Transcript of a2-l14 ========== _0:02_: Astronomy 2 Special Relativity Lecture 14. _0:11_: It was the almost the end of chapter eight. I thought I might _0:15_: manage to get to the end of Chapter 8 in time, but there was _0:18_: still a little bit leftover. _0:20_: But that's fine, because that gives us a an opportunity to _0:24_: recap the the the flood of quite sophisticated, the sequence of _0:29_: quite sophisticated ideas that we covered last time. And one of _0:33_: the key things, _0:36_: one of the key things of the whole of Chapter 8 is _0:41_: the the sequence of of restatements of successfully _0:44_: stronger restatements of the equivalence principle. _0:48_: Remember the first one? _0:50_: I was talking about uniform gravitational fields. _0:53_: The second version was talking about local free falling, non _0:57_: rotating laboratories, local and national frames and how they _1:01_: were the thing that we could understand that we already _1:05_: already in a sense understand from our pre relativistic _1:09_: physics that's that's basically nuisance laws. And they are the _1:13_: bridge to the I different way of approaching the question of what _1:17_: gravity is or how what what it means to be moving under _1:21_: gravity, _1:23_: how we go from the physics we understand to the physics we _1:26_: don't yet. _1:28_: And the third version of it was a more mathematical version of _1:31_: the of the whole responsible which said _1:33_: that if you were _1:38_: the _1:39_: that _1:40_: in general _1:42_: if you are in a local inertial frame _1:47_: you're moving under so moving under the influence of gravity _1:50_: or moving well away from all gravitating bodies _1:53_: and and physics works as _1:56_: you can special activity says it does then there's nothing a law _2:01_: that's expressed in geometrical form. Things like momentum, _2:05_: energy, momentum is conserved. _2:09_: Things like, well, let's let's stick with that argumentum is _2:12_: concerned. That's a physical law expressed in geometrical form in _2:16_: the form of a the argumentum vector that does not change its _2:20_: form when talking about a curved space-time. There's no extra _2:23_: complication that arises when you're falling through a curved _2:27_: space-time of what would they highly curved space-time _2:33_: that yeah. So the physics of that natural frame are not are _2:36_: not any more complicated And that that's actually quite a big _2:39_: deal because you could imagine that there would be extra _2:42_: physical features that would happen because you're in a _2:45_: curved the three time locally curved and they don't happen. _2:49_: I'm going to do it because it is quite important, but I _2:53_: appreciate it doesn't sound that big a deal right at this point. _2:57_: But a quick question, _3:02_: you don't. You've studied Maxwell. You've probably heard _3:04_: of, but you haven't studied Maxwell equations yet. _3:07_: But, but, but what I you you're probably aware of what I'll tell _3:10_: you now. The maximum equations are the way I I mentioned them _3:13_: in Chapter 2. _3:14_: And Maxwell's equations are the way that electromagnetism so _3:18_: late radio. All these things are described in a format which Jim _3:22_: Clark Maxwell developed at the end of the 19th century. _3:27_: And we can write those down in a form which is the right variant, _3:31_: that is a geometrical form. It's mathematical, equally exciting, _3:35_: but you can do it. _3:37_: So there is a geometrical statement of macro equations. _3:42_: Essentially says that the the direction of the electrical _3:45_: vector is family variant. _3:51_: So _3:53_: to answer the first question first, _3:56_: I'll say. So you The equivalence was telling us that the maximum _4:02_: equations that work for _4:06_: for our normal experience, our known _4:09_: not living under neutron star experience, _4:13_: those maximal equations will still work _4:15_: in _4:17_: I _4:19_: in. In general relativity you'll still work in a very highly _4:22_: gravitating environment without change. There will be a change _4:26_: appears because you're in a curved space thing. _4:31_: So if you had a radio receiver anyway, your receiver either a _4:35_: gramophone or or else you know the the the the the the GPS _4:39_: receiver on your phone. _4:41_: If you took that close to a a neutron star and and and were in _4:44_: orbit around a neutron star, would it still work? _4:48_: Whose state would still work _4:51_: was it wouldn't still work. _4:54_: Have a chat _4:56_: with that. _5:31_: OK, having thought about that, let's ask the question again. So _5:36_: you have a radio receiver and some some some radio pickup for _5:41_: example. But you're in orbit around the neutron star, so some _5:46_: highly curved _5:47_: space-time. _5:49_: Would radio receiver still work? _5:51_: Who would see it would? _5:54_: Who would say it wouldn't? _5:56_: OK. Well, it would _5:58_: because the conference, _6:00_: because the government will say that even though you are in a _6:03_: very exotic circumstance, you're in orbit around a neutron star, _6:06_: you're possibly falling into a black hole. We'll come to those _6:09_: later. _6:11_: You You're perhaps going to