Transcript of a2-l15 ========== _0:03_: E2 Relatively Gravitation, Lecture 15. _0:11_: This is the appeal _0:13_: this these 5GR lectures. They are sort of the point and the _0:18_: payoff of the 10 specific lectures, which are _0:21_: independently interesting but also to some extent lead up to _0:26_: the GR bits. _0:28_: So this is the _0:30_: last of the retribution lectures and we are bang on schedule, so _0:35_: we'll get to the end without scampering. _0:38_: And when I got to last time, _0:41_: I talked about what we covered last time was _0:47_: various _0:49_: solutions to Einstein's equation. I mentioned the weak _0:52_: field solution, which is the limit of small masses. _0:57_: So that so so or when you're far away from the from the matter in _1:02_: question _1:03_: and that _1:06_: really allows you to to rederive Newtons group love love _1:10_: universal gravitation. You discover that there's a _1:13_: potential for a function which is as part of the solution of _1:18_: equations, and you can find an equation of motion _1:23_: force in terms of the gradient of that potential function, _1:26_: which and the potential function is governed by _1:29_: integration. That's a solution _1:32_: which is essentially equivalent to _1:36_: questions _1:38_: theory of gravity. The next approximation was the _1:42_: Schwarzschild spacetime, which is the the the full relativistic _1:47_: case of a single point mass in the universe. And what is the _1:52_: space-time around that? _1:54_: We'll get more of the other ones in a moment, but I had talked _1:58_: about the 1st 2:00 and I had I and and I had emphasised the _2:02_: importance that what we're doing here, _2:05_: what the process of solving Einstein's equations consists _2:08_: of. _2:10_: It's a metric, that is, that is a set of coefficients of these _2:15_: DT, DRD, Theta, D, Phi these differential distances. The _2:20_: metric is the coefficients of those, _2:24_: and the process of solving ancient equations consists of _2:28_: finding a set of coefficients for those functional _2:31_: coefficients which depend on the position of the space-time you _2:36_: have which satisfy anxiety equation for a given right hand _2:40_: side which is dependent on the distribution of energy momentum _2:45_: in the space-time. _2:46_: And of course, the energy momentum is primarily in the _2:51_: form of large lumps of matter such as planets and stars and _2:55_: galaxies and so on. _2:58_: So that's a recap of of what what we're doing, what we're _3:01_: doing here. _3:03_: I mentioned the weak fuel solution and it's properties. It _3:08_: it was. You can see that if I were zero _3:12_: with with that that that D Sigma is the DX y ^2 plus D y ^2 + Z _3:16_: ^2 or or whatever. That's the metric of the spatial part. You _3:20_: can see that if I were zero in the limit of all of of equals _3:24_: zero. That would turn it back into the Minkowski metric, _3:30_: but it. But it's clearly not that in in general. _3:34_: And I then pulled out of the hat because I can't do otherwise. _3:39_: I pulled out of the hat the Schwarzschild solution. I _3:43_: mentioned that Schwartz had developed this solution very _3:48_: quickly in 1916, very quickly after Einstein had introduced _3:52_: the the field equations in 1950, late 1915. And as you can see, _3:57_: if that R term would disappear, that would turn back into _4:02_: the metric of special activity. It would be the t ^2 minus the _4:07_: art squared minus R-squared the Omega squared. The Omega squared _4:11_: is just the squared plus sin squared, Theta D Phi squared. So _4:15_: it's the metric of the surface of the of the area element on _4:19_: the surface of a sphere. That's that's all The Omega is just to _4:23_: love that into one place. So if that term R / R weren't present, _4:27_: this would be just _4:30_: Makovsky metric. And that's important because that is _4:33_: because one of the constraints on this being a a sane solution _4:37_: is that it reduced to the Minkowski metric both in the _4:40_: limiting case of no mass and in the limiting case of R being _4:44_: very large. _4:45_: Because you see that if our very large if. If so a little R. If _4:48_: the if the radial coordinate is very large _4:51_: and so you're far away from the central mass, _4:54_: then big R over little R is small, _4:57_: much less than one, and this return turns it back into the _5:01_: Minkowski metric. So this solution is compatible with _5:04_: Minkowski space-time at large _5:07_: coordinate R, _5:09_: which is a sanity check. It couldn't be otherwise, _5:17_: and this is this is a very important metric. This is. It is _5:22_: for the ideal idealised case of a single point mass in in your _5:27_: space-time, but that is a very good approximation to the to _5:32_: most of the. _5:35_: Practical cases we want to use General General relativity. So _5:39_: if you want to design the GPS system or get or the legal _5:42_: system this is the metric that you use for for the non _5:46_: relativistic for for for that relativistic corrections to to _5:49_: to to nutrition theory. If you want to talk about micro lensing _5:53_: which will come into very very briefly or or or or lensing of _5:57_: of galaxies by supermassive black holes is this partial _5:60_: solution that you that you use _6:04_: if you want to