerated _10:46_: away from that motion by the _10:48_: presence of the floor. _10:50_: To what the floor is doing in this _10:52_: picture is stopping us joining that _10:54_: ghost version of us in the in in _10:57_: free fall and accelerating away. _10:59_: So as our our feeling, _11:00_: the pressure on our feet or on other _11:03_: legs of our chair is not that isn't _11:05_: just a bit like a force or acceleration. _11:08_: The force of gravity isn't just _11:10_: like a force of acceleration, _11:11_: it is an acceleration. _11:12_: And the and and so that that this that _11:15_: this seems slightly equivalent equivocated. _11:17_: But in a sense if you think about _11:19_: the right way then it's clear what _11:21_: that is accelerating away from. _11:23_: And so this picture of the observer in _11:26_: freefall and the observer not in freefall. _11:30_: And you could either view this as being _11:33_: out in space and this person and the _11:35_: both out in space but this person is _11:37_: on a a platform which is accelerating _11:39_: or else you can rather a standing _11:41_: on earth and this person is on a. _11:43_: Platform this person is falling _11:45_: down and they are equivalent and _11:47_: in a sense that the, the, the, the, _11:48_: the physical statement I want to _11:50_: really get over to you by but it's _11:53_: just just repetition is is, is that. _11:56_: Any questions about that that that _11:58_: I I think I've done that today. _12:02_: It's just so nice to be able to make a _12:04_: physical statement in this course rather _12:05_: than just here's here's more maths, _12:07_: I think rather indulge, _12:08_: indulge you OK? _12:14_: So the question then becomes, _12:16_: how do we work out? How? _12:18_: What do we, how, how do we? _12:21_: Constrain what that? _12:24_: Metric or space-time should be. _12:26_: And again this will involve _12:28_: actually a physical statement, _12:29_: a statement about the universe _12:30_: which could be false. _12:31_: So we have to to guess or we don't _12:32_: have to guess because Einstein _12:34_: did the guessing on our behalf. _12:35_: So that that guessing I think happened _12:38_: more or less over the summer of 1915. _12:40_: So you've done a lot of the _12:41_: work to to to sort out the maths _12:43_: of this to learn the maths of _12:45_: different geometry over 10 years. _12:47_: You think you had a tough over 10 lectures. _12:48_: He took 10 years from 1905 to 1915 to to _12:52_: to sort that that math out in his head. _12:54_: He claimed he never understood it really. _12:57_: But it's only at the end, _12:58_: in a sense, _12:59_: that he did the guesswork to to to _13:02_: work out what the the constraints _13:05_: were on the the the space-time. _13:08_: And when we start is by going back to _13:11_: Newton and this is Poisson's equations. _13:15_: This is the. _13:17_: It's actually the the. _13:22_: The Laplacian of the _13:25_: gravitational field is that. _13:27_: The coverage of that gravitational field. _13:30_: Essentially this is just the Newtonian _13:32_: gravitational field it's governed by. _13:35_: The density of mass. _13:37_: The local density of mass. And. _13:40_: Big new gravity should constant. _13:43_: So that's a statement of Newton's _13:46_: law of gravity, if you like. _13:48_: Not in a way that Newton would recognize, _13:50_: but it's due. _13:51_: That's Newton's law of gravity. _13:52_: From that you can deduce F _13:55_: equals GM or are squared. _13:58_: And we can take and and the the the _14:00_: vacuum version of that is similar _14:02_: except obviously with with no mass, _14:04_: so, so the, the, the. _14:07_: Meet with the graphical field _14:09_: when there's no mass is well, _14:11_: it's not complicated. _14:12_: It's it's just flat. _14:15_: There's no gravitational field. _14:17_: And we could take that as inspiration. _14:19_: For what to do next. _14:23_: Because. And. _14:28_: OK, just a quick question and just to at _14:32_: least give us we probably brief pause, _14:34_: we thought what does this Phi _14:37_: comma I comma I represent? _14:39_: The diagonal of a tensor there. _14:41_: To this contraction or this? A _14:49_: construction here. Who is it? _14:50_: Was the first one? _14:54_: Who is it with the second one? _14:57_: Who was it? Was the third one? _14:59_: Well, I thought. Who was there? _15:01_: About her brief chat just to. _15:04_: This isn't saying you're wrong. _15:06_: You have reached. _15:27_: Thank you. OK, _15:30_: let's you know chatting is always brief, _15:32_: but so so the diagonal of a tensor. _15:37_: This contraction here. _15:39_: The third ring. I see a lot, _15:43_: a lot, a lot of indecisions. _15:44_: They're there, they're still, _15:46_: but it's not a big deal. _15:48_: But it's the. _15:49_: Is it just this contraction, _15:51_: it has to be the contraction _15:53_: because in the case of. _15:55_: The third one. _15:58_: We'd end up with. _16:01_: Two eyes in at the bottom if you're like, _16:05_: so it has to do in order for _16:07_: the there to be a this be a _16:09_: correct term with a sum over it. _16:11_: It has to be this contraction here. _16:13_: So the contraction over those hard _16:14_: to hard to pick out G's there. _16:17_: So I I mentioned that just because _16:18_: it's new it's about time for quick _16:20_: question but also because it's _16:22_: it looks a bit strange that you _16:24_: haven't previously seen commas. _16:27_: Upstairs if you like. _16:30_: So that's. _16:31_: That notation looks slightly strange. _16:36_: OK, so come back to that mode. And. _16:42_: So we have. Possible equation? _16:46_: I comma I comma I = 4 Pi G. Rule. _16:53_: And. Fine comma I comma I = 0. _17:01_: In the vacuum. Now we can look back. _17:06_: A bit. And again, _17:07_: lecture one who knew lecture one _17:09_: was going to be it was significant. _17:11_: We can find a thing in lecture _17:14_: one which described the you know, _17:17_: you remember this. _17:25_: Idea of the of of the _17:28_: objects falling toward um. _17:32_: Earth. _17:36_: And getting closer without experiencing _17:38_: any acceleration and we found an _17:41_: expression for that. Which were the. _17:48_: I think I I think I I think I I I _17:50_: mentioned this and so showed the result _17:52_: without so walking through through it _17:54_: step by step because because the the _17:56_: details are sort of fairly obvious, _17:58_: you walk through but it wasn't _17:60_: worth delaying which was GM over R. _18:04_: And. Uh, sorry. _18:10_: Which? Also jumping a few _18:14_: steps here is also. And. _18:23_: Not very unique side. That's better. _18:29_: OK. And that is the that was _18:32_: the 2nd derivative of the. _18:35_: This looks like I'm sort of missing a _18:38_: couple of steps in the middle here, _18:39_: but the point is, _18:40_: the point is the details of that _18:43_: then that we can get an expression _18:45_: for something that that makes _18:48_: sense in our identitarian picture. _18:52_: Which reminds us of the. _18:58_: The expression we got last at the _19:02_: end of the last chapter. Which had. _19:10_: You know, what was it? _19:24_: Which was the expression for geodesic _19:28_: deviation which involved the. _19:32_: Rementer. And the point of this? _19:35_: One can go through this _19:37_: argument in in in more steps, _19:38_: but the point of this is that it's _19:42_: that this is suggesting hinting to us. _19:45_: That the thing that corresponds to this. _19:52_: Derivative here is something to do. With the. _19:57_: With contractions of the Riemann tensor. _20:02_: And in particular. It's possible _20:06_: to do with the Richie tensor. _20:15_: Which is that that particular contraction _20:18_: of the Riemann tensor, and so one can guess. _20:22_: Is the analogue of this vacuum equation. _20:28_: Something nice and straight _20:30_: forward like our. Mute, mute. _20:34_: Alpha beta equals zero. _20:37_: Does that count as our? _20:40_: An analogue in GR of _20:43_: this vacuum equation. At. _20:49_: And the answer is yes, it does _20:51_: sort of so that that that is. _20:55_: Only the halfway at each year, _20:57_: but that that is that is true in the sense _21:00_: that that does boil down to the the. _21:02_: Those are the vacuum field equations _21:05_: for for GR so that the shape of the. _21:11_: If in a universe with no _21:13_: energy momentum in it, _21:14_: the possible shapes of the metric are _21:19_: ones where the curvature they're tensor. _21:22_: Sort of curvature is 0, _21:24_: which sort of makes sense. _21:26_: No, no mass, no curvature, _21:28_: because that's that's not _21:30_: surprising therefore. _21:31_: But we're still not there. _21:34_: And one thing we can do is we can _21:36_: add to it since it's also true. _21:41_: That if our if our alpha beta is zero _21:45_: that would imply also that G. Alpha UR. _21:52_: You. Beta would be equal to 0 trivially, _21:56_: so that that that implies also that G. _22:03_: MU equals to R MU nu plus minus 1/2. _22:09_: Or, Gee, you knew equals 0, _22:13_: so I haven't added this is equivalent _22:15_: to that, because if that's true, _22:16_: then this must be true. _22:18_: But the point is, I I'm, _22:19_: I'm you know, knowing what, _22:20_: I'm knowing what's coming here. _22:21_: And I I've I've just switched _22:23_: from our to the I sentence _22:25_: G for reasons which are, _22:27_: at this precise point, obscure. _22:32_: So that's the vacuum field equations. _22:35_: What are the you're not interested in that _22:37_: we're interested in is the field equations _22:39_: in the presence of energy momentum. _22:41_: So you've got a star, you know what's the, _22:44_: what's the field around that question? _22:49_: Or equal to zero. Yeah, we can. _22:52_: Can we immediately assume that _22:53_: the G will be the Euclidean G? _22:56_: No, because I think that. _22:59_: I think we cannot know because _23:02_: I'm not 100% opposed to this, _23:04_: but I think there would be _23:06_: non Euclidean G's which would _23:07_: have that have that property. _23:09_: I'm not sure what they would _23:10_: look like offhand, but I. _23:13_: I think the answer is probably yes, _23:16_: but I I I don't think that completely _23:19_: constrains that because the thing that. _23:23_: Yes, yes this is. _23:24_: If you start with two lines, _23:25_: they will never meet the basic. _23:30_: Yes, I think actually it might I mean I, _23:32_: I I'm, I'm hedging here because I'm _23:36_: not 100% sure but I'm fairly I think, _23:38_: I think I'm fairly confident you're _23:39_: right that that, that, that, _23:40_: that that it's probably the only _23:42_: solution of that is the Euclidean one. _23:45_: So I think I think that that would _23:46_: be the case and point for that for _23:48_: that extra auxiliary reason. Yeah. _23:52_: So. _23:55_: I guess for this for Winston in the vacuum, _23:59_: in the field equations in _24:00_: the presence of matter. _24:01_: So I guess if this is sort of right _24:04_: is to say, well how about seeing _24:08_: you you equals TU where T is you _24:11_: remember the energy momentum tensor. _24:14_: OK, that's plausible. _24:15_: That as a first step, that's so plausible. _24:19_: Or some. Proportional to that Kappa. _24:24_: And. No, but the conservation law. _24:29_: That we saw here. _24:31_: Well then tell us that so T. _24:36_: MU new semi colon. _24:39_: Neu equals zero would then imply that _24:42_: R MU nu semi colon U was equal to 0. _24:48_: Um, OK, that might be the case, _24:51_: but that would also imply _24:53_: that when we contracted it? _24:59_: That the curvature scalar. _25:02_: Would also had a 0 derivative. _25:07_: And that tells us that there's no _25:08_: coverage at all, basically. And why? _25:14_: The third one? The third one or _25:16_: the the second one? The answer _25:18_: is the second one second, yeah. _25:23_: Yeah. Yeah, which is? So um. If the. _25:39_: And. So if the coverage scaler is. _25:44_: Has zero. Derivative then that is _25:47_: essentially saying that the that, that, _25:49_: that that the universe is flat and since the. _25:56_: The. That would in turn mean that the. _26:03_: Contraction of. _26:09_: Gu Alpha T alpha. You. _26:15_: And going you would go to zero. _26:17_: That would in turn tell us that _26:20_: the universe had constant density. _26:23_: So this, this. This. _26:27_: Possibility can't be right. _26:30_: Because the the conservation law of. _26:35_: The argumentum tensor ends up leading _26:37_: us to conclude that the universe _26:38_: has constant density, which is. _26:42_: Not the case. Universe is lumpy, _26:44_: so this can't be right. _26:47_: It's clearly not far wrong. _26:48_: We're on the right track, _26:49_: but this can't be right. _26:52_: But we have been hinting _26:54_: at the right answer here. _26:55_: Because I picked this as the. As. _27:05_: Apparently trivial deduction from the. _27:09_: Richard James from the plausible _27:11_: vacuum through the equations. _27:13_: So what we can the next guess, _27:15_: the next the next step up is to guess that. _27:24_: But that is upload applicable _27:26_: applicable version. So note the _27:28_: Richie tensor being proportional _27:30_: to the argumentum tensor but this. _27:36_: Construction. Thanks. Attention _27:37_: this constructed from the Richie _27:39_: tensor being proportional to the. _27:43_: The incremental tensor, _27:44_: and that is plausible because we know. _27:47_: From when one of the correct _27:50_: contracted Bianchi identities that. _27:56_: Gu. _27:59_: The new new semi colon new is equal _28:01_: to 0 as a mathematical identity. _28:04_: Based on the properties of the tensor, _28:06_: so the conservation, _28:07_: this conservation law is satisfied. _28:10_: Mathematically, by this. _28:14_: So that's plausible, and it's right. _28:17_: It turns out turns out to _28:19_: be matched by reality, _28:20_: and this equation here is I sense equation. _28:25_: And. What I've given you is not a _28:28_: mathematical deduction, deduction of it. _28:30_: I'm seeing what I've said I said earlier on _28:33_: it's not a mathematical reduction of it, _28:35_: it's I guess it's a plausible guess that _28:37_: we have motivated by analogy with patterns. _28:40_: Equation is against the Einstein made, _28:43_: eventually offered several _28:45_: goals in the summer of 1915. _28:47_: And published in the famous paper often 15. _28:50_: And it's that that is the physical _28:54_: statement on which all of the. _28:57_: Solutions that follow have been _28:59_: based and all and which all the _29:01_: tests of GR have been based. _29:02_: So the the the tests of _29:06_: GR GPS has satellites. _29:08_: Black, you know, refugees from, _29:11_: from, from, _29:12_: from slowing down neutron stars and so on. _29:14_: Our tests of that guess. _29:16_: And it's passed every single one so far. _29:22_: So that. Through there we can just look _29:24_: at that from for that's the sort of thing, _29:28_: and that is the thing we can all look _29:30_: at and admire and put on T-shirts. _29:35_: Now that is. Attention equation. _29:40_: So it's a geometrical thing. _29:42_: But we've quoted it with. _29:46_: It in component form. _29:50_: T is our symmetric tensor. _29:53_: And we as we discovered as we as _29:56_: we reassured ourself last week, _29:60_: which means G is also symmetric tensor. _30:06_: Which means it has 4 + 3 + 2 _30:08_: + 1 independent components. _30:10_: That's 10 because if it's symmetric tensor, _30:13_: then as a matrix it's a symmetric matrix. _30:16_: I am not antisymmetric so that there _30:18_: are 10 independent components in that. _30:23_: The. And we try to Bianchi identities. _30:28_: Add 4 code. The MU equals 1230123. _30:32_: Add 4 constraints to that. _30:34_: Bring this down to six _30:37_: independent constraints. And. _30:41_: And the remaining degrees of freedom? _30:47_: Are based on the fact that we can rescale _30:50_: the coordinate functions as we wish. _30:53_: We could measure in feet rather than _30:55_: rather metres, or rotate or our axes. _30:58_: So this ends up constraining our _31:01_: the solutions as much as we need. _31:05_: Because remember, think back, unpack this. _31:08_: That GI Sentencer is composed _31:11_: of the Ritchie Center. _31:13_: The Richard Center. _31:14_: It depends on the Riemann tensor. _31:16_: The Riemann tensor depends at least _31:18_: the local natural frame on derivatives _31:21_: of the metric and particularly _31:22_: second derivatives of the metric. _31:24_: So this is a collection of of 2nd _31:29_: order of 10. Plus constraints 10. _31:34_: 2nd order. _31:35_: Differential equations in _31:37_: the components of the metric. _31:40_: The metric has it also auto metric tensor, _31:43_: so it also has ten. _31:44_: Yeah, sorry this is a said that _31:46_: I said that in the wrong order. _31:47_: This has 10 has 10 constraints, _31:49_: four of which are taking up going _31:52_: forward these if we want to. _31:54_: So it's a differential equation _31:56_: for the components of the metric, _31:58_: a second order differential equation _31:60_: for the component of the metric. _32:02_: It adds 6 constraints. _32:04_: So there are four degrees of _32:07_: freedom left unconstrained, _32:08_: they are taken up by the freedom we _32:11_: have to change our our our coordinates _32:13_: in length and and orientation. _32:15_: So this is what this does is it constrains. _32:19_: The metric. _32:21_: So the metrics that are allowed around _32:24_: our source of energy momentum are _32:26_: those which satisfy that equation. _32:29_: And those are the codes of all _32:31_: solutions to generate to generativity. _32:34_: Doing so is not trivial because you _32:36_: have 10 simultaneous differential _32:37_: equations to solve. _32:38_: But if you symmetrize the problem