be squished into nothingness _6:14_: because the black holes, they evade the situation In your _6:18_: local frame. _6:20_: You're in freefall, so material, so anything that works in _6:23_: special activity in any free falling frame still works. _6:28_: So you're you probably wouldn't be able to pick up anything _6:31_: because the radio waves that were broadcast from well outside _6:35_: would be highly blue shifted with the same you pick them up _6:39_: as you're falling into the black hole. But this thing was still _6:42_: work _6:44_: because Maxwell squeeze would be unchanged and that is _6:47_: extraordinary that that's why the this, that that vote of the _6:50_: conference is extraordinary because it's telling us that we _6:53_: already know how physics works, even though it's really exotic _6:56_: circumstance around a bit a bit of space-time at the very edge _6:59_: of the universe. _7:01_: And that it gives a big jump into how we understand the what _7:06_: we see in the in those very exotic circumstances. _7:10_: So that's why the equivalence principle, that version of it is _7:13_: such a big deal. _7:15_: It says we we already know at large chunk of how things work _7:19_: near gravity _7:21_: and as I say longer version of that in the in the state. Ohh by _7:26_: the way, talking about slides reminds me that the I don't _7:29_: think the order in the election was forwarder, but _7:35_: I hope the request. I have taken the scribbles from previous _7:38_: lectures and scanned them and they're in the lecture notes _7:41_: folder. I'm not sure how useful they are. They were intended to _7:44_: be useful but they're there and I realised I haven't put the _7:48_: notes for Chapter 9 up in the lecture notes folder. I will _7:52_: probably not today. I would have Are we check all over them to _7:56_: see if they still went bits missing before doing so. _8:01_: It's one of the things that we _8:04_: touched on lasting _8:06_: and _8:09_: is that _8:12_: we realised that because of the equivalence principle _8:15_: we could see space-time tells matter how to move. We can add _8:20_: the extra _8:22_: and _8:24_: bit of physics that _8:26_: just as when we are standing still in a national frame, we _8:29_: are moving through space-time in a very particular direction, we _8:33_: are moving along the T axis, we move along a geodesic. That's _8:36_: the the natural motion in that initial frame. _8:40_: But that that that, that geodesic that we trace out puts _8:42_: very easy trace out because we're just standing here where _8:45_: our duty is pointing straight along the same axis. _8:49_: If we turn that into a different set of coordinates then we are _8:53_: just we. It's the wrong way of saying _8:56_: it's easy to see what our geodesic is through a _8:59_: space-time. It's just pointing along our time axis _9:05_: in a curved space-time, _9:08_: just as we saw for the the an individual moving across the _9:12_: surface of the earth. A geodesic is a geometrical thing. It's a _9:16_: geometrical quantity. It's not. It's A-frame independent _9:20_: quantity. And so if it is and it is still true that we our motion _9:24_: through space-time is what happens when we go along a _9:27_: geodesic in that space-time, _9:30_: the quote straight line in that space-time. Then we understand _9:33_: how to move through a curved space-time _9:37_: because the curvature of the space-time, the shape of the _9:39_: space-time tells us you know what effect where that you would _9:42_: what points that you do actually go through. And making this _9:45_: sound very complicated. This is very clear in my head. _9:49_: It's basically a long version of space-time that might have to _9:52_: remove the the the the the shape of the space-time. This is this _9:55_: is this is the what's being mentioned in those endless _9:58_: videos of people putting weights on on rubber sheets and and and _10:02_: and seeing the rubber sheets turn into a curved sheet and _10:05_: then a a marble roll around in a in a in a in a curved path on _10:08_: that on that the marvellous discovering the geodesic in that _10:11_: curved space-time. And that's the point. _10:14_: I don't want to bang on, bang on about that. Now _10:18_: what we've been talking about is in all of this is a free fall _10:23_: under gravity. _10:25_: So we are talking about, you know, either jumping up and down _10:29_: or falling into a black hole or falling. After all these things, _10:32_: we're talking about free fall. But how does this relate to us _10:35_: standing on the ground or on the floor or whatever and feeling _10:39_: the gravitational force? Because we know there's a force we can _10:42_: feel on the soles of our shoes, _10:46_: and that's where the link back to acceleration comes in. _10:50_: Because if the floor wasn't there _10:53_: then we would move under gravity. You can imagine if the _10:57_: flows under disappeared. You could imagine out of shadow _11:00_: version of yourself following, falling, falling down, and _11:03_: you're looking at it and getting it moving fast fashion. Are we _11:07_: away from you Faster and faster? Accelerating away from you? _11:12_: Except be careful, it's not the one that's accelerating. _11:15_: Remember I said that acceleration is what you feel _11:18_: when you're pushed? _11:19_: If we feel when the train accelerates, _11:23_: whereas when you are in a local natural frame, you are not _11:26_: accelerating in that sense. _11:29_: So the _11:31_: that's a ghost version of you that's falling _11:35_: away with the Secretary, but you it is increasing without a _11:39_: nonzero second derivative. _11:42_: It's not that ghost version of this accelerating, it's you _11:44_: that's accelerating. You are the one that's accelerating away _11:47_: from what you might think of as the natural motion under gravity _11:52_: and to the pressure you feel on your feet _11:54_: is _11:56_: what's accelerating you. _11:58_: So this ghost version you is falling, is free falling under _12:01_: gravity following, following a sort of natural motion strictly _12:05_: in scapegoats. But you the fraud, accelerating away from _12:08_: that ghost version. And that's the link between acceleration _12:12_: and gravity. _12:14_: So the force of gravity that we feel in our own sort of our feet _12:17_: is the force of acceleration. _12:20_: And and that looks right back to the very beginning of of of _12:23_: Chapter 8 where I talked about the the, the, these objects in _12:26_: the box and the rocket acceleration of the box and _12:28_: people and and the objects being pushed to floor and being _12:31_: accelerated by it. So that is actually a key notion, _12:35_: although it seems like there's a rather simple thought _12:37_: experiment, _12:40_: and that's a profound point. _12:42_: And um, _12:46_: which is worth thinking about as you as you as you walk home. _12:49_: And a final point there is that there are _12:56_: if you go to the library and look at _12:59_: general relativity textbooks. So look the the body of this is. _13:07_: Get mathematically complicated quite quickly, but the _13:10_: introductory chapters of these textbooks are often very good _13:13_: about _13:15_: putting these ideas I've talked about here into into different _13:18_: and portable water laminating sequences. _13:21_: So look and Rinder, which I I think I've mentioned a couple of _13:25_: times as a more advanced general relativity textbook, is good _13:29_: about the fundamental ideas and the sequence of ideas. So if _13:32_: you're if you're if you're puzzling into that and you want _13:36_: to be prompted to think more about it then _13:40_: the introductory chapters of Advanced Books are accessible. _13:46_: And so that was I quickly recap what I I got a debate. Are there _13:50_: any questions about that, about that sort of stuff for the final _13:54_: section of this _13:56_: of this chapter? _13:59_: What I was talking about the end last time was Poisson's equation _14:03_: and the analogy between that and the equation we're going to go _14:09_: into There. I said that this was the _14:13_: that's the fire. There was the gravitational potential, _14:18_: and that's just the thing that you're familiar with from your _14:21_: your knowledge of of Newtonian gravity. It's how much _14:23_: gravitational potential is that this height, at this height, at _14:26_: this height and this height, and so on. _14:29_: And the second derivative of that _14:32_: is constrained by the amount of mass at a point _14:37_: R. _14:39_: So in the typical thing that we're interested in, there's a a _14:43_: lump of mass at the the centre, the sun see, and the the the the _14:47_: gravity potential that results from that is a solution of _14:51_: Poisson's equation. It is such that the 2nd derivative of it is _14:56_: zero everywhere except at the centre. _14:59_: OK, and a solution or personal equation _15:03_: consists of identifying what what role is the daughter of the _15:06_: centre and then mathematically you know, solving that equation _15:09_: to find what Phi is and you get a A1 over our potential from _15:12_: which you get the an inverse square force which from which _15:15_: you get nuisance gravity. _15:18_: So that is where Newton gravity comes from. _15:22_: And I mentioned that _15:25_: going to Einstein's relativity, Einstein's version of gravity, _15:28_: we you know with a version of that _15:31_: we can't talk about _15:33_: mass _15:35_: quite the same way, because as we know from Chapter _15:38_: 7, the dynamics chapter at mass is just the sort of the time _15:43_: component of an argument vector. _15:47_: So what we have to consider instead is a more general _15:51_: object. _15:52_: I talked to an energy momentum tensor and and and talked about _15:55_: the a tensor being the same as the the stress strain tensor. _15:58_: Which allows you that to say, to say things like what's the the _16:02_: component of force and the extraction across a plane _16:04_: through it, through this, this solid object and so on. _16:09_: The environmental tensor talks about the the flow or the _16:12_: momentum flux across a surface. It talks about the energy flux _16:16_: across the surface into the future. _16:19_: So it it talks about the energy and momentum, or the the energy _16:23_: momentum inside a box if you like, inside an object and _16:26_: waving my hands you're quite literally as because I don't _16:29_: want to get into