talk about _6:07_: the most power effect. _6:10_: That's the non relativistic version of _6:15_: of gravity that is used practically _6:19_: in the in the cases where where, where where was relative to. _6:21_: Corrections matter. _6:23_: So we have it, but the next thing is to examine what what _6:26_: what, what happens in here. _6:29_: Now as you can see, _6:33_: when when little is big, this turns back in and I've just said _6:37_: I emphasised little are as big, this turns back into the _6:41_: Minkowski metric. But when little gets to be equal to _6:44_: bigger, so when when you come down, you shrink our down to the _6:49_: point where it's equal to this. There's this are then the second _6:53_: term, the term on the the 2nd _6:56_: a coefficient of the second term, this one _6:59_: 1 / 1 -, 1. It blows up _7:02_: the singularity there, _7:04_: and for a long time it was thought something very exotic _7:08_: happens there. The space-time gets infinite at that point. Now _7:11_: to the long. While _7:14_: a surprisingly long while, possibly before it was realised _7:16_: that that was just a coordinate singularity, there's nothing _7:19_: actually happens there _7:21_: that you could detect from a physical experiment. _7:27_: You know that that's not obvious that that that that that's the _7:30_: case. And it was highly nonobvious people to people like _7:33_: Oppenheimer and Einstein. It it took, it took a good amount of _7:36_: work to find a set of coordinates which what didn't _7:39_: have that bad behaviour at that at that point. _7:42_: So there's nothing strange happens there. And if you were _7:46_: falling into this fall, falling radially inwards _7:50_: as you as you cross that point a little I equals bigger, _7:54_: you would do the same thing. You just be you still be threefold. _7:58_: So all of the the the the you, you, you, you have the freedom _8:01_: attached you the frame of which you are not moving would still _8:04_: be in a national freedom. And there's no experiment you could _8:07_: do that would tell you you were at that that special place your _8:11_: local experiment. You could. _8:13_: And that's and that's an important distinct, _8:15_: so that no, that inside your box _8:18_: there's nothing you could do that would let you know that _8:20_: you're crossing. _8:21_: So does anything happen there _8:24_: at another? Yes, _8:29_: and there's some sample values. _8:31_: What happened there is important _8:35_: and now the visualizer isn't doesn't want to turn on, so I'm _8:38_: going to have to do this on the board. _8:43_: I said _8:45_: that _8:46_: as you cross that regional distance R equals big R _8:52_: you are still as far as your concerned and you are still in a _8:55_: natural frame and nothing happens. So what's the? _9:01_: What does it look like? And monkey diagram. When that _9:05_: happens _9:06_: in that diagram, _9:08_: is that visible at all _9:11_: And you would find out the pain _9:21_: better. _9:23_: And so this is your. It's called the X prime _9:28_: T frame, and if you are at rest in that frame _9:32_: then your water line is going to be a long the TM access. _9:37_: Yeah, _9:40_: as usual. _9:42_: And in that in that frame, the _9:47_: no lines, which are the water lines of something moving at the _9:52_: speed of light are diagonal, as we saw back in chapter four, I _9:57_: think. And that demarcates the space-time into three regions, _10:02_: the future, the past, and elsewhere _10:06_: to the future. It all of the places that you could get to _10:09_: from this event here by moving at less than speed of light. _10:14_: The past is all the pieces you could have been and got to this _10:18_: event. _10:19_: Moving lessons, be late and in the elsewhere are all the places _10:23_: that are space like separated, _10:25_: which you can't get to moving at the speed of light or _10:30_: or or or less, and which could _10:35_: and put your such that you could find A-frame in which an event _10:39_: somewhere in the elsewhere was was simultaneous with the event _10:43_: at the origin. So you could find A-frame in which those are _10:46_: simultaneous. OK. There's space like separated _10:51_: and it is a feature of the Lorentz transformation that if I _10:55_: now ask, OK, that's that's the the motion _10:59_: in the _11:02_: in the frame _11:03_: that's attached to me that I'm stationary in. So what does that _11:08_: world line look like in our frame in which that frame is _11:13_: moving? So in another frame _11:16_: X _11:19_: T _11:20_: and I'm not telling you anything, should surprise you _11:24_: that water line would be constant speed. Look something _11:27_: like that _11:29_: because that's just the where the TPM axis ends up. _11:33_: But in this room as well _11:36_: the no lanes are still at 45° _11:40_: and this world line, _11:42_: it's still insane _11:44_: those and and you can't find a there's a Lawrence _11:47_: transformation that would get you from this frame where you're _11:51_: not moving _11:53_: to A-frame in which you were moving faster than the speed of _11:56_: light in which that world lane moved outside of those of that _11:60_: light gone _12:01_: OK. So that's a bit of special activity, _12:04_: and it is a feature of the Lorentz transformation, _12:07_: those diagonal lines _12:09_: trying to do diagonal