Transcript of a2-l13 ========== _0:02_: Protecting Gravitation Vector 13 _0:10_: This is a peaceful lecture. This is where we're all your hard _0:14_: work thinking about things Start to pay off and you'll come back _0:18_: to this one, I hope, and think more deeply about the details _0:23_: as before. Good. I'm coming from _0:28_: right _0:30_: as I said before, _0:32_: but before we get going with that couple of details _0:39_: is that the after some investigation, _0:43_: the _0:45_: Echo 360 recording of the lectures are now visible. _0:49_: I thought they were before, but it turns out that what I was _0:51_: seeing was what you were seeing. And I cheered them. I carefully, _0:54_: you know, assembled them all to one place carefully. Shield _0:57_: shared them with a null set of people _1:00_: so they were invisible to everyone except me. _1:03_: Turns out that has now been rectified. So if you go to the _1:06_: link _1:07_: in the in the middle to the E360 lectures you should see _1:12_: is regarding _1:15_: Now. Also on the Moodle there's a link to our collection of _1:19_: recordings assembled as a podcast. _1:23_: That's my recordings. The audio is better for those for reasons _1:28_: I don't know why, but they don't link to the slides at all. So _1:33_: it's whichever you you you feel is more useful. If the _1:37_: recordings equities, the recordings are not visible. I _1:42_: have had at least one useful report that they are. _1:47_: If you don't see them when you expect to see them, let me know. _1:50_: The fact that no one in _1:52_: but how many weeks is has mentioned to me before that _1:55_: they're not visible suggest they're not heavily used, _1:59_: but I am interested in feedback on how useful those either those _2:03_: sets of recordings is. _2:06_: Another thing is to reiterate what I mentioned I think last _2:10_: couple of lectures and also one of the title that posts is that _2:14_: the material that I deemed to be in scope for the class test is _2:19_: chapters one to six that everything up to Kinematics _2:24_: Max is really interesting and there's lots of nice questions, _2:28_: there's lots of nicely accessible material in there _2:32_: that's a hint for later assessment opportunities. But _2:36_: for the class test it up Chapter 6. Kinematics. _2:41_: Any other questions _2:43_: about organisational things of that type? _2:46_: OK, now on to astronomy. _2:49_: As I mentioned last time, a time before the GR bit is _2:56_: conceptually _2:58_: challenging _2:60_: because it's also _3:02_: mathematically very challenging. We can't go into very many of _3:06_: the details, which means that I can't that that there are fewer _3:10_: things _3:12_: I can assess in these last five lectures. _3:15_: Pay attention to what it says in the _3:19_: objectives, aims, objectives at the beginning of the of the _3:22_: section because all the things and the objectives as I repeat _3:26_: are the things I will deem to be in scope for the _3:29_: degree exam in June. So. So those are the things that I deem _3:33_: to be accessible, _3:35_: but they are rather thinner, if you like, in these last five _3:39_: lectures than in the previous ten, simply because it's hard to _3:43_: assemble accessible things. _3:45_: But that should not to tell you from paying the most rapt _3:49_: attention to these five lectures, because they are, in a _3:52_: sense, the really interesting bit that the the special _3:55_: activity _3:57_: is paying off in. _3:58_: So enough pep talk. Let's get going. _4:03_: You remember that last time I was talking about _4:07_: I? I just got onto _4:08_: talking about _4:10_: metrics _4:12_: and the idea that the the definition of distance inner _4:16_: space is given by the metric and I mentioned the Euclidean _4:20_: metric. I think I _4:23_: the including metric which was _4:27_: seem not to have written down in a Slade but it's _4:34_: go back to where I was it's. Well I'll leave you a bit of _4:38_: chalk just for it's the main straight forward the squared _4:43_: equals DX squared plus D y ^2 and that's that's Pythagoras's _4:47_: theorem. OK it's Pythagoras Pythagoras theorem written in _4:52_: differential form. You go a little bit in the extraction a _4:56_: little bit in the