enough, _32:41_: then you can solve that problem. _32:46_: And the first. Predictions of um. _32:52_: Which come up come ohgr, where? _32:55_: I Einsteins in sort of 19 fourteen _32:59_: 913 I I believe but they were based _33:03_: on the gravitational redshift that _33:06_: we talked briefly last in lecture _33:09_: one again the idea that a photon _33:11_: claims through gravitational field _33:13_: it changes frequency and to the _33:15_: extent that frequency is the type _33:17_: of clock that means that that that _33:20_: time moves differently at higher _33:21_: and the shield photon argument said _33:23_: that time moved differently at a _33:25_: higher gravitational potential. _33:26_: From at a lower one. _33:28_: And you can deduce from that _33:31_: version of the metric of. _33:34_: Of a weak field space-time. _33:37_: Which is what you and from that you _33:40_: can deduce things like that there _33:43_: will be some deflection of Starlight _33:45_: as it goes past a large mass. _33:47_: But I distract myself because _33:49_: that is not that. _33:50_: That although it's significant _33:51_: and certainly significant, _33:52_: and was one of the things that the famous _33:55_: Eddington Dyson measurement of of of _33:57_: the Solar Eclipse aimed to rule out, _33:59_: it's not by itself a solution _34:02_: of intense equations. _34:03_: So what are the solutions of _34:06_: intense equations? Can I think? _34:10_: And. _34:12_: Yeah, _34:13_: I think we actually we're actually _34:14_: going ahead of time here. _34:15_: So I think we can we we'll we'll move on. _34:19_: There may even finish early next next week, _34:21_: but. _34:24_: Before you do I do move on to the _34:27_: first simple solution of this. _34:29_: Other questions on what we've done so far? _34:32_: Are you happy or perplexed or a question? _34:35_: There were two questions here so. _34:40_: Right, so here this implies _34:42_: that the density is constant. _34:45_: But why does it notify? _34:48_: And so the question was. _34:52_: How does this imply that density is constant? _34:57_: But this doesn't? _34:58_: And I think the answer to _35:01_: that is that if you if you. _35:03_: Stick to just that. _35:06_: Then that's. _35:11_: Our function of second _35:12_: derivatives of the metric. _35:14_: So that's a constraint on 2nd _35:16_: derivative of the metric. But the. _35:20_: The the mathematical remark _35:22_: that the argumentum tensor that _35:24_: the Andrew Mentum is conserved. _35:27_: If this is true, _35:29_: forces the Ritchie Dancer. _35:31_: To be have have A to be _35:34_: conserved as well if you like. _35:36_: Which in turn implies that the _35:39_: curvature scalar is conserved and. _35:43_: The other but if if that's true, _35:46_: then the this contraction if _35:48_: that's true and that's true. _35:51_: Then this. _35:53_: This could be written as _35:57_: this contraction of the. _35:59_: Andrew Mentum, _36:01_: tensor having with you interactive _36:03_: since G is. _36:05_: And. _36:12_: I think I'm saying missing a step here. _36:15_: Umm. If you could see me going _36:17_: on struggles at this point. _36:25_: OK. I I think I may have. _36:30_: I have a comment misplaced the _36:32_: careful description of this of this _36:34_: step in the notes, but the the the _36:36_: to the question of why does this up? _36:39_: It's gonna be like, why? Why does this? _36:44_: Make. _36:47_: Give the got the argument rate, _36:48_: why does this make predict that the universe _36:51_: is constant density and this one doesn't? _36:54_: I think it's basically because there's _36:56_: this extra freedom in here. So the. Umm. _37:02_: If if that's zero in a in a vacuum. _37:07_: Then you have the extra freedom _37:10_: of of changing G if you're like. _37:13_: Without changing the Richie changer this is, _37:15_: this is rather handwaving argument, but _37:16_: presumably that's a more complicated object. _37:18_: I think is is a slightly _37:20_: more complicated object. _37:21_: This, this argument that that, that, _37:23_: that, that that the potential ends up _37:25_: being being constant doesn't work out. _37:27_: OK, I'm going I'm annoyed that I'm _37:30_: going to have to go to reassure myself _37:32_: that's rather hand waving argument, _37:35_: but I think I think the key, _37:36_: the key point there is this is _37:38_: simply a slightly richer object _37:39_: so it has slightly more freedom. _37:43_: You ask questions. _37:49_: Yeah. You write. On the north you're _37:52_: right that is divided over RI was _37:55_: wondering if because I I was looking _37:56_: back at the nodes like set first _37:58_: set of notes and it was divided by _38:00_: zero Q and I was wondering it. Ohh. _38:03_: And that maybe just my mistake. _38:06_: Yeah. Yeah. And in fact, yeah, yes. _38:09_: When I was writing that up, _38:10_: I thought, is this right? _38:12_: But I think our cubed bread make a lot _38:14_: more sense because I think that's right. _38:16_: I think that should be. _38:21_: That, that, that, that probably _38:23_: should be an an arc cubed there. _38:27_: All right to him. _38:31_: But. OK. _38:36_: Thank you for that because that's that _38:38_: could have been a distraction. And the. _38:43_: In the beginning of this section, _38:46_: we can approximate the perfect _38:48_: fluid with a delta function, yes. _38:53_: Chronic delta, yeah. _38:54_: Is that what gives us? _38:57_: Is that the only case where we _38:59_: can have one solution to those _39:01_: 10 differential equations? _39:05_: Perfect fluids or can you _39:09_: parameterize it first? _39:10_: I think in that case we're jumping _39:12_: back a bit and alright so, _39:14_: so I I think the question _39:17_: if we go back to. And. _39:22_: Basically here. _39:27_: Yeah. And so. And this is the energy metric _39:32_: tensor or the components of the metric _39:34_: tensor for the perfect fluid which we have, _39:37_: and we persuaded itself of that because. _39:43_: The answer we we, we, _39:44_: we guessed there has to be independent _39:46_: of what direction we're facing, _39:47_: what what, what, what we're picking on. _39:50_: So that isn't a solution to anything. _39:52_: That's. That's our input to the to to _39:54_: to to the to to to to to Einstein's _39:57_: equation we're seeing this is how. _39:59_: This is what the arguments must be. _40:01_: Must look like this. _40:03_: So the only parameter there is _40:05_: is the pressure so that the. _40:14_: I think this will be. _40:18_: Well, that, that, _40:19_: that I I think in in local natural _40:21_: from that would also be a diagonal. _40:24_: That, yeah, that would be a diagonal _40:27_: matrix of coordinates in the local frame. _40:29_: Is that what I'm saying? _40:31_: I said what you're asking. _40:34_: Be part of the left hand side _40:36_: of the equation of this yes then _40:39_: financial equation yes means. _40:40_: It doesn't mean that they _40:42_: are have one solution, yes, _40:44_: but at least in a perfect fluid. _40:46_: Since we have that, _40:47_: we know that there has to be _40:48_: only one solution to those 10. _40:53_: And yes. Yes, and and and and _40:57_: and and we won't cover this, _40:59_: but in the case of a, _41:02_: because they're recovered next in in G2, _41:05_: so G2 is picking up where this leaves off. _41:08_: I will mention a solution of incense _41:10_: equations just because I show you one, _41:12_: but G2 is essentially solutions, _41:14_: plural, of Einstein's equation. _41:16_: And solution number one that you'll teach, _41:19_: you'll learn. _41:20_: Is this the Schwarzschild metric? _41:22_: And the structural metric is the solution _41:25_: to the case where you have a central, _41:28_: a single isolated source of mass, _41:30_: so just a lump of mass in the alone in _41:33_: the universe and the and and and that. _41:35_: So that's a very specific distribution _41:38_: or argumentum is a lump there. _41:41_: And the the solution to that _41:42_: is a particular metric which is _41:44_: called the structural solution, _41:45_: which has properties which _41:46_: are are of interest. _41:48_: But that is the solution to the case where _41:51_: there's only one dot in your universe. _41:54_: So the and and and jumping ahead _41:56_: of myself because we have a whole _41:59_: 5 minutes of time jumping myself. _42:03_: Or do you have Twitch? The. That was. _42:11_: I seen the the 1915 people _42:13_: was in November 1915. _42:14_: And Schwarzschild, who was, _42:18_: I think in the German army at that point _42:20_: in the middle of the First World War, _42:22_: you know, piles Louise time _42:24_: under shell fire, you know, _42:26_: solving Einstein's equations as you do. _42:28_: And he did it remarkably quickly. _42:30_: And I think in early 1916 he wrote Einstein _42:34_: saying this appears to be a solution. _42:36_: And I said went ohh, I didn't think _42:38_: I thought take longer than