lines. _12:12_: OK, _12:15_: but that's a feature of the Lorentz transformation in _12:18_: special relativity. _12:20_: In _12:23_: the special metric, _12:26_: it's a little different, _12:27_: so let's look at this the right hand side here. First of all, _12:31_: that's intended to represent the _12:35_: the _12:37_: Bitcoin _12:39_: of the observer who is _12:42_: free falling, who? Yeah, we're moving purely under the _12:46_: influence of gravity. Notice that the that's the radial _12:50_: direction, that's the the the time, the time direction. So _12:54_: basically and that's the the late cone of an observer who is _12:60_: in freefall moving in some direction rather. _13:04_: But when we turned that from _13:07_: their frame _13:09_: into the _13:13_: into this frame _13:15_: in this metric _13:18_: the the, the, the right cone that's 45 years here ends up not _13:22_: at 45 read but a bit a bit a bit a bit twisted _13:27_: but in a slightly different direction _13:29_: which is you know OK that's that's that's fine. It just _13:33_: looking a bit strange. But that dash lane is still a viable _13:38_: worldwide for that observer. It doesn't go outside of the lake _13:42_: on the fact the lake was a little twisted. It's fine, _13:47_: but as you move further in toward the centre, toward lower _13:53_: R _13:54_: that rotation, that distortion ends up being more pronounced. _13:58_: So once you're quite close, that thing is is is turned over a _14:02_: bit. _14:03_: So the the set of possible _14:07_: future, the set of of trajectories from the point here _14:12_: is still has to see in that league. But it's. You can see _14:16_: it's it's it's somewhat constrained and the time you get _14:20_: to r = 2 GM, this functional radius that's turned over to the _14:24_: point where the light cone is _14:27_: vertical in this diagram. _14:30_: No, _14:32_: I found it. The the the the person in the in the in the in _14:35_: that frame is concerned it's still there. They're they're _14:38_: Mitkowski, Diane still looks like this. They're they're past. _14:41_: The future still looks like this. They can only move in the _14:46_: in in the future cone _14:48_: the the the only possible trajectories are ones which are _14:50_: in the future code. But because of the way that this has changed _14:54_: in these coordinates, _14:56_: the future _14:57_: is entirely inward pointing, _14:60_: because there's no worldly going from this point here. _15:05_: The new world lane that stayed within that light cone that goes _15:08_: to increasing our _15:10_: in other words, the future of that observer at that point is _15:15_: inward pointing _15:17_: and as you go further in, it becomes even more extreme. _15:22_: So there's nothing that feels different at that point. _15:27_: It still feels like you're in inertial frame. There's no local _15:31_: measurement you can make. But your future is very different _15:35_: and rather bleak _15:37_: because at that point there is no way _15:40_: you can get out of the black hole if you're in freefall _15:46_: and _15:47_: so so and that point, that radius there is the, it's called _15:50_: the event horizon, it's got, it's the it's the size of the _15:54_: black hole. The black hole itself is I think a larity at _15:57_: the very centre. _15:59_: OK _16:00_: so so the black hole if you like is an edge of space-time which _16:04_: is at R = 0 and I said ohh mathematical bets are off there _16:07_: because it's it discontinues. _16:10_: But the the size of a black hole is the size that this radius _16:15_: here or two GM which as I've said for the Sun, _16:19_: if the Sun were compressed into a radius of 3 kilometres, just _16:22_: three kilometres, it would be a black hole. If the Earth _16:25_: compressed into account the figure, it would be a black _16:28_: hole, _16:31_: and so that's the the distinctive feature of black of _16:34_: black holes and other things you can talk about with black holes. _16:37_: The fact that there are tidal effects, if you remember _16:41_: way back in I think Lecture 2 I talked with an example of two _16:47_: people falling toward the centre of the Earth and they are _16:53_: a separation between them _16:56_: because they're they're they're falling except they are they're _16:59_: moving, pulling into gravity. So as far as they're concerned _17:03_: they're not accelerating but the separation between them is _17:06_: changing as time goes on. So the the and that's a title effect. _17:10_: It's not a a local effect. It's purely an effect discernible _17:14_: with a a measurement over an extended non local part of _17:17_: space-time. And another sort of tidal effect is that _17:21_: in the high gravitational the rapidly changing gravitational _17:26_: field, if you like. Here, if you were falling in feet first, see, _17:30_: your feet would be accelerated more than your head would, and _17:34_: so a tidal effect. You'd be stretched out _17:37_: and put it and and and and that's for accretion discs _17:42_: that one of the things that happens in accretion discs, it's _17:46_: partly because of the the the the extreme tidal effects on the _17:50_: creation discount black hole that cause all of the some of _17:54_: the physical features there. Just looking at the light cone _17:57_: on the slide, _17:60_: Yeah. Does that mean