the details of what that is. But the point is _16:33_: it is the the relativistic analogue of that matched term. _16:38_: So the right hand side of the equation we're we're gonna be _16:41_: looking but we we can't study but we're gonna be quote is a _16:44_: mass related thing in the same way _16:47_: towards the left hand side _16:51_: if you _16:53_: I I have repeatedly quoted. _17:00_: Things like the _17:07_: the s ^2 equals the X + y ^2 as the differential version of _17:10_: Pythagoras theorem. _17:13_: I've talked about the _17:16_: describe minus DX squared, a sort of differential version of _17:23_: right _17:27_: of the of the interval and has said that this that both of _17:30_: these are are metrics _17:32_: in the sense that they are the the things that define distance _17:37_: in the respective spaces. _17:41_: With this _17:43_: now the _17:48_: the the thing that's called the metric _17:51_: is _17:54_: from maps so that for example also in _18:03_: spherical pullers that would be the r ^2 + r D Theta squared _18:09_: plus _18:10_: sine squared, Theta D Phi squared. _18:15_: Just going to quote that just to say that that that that's still _18:18_: the metric of of of flat 3D space but it looks a bit more _18:22_: complicated because we could we're doing it in in in in _18:25_: spherical Polaris. But the point is you can see there are _18:31_: crawfish is multiplying the various, _18:33_: you know, one outward outward change word, multiplying the _18:38_: various differential elements there and the _18:42_: those are simple versions of of a matrix essentially _18:48_: are another tangible object which is called the metric. _18:55_: Again _18:57_: elating some details, _18:59_: the point is the metric. This definition of distance d ^2 _19:06_: is the thing that corresponds to the _19:11_: potential in _19:13_: Poisson's equation _19:15_: with with the metric is the thing that defines the shape of _19:18_: the of the curved surface. Remember it was because we had a _19:22_: a metric like that on the surface of the of the of the _19:25_: sphere that we that we we end up discovering spherical _19:28_: trigonometry with with with these different rules about _19:31_: internal angles of triangles. _19:34_: So jumping. So this is all _19:38_: seeing that _19:42_: to make plausible the observation _19:44_: that the thing that ends up being. _19:48_: Instead the equation is _19:51_: and Andrew Mentum tensor, _19:53_: which is a way of encapsulating the amount of energy momentum in _19:57_: a in a box. If you like please. And another thing, sort of G _20:02_: big. _20:03_: Which involves the 2nd derivatives of the _20:09_: the _20:11_: of the coefficients of the metric. _20:15_: I am missing a lot out there, _20:17_: but but the the the key thing is this in a way looks like _20:20_: Poisson's equation. _20:22_: In both cases you have a thing on the left, on the right, _20:25_: sorry, which is related to the mass, _20:29_: and I think on the left which is related to a second derivative _20:32_: of the shape, if you like. _20:35_: In that case the shape is just a potential. In the case of _20:38_: Einstein's equations, the shape a second derivatives of the _20:41_: metric of the coefficients of the metric. _20:45_: There's not much I can say about that before, without going into _20:49_: way too much mathematical detail that that was occupied for _20:53_: another few months. _20:55_: But the the key thing is that analogy is _21:00_: it is real. It is about analogy and to point out that this _21:05_: equation, this is what I say published in 1915 _21:10_: plus definitions were GS because it's so that that expands to A _21:14_: to A to a big set of equations. But it's basically a simple _21:17_: idea, and that is another of these physical statements. _21:22_: You can imagine that being otherwise, it is mathematically _21:24_: reasonable for that to be otherwise. And there were some _21:27_: other alternatives that Einstein tried first to think, oh it's _21:30_: going to be this is going to be the physical law that's going to _21:33_: explain everything and it didn't work out. _21:36_: So over the course of of some years and months leading up to _21:39_: the end of 20/19/15 he eventually discovered that the _21:43_: there was a way of of putting things together that identified _21:47_: A quantity of G and quantity T which for which this equation is _21:51_: equality _21:52_: explained everything. I would have discovered how it explains _21:55_: things in in a moment, _21:58_: So there's not much you can do with that fact, but nothing you _22:02_: can do with that because maths. But the the reason I'm showing _22:05_: you is simply true that it that in a sense it's quite simple, _22:10_: but in a sense it's also very hard _22:12_: because these are equations linking multiple components of _22:16_: these matrices. So those are both _22:20_: that that that equation there represents 10 simultaneous _22:23_: second order differential equations and that's not easy to _22:27_: solve _22:28_: and they have not been solved in general. _22:31_: What has happened is people have solved them in particular cases. _22:35_: Particular special cases, particularly simple simple cases _22:38_: could