that. _42:40_: And said, well, _42:41_: it's quite clearly classically right. _42:44_: So um. _42:47_: That so that's that's an example _42:48_: of the case where although you _42:50_: have ten couple differential _42:52_: equations if you make the problem _42:54_: symmetric symmetrical enough. _42:55_: And you're you're move. _42:59_: Complications by saying we _43:00_: only have one bit of mass, _43:03_: then you can then half of these equations. _43:05_: You know the large chunks of the different _43:08_: equations of instructions fall away, _43:11_: and you're left with much less to solve _43:13_: than you would start off with more _43:15_: complicated things like other solutions of. _43:19_: And. _43:22_: And wrong what I said before. Because the. _43:28_: The the free space. _43:31_: The the the the vacuum _43:34_: form of instantiations, _43:36_: which is what that really is. _43:38_: Or or that has another solution which _43:42_: isn't a constant flat universe. _43:46_: That solution is gravitational waves. _43:49_: The gravitational waves are _43:50_: a solution to that equation. _43:54_: In the presence of no matter at all. _43:57_: So graph you could have graphical waves in _43:59_: a universe with no matter at all, because _44:01_: what graphical waves are is a dynamic. _44:07_: Perturbation in the metric. _44:10_: This metric is perturbed, _44:12_: perturbed the next bit and and that _44:14_: and that propagates. So that's all. _44:16_: Gravitational waves are one _44:18_: another solution to up to I said _44:20_: equations in this particular case _44:22_: of a 0 right hand side. Question. _44:29_: So again, how would there be a _44:30_: perturbation if there isn't any master? _44:32_: And well, the question of how you how _44:34_: that is started is a different question. _44:36_: But so, so that solution says that _44:40_: if given that it was perturbed, _44:43_: then the perturbation, it's like Maxwell _44:46_: equations support a perturbation of _44:49_: the electric and magnetic fields. _44:52_: They don't sort of care how _44:53_: that perturbation was started, _44:54_: that's what. That that's. _44:58_: Another another question if you like, _44:59_: but they support a perturbation, yeah. _45:04_: Thank you. _45:08_: Good. So that's a. That's a fine thing. _45:11_: So and and that gives the _45:12_: other half of the slogan. _45:13_: So space tells matter, _45:15_: how to curve, how to move. _45:17_: Think we're lunatics, Mattel speed, _45:19_: how to curve instance field equations. _45:21_: So you have doubtless seen that slogan _45:24_: in various popular accounts of of general _45:27_: activity or possibly or or whatever. _45:29_: Or you may have heard of this, _45:31_: but that's that's where those two, _45:34_: those 222 parts of that slogan _45:37_: are both deep physical statements. _45:40_: Expression or jocular form, _45:41_: but they are both physical statements. _45:44_: Push, you know no the underlying. _45:47_: Underlying and that's just _45:48_: a nice picture really. _45:50_: The point being that once _45:52_: you have a curved space-time, _45:55_: then doing going in a straight line _45:57_: in that course space-time can end up. _46:01_: Producing port. As it were of youth, _46:04_: most eight would be a a closed loop. _46:10_: Key points. Now I I, I, I, I. _46:14_: OK. Have you been ahead of time? _46:17_: I have. Look up the last of _46:21_: the stuff before section 4.3. _46:22_: But second 4.3 is fairly self-contained _46:25_: and so next time what we'll do is _46:28_: we'll go through section 4.3 which _46:30_: is about doing most of the solution _46:34_: of Einsteins equations minus chunk _46:37_: of boring algebra in the very _46:40_: simple case of attaining mass, _46:42_: attaining central mass and and we _46:44_: discover a nice thing at that point. _46:46_: And time is the last. _46:49_: Let the last lecture, _46:50_: I think it's the Friday of next week _46:53_: that we have the next supervision. _46:55_: What? Sorry, this week is OK, right? _46:59_: In that case I will. _47:03_: And I haven't yet, _47:04_: but I will put up a a a little _47:06_: heading on on on the on the pilot for _47:09_: Supervision 2 and I exhort you to add _47:11_: questions or puzzles or things there. _47:14_: Comment on what someone puts up _47:16_: something which is almost what you want. _47:17_: Then add comments like things you know. _47:21_: I will use that as the as the _47:23_: skeleton of the supervision this week.