that because it like sort of like was _18:03_: taught so much that it comes above the access, does that mean _18:07_: that your past could have been in the future? _18:13_: Right now that's a very good question and we're gonna get _18:16_: rather evasive answer. _18:18_: Um, because inside the black hole the _18:23_: and _18:25_: the the the. The problem is that the _18:30_: the _18:32_: so speed access and the time axis sort of swap over _18:36_: and so the the the the space axis becomes your time like _18:38_: direction _18:40_: in in its own way. So that _18:45_: at this point you can have a discussion of what does it mean _18:47_: to move forward in time. But in inside the black, inside they _18:50_: went horizon black called moving forward in time means moving _18:53_: inwards in radius. So the radio direction becomes time _18:57_: in a way. _18:58_: And so _18:59_: I have never found a satisfy our intuition for this picture for _19:03_: this entire satisfactory to me or what's actually happening in _19:07_: there. I think it's something that is requires careful _19:11_: thought, but it it does get quite exotic there and things _19:15_: like that you could have been. So I think that that time _19:18_: becomes a sort of Space Flight direction. So you'll watch _19:22_: Google backwards and forwards. You know Mumble, you know, _19:27_: And that's not a very satisfactory answer, but I don't _19:30_: really have a very satisfactory answer. _19:33_: Just only you're saying there about the the time Max coming. _19:38_: You're like spatial access that effectively being made the _19:42_: bottom, like the singularity is just always in your future. Yes. _19:46_: So you you never get to the singularity _19:50_: and I you understand now keep on opening Rd, never and never _19:54_: never get there. And we've talked. We've talked here about _19:58_: the what happens to you as you you're going in. There's a whole _20:03_: separate thing we could talk about, about what someone _20:06_: looking at you sees. _20:09_: Because what someone looking at you sees is _20:12_: light coming from you _20:14_: and so some some light emitted. So see you. You're here _20:19_: and managed to send a a sort of goodbye forever, sweet world _20:23_: message out of but but but but by pointing a laser you know, _20:27_: directly outwards? That's good dispute. Later it would get out _20:31_: moving the spotlight if it was sent from outside the event _20:34_: horizon, _20:36_: but that then has to climb through a huge gravitational _20:39_: field in order to get to the observer. _20:42_: We are here _20:44_: and so it would be hugely red shifted and the the time _20:50_: and another one of the effects that we could talk about if we _20:54_: had more time is what you know related to the idea of of of red _20:58_: shifting light is how much would time slow down as you watch _21:03_: someone _21:05_: falling into a black hole. So you would see their watch tick _21:08_: slower _21:09_: for for a number of reasons, _21:12_: but you would never see. So one of the things that _21:16_: to reflect on it, you would never see anyone cross into _21:19_: across the event horizon. _21:21_: Someone falling in _21:23_: just falls in and they cross the road and they don't locally _21:26_: notice. But if you are looking at that from our side, you would _21:30_: never see them actually cross because the time that you as _21:32_: you're watching there watch, you're looking at their clock. _21:36_: You'd never the time was fluent enough so that pertained to get _21:40_: the crossed the event horizon. It would have slowed down _21:43_: infinitely from your in your measurement, your observation of _21:46_: them. _21:48_: So all sorts of exotic effects what you what you're talking _21:51_: about here. But that event right? The thing is the the the _21:54_: key one if you like. _21:57_: Umm, _22:01_: so good. The other thing that that can happen here is let's _22:06_: not worry about the _22:08_: let's move back a bit from the exotic behaviour at the centre _22:12_: and concentrate on things a little bit further. _22:17_: So you can imagine _22:18_: that that's just the RT plane of this space-time diagram. _22:24_: There's also the the the Theta and Phi coordinates that are _22:28_: suppressed here, _22:31_: but _22:32_: amongst the SO so so we've talked about the the metric. _22:37_: Once you put the metric you can talk about geodesics, and _22:41_: geodesics are the trajectories that are _22:46_: a body in free fall could potentially follow, so the the _22:49_: the. So if if I if I'm if I'm on the surface of a sphere and I'm _22:52_: facing in this direction, then there's a geodesic that I I will _22:56_: fall follow if I just head off in that direction. So depending _22:60_: on what direction I'm I'm facing in on the surface of a sphere, _23:03_: I'll pick out a different geodesic. _23:06_: Really, obviously, but there will be geodesics. But the _23:08_: initial conditions, _23:10_: what direction of fitting we'll select which duty it is. _23:14_: Similarly, geodesics in our our _23:18_: which includes time. _23:21_: All the possible motions from a particular point or point of _23:24_: water lines will be geodesics. But which one you follow, in _23:28_: other words, how fast you're moving and in which direction _23:31_: depends on the