discover those. _22:40_: So no general solution of that exists, _22:43_: only _22:44_: situations only, only analytic solutions of special cases _22:47_: critical and. _22:49_: But the point here then is this allows us to complete the other _22:53_: half of the female slogan _22:56_: speech themselves. Material curve _22:59_: The culture of the space-time government. The geodesics in the _23:02_: space-time and the geodesics are how we move through things _23:06_: and matter to your speech temperature curve. _23:08_: The distribution of energy, momentum in space governs. That _23:13_: is what governs the shape of that space-time. _23:18_: So the two things linked together _23:22_: and and that is the key insight _23:25_: of general relativity if you like _23:29_: masters in the details. But that is that is the key into and and _23:35_: and and it links very smoothly to special relativity. _23:38_: So one of the of of the questions that came up I think _23:43_: I think on the, on the, on the on the padlet was why studying _23:47_: special video. I'm not sure if that was how it was explained, _23:50_: but what would somebody if the application, you're special _23:54_: activity, and there are applications, special activity _23:57_: within particle physics, You can't design CERN without _24:00_: knowing lots of special activity. _24:03_: But the other big thing, that very important thing, and The _24:06_: thing is very important for you as astronomers is to is that I _24:09_: want you to have a very clear picture in your head of how _24:12_: smoothly special relativity turns into general activity. _24:16_: Specialty is the the key link that that goes from the physics _24:19_: that we understand to the physics that we don't yet. _24:27_: We'll come back to that picture later, _24:31_: OK? _24:33_: On to Chapter 9 _24:35_: and breathe. _24:45_: Are there any questions about that that certainly caught _24:48_: anybody? _24:51_: There were plenty of questions as you as you mould this over in _24:55_: your head and so I do ask those, But these are some of the most _24:58_: exciting ideas in physics of the last of the 20th century. And _25:02_: the thing that's more exciting than these is in 21st century _25:05_: things like spring cleaning and quantum gravity and stuff. But _25:08_: this links directly to directly to that. _25:11_: So this is all a classical theory. _25:15_: OK, moving on. _25:18_: So in this chapter there's the final chapter and we'll have a _25:21_: lecture and a half to get through. _25:26_: I'm going to just talk about solutions to ancient equation. _25:30_: So I I quoted intense equation there and said details _25:34_: complicated, _25:35_: but a solution to Einstein's equation is when you you you _25:39_: take a configuration of _25:42_: of matter and energy, whatever on the right hand. You can write _25:45_: down fairly easily what the right hand side of that equation _25:48_: is, what the energy metric tensor is, _25:51_: and then the key to finding what is the metric, _25:55_: what are the coefficients of the of all these coefficients of the _25:60_: metrics, such as the? What's the factors in front of of DRD, _26:04_: Theta, and Phi? What are, what are those coefficients of that _26:08_: which satisfy itens equations? And we've got that. That's a _26:12_: solution to _26:13_: equation. It gives you a space-time which is curved in _26:16_: the right way. _26:18_: OK, _26:20_: the so the objective here are rather hand waving ones, you _26:23_: know give a qualitative account and and and so on and so on. _26:28_: What we're going to cover is _26:31_: the weak field solution, and the weak field solution is the case _26:35_: of the the extreme case where the engine intends to. The right _26:39_: hand side is is small _26:41_: and as you will be familiar from other basic but your education, _26:46_: extreme cases such as you know, small mass are solvable in a way _26:50_: that the more general things are. _26:54_: We'll discover the special space-time _26:56_: which is extreme in the case that there is only one mass in _26:59_: the universe. _27:01_: That's another extreme case which happened to be solvable. _27:05_: Will briefly touch on gravitational waves, which are _27:08_: interesting because they are source less _27:11_: solution in the sense that gravitational waves don't depend _27:15_: on any masses being there to propagate. They're all about _27:18_: gravitation wave propagating through an otherwise empty _27:21_: space. Time, and we'll briefly touch on if we have time, on the _27:25_: solution of the of Einstein's equations, which refers to the _27:28_: entire universe, _27:30_: which is another extreme case. _27:33_: So we can see a little bit about all of these things, _27:38_: no? _27:40_: Here's another metric, _27:43_: and this is actually similar to what we've seen before, _27:46_: because if you _27:47_: ignore that bit, which I'll come to the moment, that metric there _27:52_: is just DT squared minus DX squared and 100 X squared _27:55_: written D Sigma squared and D Sigma squared is. I will _27:59_: occasionally refer to it as _28:02_: is DX squared plus dy _28:05_: squared plus. _28:07_: These are just the the, the, the, the the distance element of _28:12_: Euclidean space. Texas, that's flat space that's what would be _28:17_: segment. So without that there that's just the metric of _28:23_: of special activities Explained it. DT squared minus DX squared _28:28_: basically. _28:31_: But if you think back to the beginning of Chapter 8, _28:35_: one of the of thought experiments I talked about was _28:38_: this idea of our a mass following, following being _28:42_: turned into a photon and and writing again. And we were able _28:46_: to deduce that there was a _28:48_: are our redshift which happens to be found at that point. _28:52_: And _28:53_: I showed you, I I I fairly quickly. I showed you that you _28:57_: could characterise that red shift by the gravitational _29:01_: potential. _29:03_: A Newton gravitational potential. How? How? How high _29:07_: above the the mass are you _29:10_: and _29:12_: you know little did little little detail _29:14_: you can _29:18_: and _29:22_: I'm trying to use the free It can be shown that as little as _29:24_: possible but I'm not going to succeed very much in the in this _29:27_: final section. _29:29_: So I think that I haven't expanded that in the notes and _29:32_: I've just said it. It can be shown that _29:35_: that the the way that that that observation of a red shift _29:40_: coming through a gravitational field can be held together is if _29:45_: the distance between that's it. You remember the Shields photon _29:49_: thing that I've briefly mentioned we had. You had two of _29:54_: four _29:55_: diagram with two photons going vertically upwards after a delay _29:59_: and we discovered and I _30:02_: very quickly mentioned that you could see because of the red _30:06_: shift of the photon climbing through the field, that the _30:10_: lengths times between the end point of those arriving photons _30:14_: were and the departing photons were different. Blah blah blah, _30:18_: doesn't matter. The point is you can justify this as A _30:23_: at a metric _30:25_: for the space-time where this _30:29_: function Phi this coefficient in front of the the DTSA component. _30:37_: Is this recognisable expression here GM over R? _30:44_: And this is in natural units, so this ends up being a very small _30:49_: number in these units, so this is 1 + a small number. In other _30:53_: words, this metric is very nearly the Minkowski metric, _30:59_: but with an extra term here, an extra small term here which _31:03_: allows you to. _31:09_: Slightly deviates from that _31:13_: and _31:19_: no _31:23_: the _31:27_: there's movement could see about that, but I'm going to move on _31:32_: to the next to the to quickly to the next. The the reason I _31:37_: mentioned that is because you can go from just to. Note that _31:41_: you can go from essentially the _31:45_: the arguments in the _31:48_: the beginning of chapter 8, the regift argument and the shows _31:53_: photons arguments to that metric fairly directly. You don't have _31:58_: to use any general activity I and and. _32:04_: I'm not seeing any more of that. _32:10_: What I am going to see instead _32:12_: is that. But what? But the point of that is that all of that is _32:17_: that you don't use Einstein's equation in that you can _32:21_: slightly clever way get teacher tiptoe into talking about curved _32:25_: space times without actually using items. _32:29_: If however you start with ancient equation, the G equals _32:34_: Kappa T _32:37_: and look at it in the extreme case of _32:42_: a very small mass. _32:44_: And a small mass here is something less dense than a _32:47_: neutron star, _32:49_: so a star or a or a Galaxy or something like that. So it's a _32:52_: small, a small mass, _32:56_: and if if in that case a Galaxy probably wouldn't have it was _33:01_: dark, but it's something something anonymously small, _33:06_: then you can do. Then a lot of the of the mathematical _33:09_: complication at that point falls away and you end up with a _33:15_: aversion, _33:18_: a low mass version of Einstein's equation, which is solvable. _33:24_: And it's and it's solvable in the sense that you can find _33:28_: a metric which which satisfies that equation and that _33:33_: that solution is this one. It's called the weak field solution, _33:37_: and again you can see that _33:40_: if I ignore bits of it, _33:43_: that's still DT squared minus DX squared. So that's still vague. _33:47_: That's still close to being the Minkowski metric. _33:52_: So the the, the, the the space-time in our around our _33:57_: a small mass object is still. This is seeing basically that of _34:02_: special activity _34:04_: but with some perturbations from it _34:07_: Which and this and this pops out from pops out from Einstein's _34:11_: equation _34:12_: which have this form this GM over R which is the potential _34:16_: that you find in using gravity _34:20_: and this if you like is the first Test of general activity _34:23_: and and so I think Einstein had a happy day when he he he he _34:26_: worked this out _34:28_: that if you go from Einstein's equations _34:32_: take the low mass limit of them _34:35_: then what you get is an expression which involves _34:38_: something which is very clearly identifiable as the interim _34:42_: potential. And if you