initial conditions, _23:33_: but they're all geodesic _23:35_: amongst the eugenics _23:37_: that that happened in this space. Like this is 1. Where the _23:42_: where you moving mostly in the _23:46_: and _23:47_: find direction for example and that what that will look like is _23:52_: a spiral on that in in that space-time as as time goes on _23:56_: your your your _23:58_: you you you can you can imagine an Arteta plane _24:03_: and a time axis here and and and then one of the geodesics is _24:08_: something like that in a spiral and that is basically an orbit _24:12_: that's when orbit looks like in the _24:15_: because in the in the in the in the space-time. _24:21_: And if you work that out and look at and do the calculations _24:25_: you get that what that duty looks like you discover that _24:29_: dudzic traces out something which looks very like and lips _24:33_: of course because you have to reproduce the the the results _24:37_: that you get from sodium analysis the the the two body _24:40_: problem so you get something which is very like an lips _24:44_: except _24:45_: that is an ellipse which slowly processes _24:47_: over the course of time as so So this this spiral isn't quite a _24:52_: spiral it's slightly elliptical spiral and the axis rotates as _24:56_: time moves as it moves up the the T axis _25:01_: So the orbit processes _25:04_: no step back a bit _25:06_: and _25:09_: and I think I have _25:11_: it _25:15_: and and the amount of procession _25:19_: ignore the second line there for the moment the amount of _25:22_: procession how much the direction. Of the semi drags is _25:26_: moves per orbit is a function you can you can work it out you _25:30_: can calculate it it depends on the mass the semi draxis and the _25:34_: the eccentricity. _25:37_: Now the nearest planet to the sun is Mercury and it was well _25:41_: known as long we well known that the orbit of Mercury's processes _25:47_: it processes at _25:49_: 574 arc seconds per century. _25:53_: OK so not a lot but it's you know straight forward and _25:55_: measurable _25:57_: and almost all of that is explicable by the gravitational _26:00_: influence of the other planets and solar system _26:04_: you know 99 point something percent of the masters or system _26:08_: is in the sun but the other planets are not are not _26:12_: negligible and the affect the each other. So solving the _26:16_: equations of motion of the planets in classical celestial _26:20_: mechanics is _26:21_: complicated but doable. And _26:25_: by doing things like that people were able to predict that there _26:29_: must be a planet beyond Uranus and and and and who was said, if _26:34_: you look there you're bound to find a planet and and Neptune _26:38_: was found so that that that works. But in that calculation _26:42_: there remained A stubbornly unexplained bit of the the the _26:46_: procession of of Mercury. _26:48_: And Mercury was processing at 43 arcseconds per century. There _26:53_: was unexplained by this process _26:58_: and the thought was perhaps Newton's gravity is wrong. Which _27:02_: of course is the case. _27:04_: And if you work out what that procession is _27:09_: that that _27:10_: particular from from this relativity correction, calculate _27:14_: calculation, put the numbers in, what you get is a procession of _27:18_: that orbit of 43 arcseconds per century, exactly what was _27:22_: observed. _27:24_: And that was one of the that's one of the classical tests of GR _27:28_: that GR does predict, or this financial solution does predict _27:32_: exactly the missing procession of the the orbit of Mercury, _27:37_: you know for three hours seconds is again not much. For three _27:40_: seconds I think it's a. It's a metre _27:43_: 4 1/2 kilometres away so it's not much but it's detected _27:48_: and would long detectable before general activity. So this is a a _27:51_: classic case of there being an anomaly which and explained _27:54_: their new theory comes along and says Ohh I can explain that _27:58_: and we could talk more about that _28:01_: but we won't. _28:03_: So what that means is that OK, you can have orbit, _28:08_: but you also have _28:11_: and as we learned at the beginning of the of the of of _28:15_: chapter chapter 8, we also have light being bent by a _28:19_: gravitational field, _28:21_: and you can come up with fairly based heuristic measurements of _28:25_: how much that is. But also just by looking at the small solution _28:30_: you can work out how much light ray is bent as it goes past an _28:34_: object. And here, for example, there's a a star A _28:39_: an array of light which comes from. The star will be bent as _28:44_: it goes near a mass _28:47_: in such a way that when it arrives at the observer we see _28:52_: it coming from a star _28:56_: the the the star appears to be a position B there. _28:59_: So the star has moved its position _29:01_: because of the presence of because the late we had gone _29:05_: past a large pass _29:09_: and _29:13_: I think in and in 1919 _29:17_: through just after the First World War Eddington and Dyson _29:25_: an expedition to Principe in _29:31_: UM, _29:33_: Brazil. And _29:37_: no no found that precipitate is is an island off Africa and _29:41_: another one with Brazil. I can't remember where it was, but the _29:44_: point because there was going to be a total eclipse of the sun _29:49_: at _29:51_: over there, visible in those places _29:54_: at that point. So what they did, _29:56_: they took, did careful astrometry, got the _30:01_: put, confirmed the positions of of of, of the stars at night. _30:06_: Then during the eclipse the next day, when of course, they when _30:10_: when the sun and the moon were between what proof of the sun _30:15_: was between them and the stars. They remeasured the positions of _30:19_: the stars and found that the stars were coming, were in a _30:23_: different position, observed to be in different position from _30:28_: what they were at night when the sun wasn't. What wasn't there, _30:32_: of course, _30:33_: had away from eclipse, because the Sun had to be occluded by by _30:36_: the moon to make the observation, _30:39_: and with a very fine, very fine, fine and difficult measurement. _30:42_: And there's a lot more one can talk about that particular _30:45_: measurement. It's very interesting historically, _30:49_: but they found a TV ad, a deflection of exactly the right _30:52_: amount. _30:54_: Another the the second classical test of of GR that the light _30:58_: from the from the start was indeed been deflected at a _31:01_: passed by the mass of the _31:04_: SO _31:05_: and now _31:07_: that that was an extreme, a fairly extreme measurement which _31:10_: had to wait for an eclipse to happen. _31:12_: But if you're a radio astronomer now, _31:15_: radio radio telescopes have extremely good angular _31:18_: resolution, and they don't care about the sunbeam being up, and _31:23_: so this deflection is a routine experimental _31:27_: detail when you're making visual observations. If you're making _31:31_: radio observations near the the you near the limit of the sun, _31:34_: then you have to correct for it routinely, otherwise you're sums _31:37_: won't add up. _31:39_: Another place where this happens is if you have a. _31:45_: If that mass there is _31:48_: the Galaxy, _31:49_: and what you're looking at is a quasar on the far side of it, _31:52_: then _31:53_: that guitar will appear to be a different place and a different _31:58_: shape _31:59_: from what it would be if you're looking at it as _32:02_: straightforward. So you can detect the Galaxy in the _32:05_: foreground by looking at the distortions in the shape of the _32:09_: equator in the background. So again, a routine observational _32:12_: effect which is purely realistic. _32:17_: OK. _32:19_: And _32:21_: there's the the figure, _32:24_: no, _32:28_: I'm not going to talk about that. We're gonna talk about _32:31_: Shapiro Delay and the change in _32:35_: frequency, but we won't because that extra detail there. _32:46_: The next _32:49_: possibility is that the next solution we're going to talk _32:53_: about is our _32:56_: I didn't Amic 1. _32:58_: In both weaker Solution and the Charter Solution, _33:02_: the setup has been amassed in the centre of the universe, but _33:05_: you know, amassed by itself in the universe. And what's the _33:08_: shape of the space-time around that? _33:11_: But the _33:17_: Einstein's equation, as I mentioned, is on the left hand _33:20_: side _33:22_: term involving the 2nd derivatives of coefficients of _33:25_: the metric, and the right hand side are term which is the _33:31_: which characterises the _33:34_: energy momentum _33:36_: at that point. _33:37_: But if the right hand side is 0, so if there's no mass at a _33:42_: point, you can still get a solution to _33:47_: to that question. It's and it's a wave. A wave solution. _33:51_: You can still resolution which is. _33:58_: Possible _34:00_: I've I've lost quite a lot of thought and and a solitary _34:03_: solution. _34:07_: And our solitary solution exists and and that solution is _34:12_: the solution of a wave equation and that and that is the _34:15_: gravitational waves. _34:17_: What do gravitational waves look like? _34:22_: Something like this. _34:24_: So if you imagine that the _34:29_: that that that's a two-dimensional space, A2 _34:31_: dimensional space for example the top of a drum for example, _34:36_: then _34:38_: that there there are various solitary modes in in in that _34:43_: drum skin in in that in that surface and and that that that _34:48_: that illustrates 2 _34:52_: phases _34:54_: in a portable oscillation. _34:56_: But if you were at. _35:02_: If you were on on the outside edge of that, _35:04_: you could walk around the perimeter and get a length for _35:08_: it. You could you could you could measure the size of the _35:12_: circumference of that _35:14_: of that space. And if you walked across from why _35:18_: one way to the other, why this bottom case? What you see is you _35:23_: you you measure a diameter which was 1 / 2π times the _35:27_: circumference. _35:30_: But if you walked instead from X to X, _35:34_: because of the distortions in in the space you'd be walking _35:38_: longer would take longer to get from X to X than to Y. So the _35:42_: distance across that the diameter of that space would be _35:46_: larger in that direction than in that direction. If if you were _35:50_: thinking about it, you would think the cleanliness and the _35:54_: lips, _35:57_: but then at a different phase of the oscillation it would be like _36:02_: at the top. And now the distance from