then turn the different handle _34:46_: and ask OK what are the geodesics _34:48_: in this _34:51_: in in this space-time do I have a picture of those Yeah ohh _34:55_: those are the those those are the the other dude the equations _34:58_: of motion for the duo D6 OK does that help us. It does help us _35:03_: because those that that that P there is the energy momentum _35:08_: vector. _35:10_: If you remember, that's the thing with ease in the time _35:13_: component and the spatial momentum in the spatial _35:15_: components. _35:17_: And what that first equation is saying is that the rate of _35:21_: change of the 0 component of the energy momentum of our test _35:24_: particle _35:26_: is proportional to the rate of change of Phi with time. And so _35:30_: if if the the the the the the central mass is a star we're _35:33_: talking about isn't changing mass which it won't be _35:37_: very high stable to 0, so there's zero component of the _35:40_: energy, momentum of our test particle will will be constant. _35:44_: In other words, energy is conserved in as things move _35:47_: around in this space-time, _35:51_: and this _35:52_: second equation, which is I in the in the in the spatial sector _35:58_: is a way of rating F = -, M _36:05_: del Phi, _36:07_: which is Newton's law of universal gravitation. _36:11_: In other words, _36:14_: the the low mass limit of isange equations _36:20_: is solvable to give _36:22_: a metric to give a a space-time which is curved slightly curved. _36:26_: Because again Phi is much smaller than one, it's slightly _36:29_: curved in a particular way. _36:32_: And when we ask how do part, what are the geodesics in that _36:36_: space-time? How do particles move in that space-time? We _36:39_: discovered they move in the same way. Excuse me that Newton says _36:43_: they _36:44_: So we have invented Newton's law of universal gravitation. _36:48_: So this love universal competition is a pretty direct _36:52_: consequence of one of the simple simplest, _36:57_: one of the simplification #1 of Einstein's equations. _37:02_: That is a solution that maintains equation, _37:04_: and that is pretty mind blowing I think. _37:08_: And that is one of the first tests of generativity. That's _37:12_: one of the first reassurances that you're onto something with _37:16_: that, that that Einstein's guess of G equals Kappa T _37:20_: That was right at her physical state, _37:25_: which just meditate on that just for for for for 10 seconds, I _37:28_: think. How wonderful that is. _37:39_: Key points. _37:43_: So that was the. _37:48_: We've looked at 2 simplifications there. _37:52_: We looked at the very first thing I mentioned, which was the _37:56_: slightly partial missed out. Next time the a version of what _38:00_: we are getting _38:03_: a curve to be saying which doesn't use instant equation but _38:07_: is fairly fairly directly follows from the thought _38:11_: experiments we had at the beginning of chapter 8 And it's _38:15_: one thing the simple case of very low mass and non _38:18_: relativistic motion. I should I should also mention gives us a _38:23_: nice agree which is solvable feel straightforwardly to give _38:27_: nuisance laws. _38:29_: But the other simplification that we can make to a scenario _38:33_: is not that different from that weak field solution. It's to see _38:37_: what happens if we have only a single mass in the universe, _38:41_: only a single massive object in the universe, and the universe _38:45_: with single message. What is the shape of the space-time _38:50_: Miranda _38:51_: and we're going to not talk about things moving at non _38:54_: rustic speeds. What we'll we'll talk things moving as fast _38:57_: rewards. We won't have that restriction that was that _38:60_: restriction is implicit in the week. The previous weekly _39:03_: version and _39:04_: I said equations were actually published those as equations in _39:08_: 1815, _39:09_: in 1916. So only a year later _39:13_: Carl Starshield _39:15_: vote solution to that other simple case where there's only a _39:19_: single mother universe. And there's all the more impressive _39:22_: because Churchill was an officer in the in the army and was in _39:26_: the middle of the First World War. So there there were _39:29_: literally shells going overhead when he was trying to work this _39:33_: out. You know there's a variety of ways of distracting yourself, _39:37_: but solving 19 equations is is one of the most stylish ones, I _39:40_: think. _39:41_: Anyway, Schwarzschild discovered that this was a solution to the _39:47_: twins equations in that special case of a single single mass, _39:52_: and this also _39:55_: if you were to take R to be 0, _39:59_: you see this also is DT squared. _40:03_: Um minus _40:05_: at the artwork, _40:07_: minus R ^2 D Omega squared. The Omega squared there is is _40:12_: is this the Theta squared plus sine squared D Phi squared? _40:15_: That's the the the the the area element on the surface of a _40:19_: sphere is what is all the Omega is there. _40:22_: So you can see that if R is zero, that also reduces to _40:26_: Minkowski Minkowski metric, _40:29_: so that also is compatible with _40:33_: special activity. _40:35_: You can also see that