X to X _36:07_: in the top diagram is less. The distance from _36:11_: quite away is more _36:14_: so. _36:16_: In both cases, the separation between these two positions has _36:20_: changed _36:22_: without acceleration, _36:24_: so observers at excellent at X&Y in these two cases have _36:28_: not accelerated. They don't feel any movement, _36:33_: but the separation between X&Y has changed _36:40_: and that is a _36:44_: right _36:45_: hold on to that thought for a moment. So that's that's roughly _36:49_: progression we've looked like 2 points in _36:53_: in a speech team as a gravity wave goes past the separation _36:56_: between the changes without them moving _36:59_: to to to to the speech that there's more space between them _37:02_: without _37:04_: without _37:06_: demonstration. _37:07_: Can you observe these? _37:10_: They would be they observed indirectly before they were _37:13_: observed directly. _37:15_: And this is the diagram of the _37:20_: A a binary pulsar called the whole tailor binary _37:24_: PR for the British can't remember and _37:28_: it being a pulsar, the _37:31_: orbital frequency of the two stars in the binary. We're very _37:35_: well characterised but over time _37:39_: the period _37:40_: changed, _37:41_: it slowed down. _37:44_: Why was it slowing down? _37:46_: A number of possibilities exist, but _37:48_: if you _37:50_: plot _37:51_: the PD shift over those years 1955 to 2005 _37:57_: and ask what would the how much energy we would be rotated would _38:00_: be emitted in the formal gravitational waves by these two _38:04_: accelerating masses operating each other. And you plotted that _38:07_: on the same diagram, _38:12_: it worked rather well. _38:13_: Those red dots have error bars _38:18_: with arabad are the side of the of of the line in in in the _38:22_: graph. So that is a magnificent bit of experimental observation. _38:28_: The observation is perfectly match the what what the change _38:32_: in the period would be if this binary were emitting _38:37_: gravitational waves. _38:39_: So that's an indirect but very convincing account of what _38:47_: evidence for the existence of gravitational waves. _38:50_: But going back to this, _38:52_: I mentioned that here what you have is if you had test masses _38:57_: at these four points from the edge _39:01_: and measured the distance between them _39:05_: just by whatever means, _39:08_: then as a gravitational wave went past, the distance between _39:11_: these test masses would change without the test masses being _39:14_: accelerated. _39:16_: And what that is, is a description of _39:20_: an interferometer. _39:23_: So here you have a laser _39:26_: being fitter, _39:28_: being reflected from a mirror _39:30_: and possibly several times. And we we observed and that, _39:38_: ironically it turns out, is the same setup as was used in the _39:42_: Michael Morley experiments, which were one of the famous _39:46_: null results which were one of the puzzles leading up to the. _39:53_: Development of or special activity in the late 9th _39:57_: century, but it's also the the layout of our interferometers, _40:02_: such as the Legal or Virgo or Geo 600 gravitational wave _40:07_: interferometers which were. _40:09_: Built, Built over a number of decades in the US and in Italy _40:15_: and in Germany. _40:17_: So these masses May 123 and four are _40:23_: just suspend it the the, the, the, the, the, the quartz _40:27_: mirrors that sort of size. They're big things and the _40:31_: suspended very carefully so that they are not vibrating. _40:36_: You measure the distance between them by shining a laser back and _40:39_: forth and looking for interference fringes. _40:43_: You you'll have heard about this and _40:47_: it would be formed I'm sure. And this is an interesting _40:51_: experiment because the this is the the the change in the what _40:56_: you're measuring is the change in the distance between these _41:01_: two masses, which _41:03_: because the the the pure suspended they are not _41:07_: accelerating. And that change in difference change in distance is _41:11_: around 10 to the -19 metres _41:14_: through 10 thousandth of the of the diameter of a nucleus. _41:19_: So it's not much _41:22_: so and it was in 2014 and a lot of the the, the and and Glasgow _41:27_: played an important part in this. Glasgow particular _41:31_: contribution was the was very much the experimental _41:37_: details of this. I think Glasgow is big on the details of how the _41:41_: how these lasers work in her condition and on on how the _41:44_: these test matches are suspended. So that Glasgow's _41:47_: contribution to this very experimental part, _41:50_: like we need a lot of of data analysis for gravitational _41:54_: waves. And that's the other the other end of the of the whole _41:59_: sausage, if you like. _42:01_: And as I say, it was in 2014 that this was trevally announced _42:05_: as a as amendment, a direct measurement of the existence of _42:09_: traditional waves, which is rather beautiful. _42:14_: OK, Neil there. _42:17_: Umm, _42:21_: so I won't put for questions, just go straight on another _42:25_: metric, _42:27_: and this metric is _42:30_: mathematically inspired. _42:32_: It's the answer to the question What is the most general measure _42:36_: you can have which is homogeneous, that is the same _42:39_: everywhere and isotropic, that is the same in all directions. _42:45_: The idea being that where we are now isn't special, it's called _42:48_: the Copernican principle. _42:50_: OK, and _42:52_: details the details omitted. The more general metric that you can _42:56_: find that has those probability properties is the Friedman, _42:60_: Levitra, Robertson Walker metric FRW _43:03_: which is that is that one. _43:06_: And you can see that on the this part here you've got that the _43:10_: Omega that's the the angular angular bit and _43:15_: a different coefficient in front of the Dr term. And the the the _43:20_: radial parameter R is scaled so that Kappa is either -1 zero or _43:25_: plus one for different different cases _43:29_: And that E prompter. The coefficient sitting out at the _43:34_: front is an overall scale factor, which _43:39_: is time dependent, time dependent, but not dependent on _43:42_: anything else, _43:44_: and you plug that into the. _43:49_: Thanks to Einstein's question, _43:51_: turn the handle not not. Again, not trivial, and what you get _43:55_: out is a solution for the universe as a whole, _43:60_: which has the property that a dot _44:04_: is not zero and it a double dot is not zero. So the 1st and 2nd _44:08_: derivatives of A are not constrained to be 0, _44:12_: and that solution was thought to be impossible. You can't have an _44:16_: expanding universe. That's clearly silly. And I'm saying _44:19_: then that's OK. Perhaps my part of the island equation is wrong. _44:23_: Well, another term to it you can add another term which doesn't, _44:27_: which is plausible, which has a a constant in front of it times _44:30_: the details that matter. But there's a constant in there _44:34_: called big Lambda logical constant. When you solve that _44:37_: version of the Einstein's equations with this metric, you _44:40_: get you can you can pick the constant Lambda. _44:44_: You know, another white, unmotivated way so the universe _44:47_: isn't expanding. _44:48_: And that was fine. That was good, _44:51_: but then the Hubble expansion was detected. It appeared the _44:55_: universe was actually expanding and so the IT became unnecessary _44:60_: for that extra term to be in in any sense equation _45:04_: and and I I think also his biggest blunder Oh my God from _45:07_: Queen. _45:10_: But subsequent to that it turns out that the that the _45:17_: that me will be _45:20_: rule for that that that constant which which adds what is _45:24_: effectively a negative pressure through universe details essence _45:28_: to follow. And to that that time has come back in as a plausible _45:32_: bit of physics which explains the observed behaviour of of the _45:36_: actual universe. And the last thing to see there is that what _45:40_: that that that that that parameter E the overall scale _45:44_: the overall size of the universe is independent. It's changing, _45:48_: it's increasing. _45:50_: And what that means is that if you track the universe back, _45:53_: there was a point where that A _45:56_: was 0 _45:57_: universal 0 size _45:59_: and that is and and then really, really playing forward from _46:04_: there you have the universe expanding from that point. And _46:08_: this of course is the, the, the, The Big Bang. But The Big Bang _46:12_: is a _46:13_: E dot being positive _46:15_: and a dot and a being zero at some time, _46:20_: which is a big deal. _46:21_: But the _46:23_: last, the very last point to make in the 20 seconds remaining _46:27_: to us _46:28_: is that _46:30_: at _46:32_: that initial time _46:33_: universe was very small _46:36_: obviously. _46:37_: And that means that it was high. They highly curved, _46:41_: the extremely curved _46:44_: to the point where and. And part of the nonlinearity of _46:47_: Einstein's equations is that there is also that the the the _46:50_: curvature of space-time is also a source of curvature _46:55_: that that's one reason why institutions are very hard to _46:57_: solve. _46:59_: And and what that means is the energy density gets very high _47:03_: when the universe is very small, to the point where the energy _47:08_: density in curvature _47:11_: is big enough that you can get particles being created from the _47:16_: vacuum. At that point you have to worry about the quantum _47:20_: mechanics of space-time _47:22_: and that it's quantum gravity and that is still very much up _47:27_: in the air. But that's the that that intense is the the next _47:32_: step after this which is not beyond the scope. _47:37_: And that is the end of of the 15 lectures. There's an inspiring _47:42_: remark from the great ocean of truth that all discovered before _47:46_: me. But and so we still know there's still vast quantities of _47:51_: don't know and the quantum gravity is a large chunk of _47:55_: that. But from _47:57_: starting with the two axioms of spectral activity 15 Electrical, _48:01_: we have come a very long way. I've there's been a lot of hand _48:05_: waving in the last five lectures a lot of it can be shown that. _48:09_: But I hope that I have connected the GR stuff to the special _48:13_: edition stuff well enough that you have some idea of the shape _48:17_: of the mathematical ideas that are that make GR the the _48:21_: description of gravity. _48:23_: We'll stop there,