in the spatial sector, that's the D _40:39_: R&D, Phi and the Omega. So that's that's on the surface of _40:43_: the sphere, that's the radial component _40:47_: that the that they are no longer flat. _40:50_: So this is spatially curved as well as curved in space-time _40:58_: and other things that for that are worth pointing out. There _41:03_: are _41:06_: ohh yes that, this _41:09_: and the reason I haven't written up. _41:12_: Ohh, yes. And and this _41:15_: them. _41:17_: There's this big art. There is _41:21_: parameter which has the same which has the same dimensions as _41:25_: a radius. _41:26_: So the the, the the the. The radial scale here is governed by _41:31_: this parameter big R _41:34_: and that _41:35_: that that that are is called the short short radius. _41:40_: What is that parameter R? _41:43_: But in physical units, it's two GM over C ^2. _41:48_: So again, there's something reminiscent of _41:51_: Newtons gravity _41:54_: in there, _41:56_: and that really is there is the radius on which this functional _42:03_: solution _42:06_: deviates from _42:09_: from from a flat that that that characterises the the the the _42:13_: length scale on which the spatial solution deviates from _42:17_: flat space. _42:19_: What number does that produce _42:21_: if you put in the numbers there for that? That G, by the way, is _42:25_: not confusingly isolated intensive, but but but but the _42:28_: but nuisance constant use constant coverage of of of of _42:31_: gravitation. You put the numbers in there with, for example, the _42:34_: mass of the Sun. _42:36_: Then the structural radius of the Sun is 1 1/2 kilometres. _42:42_: So _42:45_: if the mass of the sun were compressed into something _42:49_: smaller than 1/2 kilometres, _42:53_: because it would be like who and and and and that's what we're _42:57_: about to come to. But _43:00_: that that that radius are we 1/2 kilometres and _43:05_: whenever art was bigger, was much bigger. _43:08_: The ratio between that slash rates are and the regional _43:12_: parameter little R is how much this deviates from being just _43:17_: flat. Space-time flatmate office space-time. So out here _43:22_: when we are 500 light seconds from the sun, _43:26_: that R here is much bigger than _43:30_: the the 1/2 kilometres. _43:33_: So out here, space M isn't curved at all really, _43:36_: basically not covered at all. _43:39_: It is but but it but it it's very, very small. It's small by _43:42_: a factor of 1/2 kilometres. Over 500 light second _43:47_: means not very much _43:51_: and and and the I can't offhand but the the special region of _43:54_: the Earth is or you can work at your own spatial radius _44:00_: if you wanted to _44:04_: and _44:06_: and I'll also mention just in the last few minutes that those _44:11_: are _44:13_: some weight of of of writing down _44:15_: the _44:19_: some sample version so that the _44:23_: GM is that the coverage of constant times the mass of the _44:28_: sun is that or 1 1/2 kilometres in natural units _44:33_: and _44:36_: that number at the top there 1.327 at 10% twenty actually had _44:40_: a lot more decimal places _44:43_: add to it because it can be measured quite very precisely _44:46_: because you can make very precise measurements of things _44:49_: like artificial satellites in orbit around the around the sun _44:52_: and make and measure that very carefully. _44:55_: So you can characterise that's called the gravitational _44:58_: parameter of the sun, and this is a Metal Gear. Here we're just _45:02_: that parameter. You characterise that very, very precisely. So I _45:05_: think to sort of like _45:07_: 8 significant figure, 7-8 significant figures, _45:10_: but then to look at what the mass of the sun is. What you do _45:14_: is you take that gravitational parameter GM and you divide by G _45:18_: How do you get G? You make experiments on Earth with things _45:22_: like torsion balances and so on, but the _45:25_: those _45:26_: figures for big G, the gravitational constant, you know _45:29_: it's a hard measurement to do, so you only know those to to to _45:33_: to perhaps three significant significant figures. _45:37_: So you're dividing this quite accurate number by a number _45:39_: which is not only three significant figures. So you get _45:42_: the mass of the sun in kilogrammes _45:44_: to only three significant figures. _45:46_: So we know that this factual radius of the of the Sun _45:49_: GM _45:51_: to about 8 significant figures. _45:53_: But we know the mass of the Sun kilogrammes. So that's the mass _45:57_: of the sun in metres is what we know. _45:60_: But to convert that to the mass of sun in kilogrammes we have _46:03_: divided by number. We don't know very well. So we know that the _46:07_: radius of the sun the matter sun metres better than we know the _46:10_: start. Again, we know the mass of the Sun in metres _46:14_: better than we do with the mass of the Sun in kilogrammes, _46:18_: and that's a striking thing I think. _46:21_: And and and that's the the official radius of the of the _46:25_: Earth there. _46:29_: Before we go, I think next time that's a good place to stop and _46:32_: next time we'll pick things up until you stop active at _46:36_: Blackpool.