Y direction. You square the both of those and _5:03_: the answer is the distance you've gone in Euclidean space, _5:08_: just by that's theorem. _5:10_: The reason I'm calling it the metric is because that _5:14_: relationship between _5:16_: and a difference, the next coordinate difference, the Y _5:19_: coordinate and the distance travelled, is characteristic of _5:23_: flat Euclidean space. _5:25_: That's the space on a sheet of paper or drawn in the sand. _5:30_: And the three-dimensional version of that, d = + y ^2 plus _5:34_: DZ squared is the the way that distance is defined in 3D space. _5:38_: If you go from here to here, the distance you've travelled is _5:42_: that squared plus that squared plus that squared square rooted. _5:46_: OK, and that is completely characterising Euclidean space. _5:52_: OK, either in two dimensions or three dimensions. _5:56_: And I I very quickly the at the end of last lecture talked _5:59_: showed how you could turn that into what looked like a very _6:03_: complicated way of answering a simple question, the distance _6:06_: between two points directly from the metric. _6:09_: And there's other things I could, I could you could talk _6:12_: about there. But the point is just drive home. That is _6:16_: this the full story of Euclidean space, if you like. Everything _6:19_: about Euclidian space is in there. _6:23_: But that's not the only space you are familiar with that _6:26_: you're familiar with, _6:31_: because you're also, as astronomers, familiar with _6:35_: spherical, spherical trigonometry and the geometry of _6:38_: things on the surface of the sky, on the celestial sphere. _6:43_: You remember that from last year, _6:47_: and there are coordinates just with other X&Y or R and _6:50_: Theta coordinates. _6:52_: On the plane _6:54_: there are coordinates on the sky, _6:57_: latitude, longitude, or or the the various variants of that _7:00_: that you know about, something like that. So there's a a _7:04_: longitude coordinate, and there's a there's a cool _7:07_: attitude coordinate, and you can identify any point on the on the _7:11_: surface of the celestial sphere with those with those two _7:15_: coordinates. _7:16_: But one of the other things you're taught in first astronomy _7:20_: is how to get _7:23_: distances, calculated distances between two points on the _7:27_: celestial sphere. And what you don't do _7:30_: is take the if you've got two points which are separated by by _7:34_: D Theta and and and and and and D Phi. What you don't do _7:38_: each calculate do theatre squared plus D Phi squared and _7:42_: squared for that, as you recall, because that's not what that _7:45_: give you the wrong answer. On the celestial sphere _7:50_: do I have a? I don't have a a. But what you do have instead is _7:56_: that the the s ^2 is equal to _8:00_: are able to speculate memory the Theta squared plus _8:05_: sine squared. Theta D Phi squared _8:10_: I I'm _8:13_: how will they? Haven't they? Haven't they? _8:16_: Yeah. So that that's the distance between two points on _8:19_: the celestial sphere, _8:22_: and it's not the same. _8:25_: I'm not telling you you don't know here, but I am pointing out _8:29_: that the difference between the geometry of things in flat space _8:32_: and the geometry of things on the celestial sphere is all in _8:35_: that. _8:37_: How does that manifest itself? _8:40_: It manifests itself in things like this. A spherical triangle _8:43_: on the surface of a sphere. _8:46_: OK, you remember spherical triangle. You go from that, the _8:49_: other three points on the on the sphere, and you take great _8:52_: circle here, here and here. _8:54_: And you can do things like ask what are the internal angles, _8:58_: Look at the internal angle of this triangle and ask things _9:02_: like what's the given two of these angles, what's the third? _9:05_: You remember that sort of stuff. _9:08_: And the rules for that special trigonometry are not the rules. _9:11_: You learned about triangles when you're doing soccer too, and all _9:14_: that sort of stuff in school. _9:18_: And some of the things that you are familiar with from plane _9:22_: geometry are not true here. _9:25_: So for example, the on plane geometry, the internal angles of _9:29_: a triangle add up to 100 degrees or π radians. You remember that _9:34_: and that's not true here _9:38_: on in plane geometry the the the area of our _9:43_: of a triangle is given by some something to do with the the _9:46_: internal angles and that's that's a different thing here. _9:51_: And what we again are. _9:55_: I would like to have more slides on more equations than this. _10:02_: I'm not gonna write it up, but the _10:06_: there is an expression for the _10:12_: Yeah, there's an expression for the deviation _10:16_: between the sum of those angles, AB&C and π radians, which is _10:19_: a complicated thing. We know that's the point really having _10:23_: to say because just a bit of a mess but it's something you can _10:26_: work out _10:27_: and it depends on the _10:32_: the length of the side of that triangle AB, little AB and C and _10:37_: it depends on the radius of that of that of the sphere there now _10:41_: in _10:43_: celestial is is is special trick you you've learned the celestial _10:47_: sphere is deemed to be of radius one. _10:50_: But in this case we're going to say it's it's of the the sphere _10:54_: we're talking about is of radius R _10:57_: and we can do our calculations and get and learn that the _11:01_: trigonometrical things on stuff this year. Just to to remind _11:04_: you, this is the surface of the sphere I'm talking about, not _11:08_: the sphere as a whole embedded in 3D space. _11:12_: And an important point _11:15_: is that although we're looking at that _11:18_: from outside, we're looking at that sphere from outside We're _11:20_: looking down on the sphere, if you like, _11:23_: from our position above it, _11:26_: we we could do this, the same calculations on the surface of _11:30_: the Earth and discover that the Earth is round so we can do _11:34_: things like add up the one, can you do that? I'm sure surveyors _11:38_: have to do this sort of calculation, at least at some _11:41_: approximation. We can draw lines on the surface of the Earth, _11:46_: carefully measure the angles between the three angles of that _11:50_: triangle, and discover that you add up to 180 degrees. They very _11:54_: nearly do, _11:55_: but not quite. _11:57_: In other words, all this geometry that we're doing on the _11:60_: surface of the earth of surface the sphere, _12:03_: is intrinsic to the surface. It's not just an artefact of us _12:07_: looking down on it from at the motor side. It's not just an _12:10_: artefact of the coordinates we're choosing, _12:14_: it is intrinsic to the sphere and we can learn about it _12:19_: without stepping outside. _12:21_: Is an important point. So some ants could live on that sphere _12:25_: and work out geometry and discover they were on a sphere _12:28_: of radius R. _12:30_: Just as we can make suitably accurate measurements on the _12:33_: surface of the Earth and discover were not on a sphere, _12:38_: we are on fear. I'm not on a plane _12:41_: and this can also be fairly obvious in some circumstances _12:44_: mean that this is probably going to be fairly _12:48_: fine calculation. But if I do something like I'm telling you _12:52_: have made a decision to the board a lot today _12:55_: I had planned I'd expected not to have to write anything but I _12:59_: shouldn't be much angle. And you can imagine our _13:05_: but sort of the Earth, sea _13:12_: and the equator. _13:14_: And if we took a start at the pole, _13:17_: Andrew, _13:19_: a great circle down to the equator _13:23_: and another great circle down to the equator, _13:27_: what we would see _13:30_: if you get 2 right angles there _13:33_: and an arbitrary angle at the top. So obviously _13:38_: the the internal angles of a triangle on the surface of a _13:41_: sphere don't add up to 180 degrees. _13:44_: Is very clear _13:46_: and obviously _13:48_: the circumference _13:51_: on a on a on the on the sphere. Here a circle on the plane, _13:57_: a circle in the set of points which are a constant distance _14:00_: away from ascent from the centre. That's how you define a _14:03_: circle. _14:05_: OK, on the surface of a sphere, _14:09_: the circle is the set of points which are a constant distance _14:14_: away from, for example, the pole. So the equator is a circle _14:18_: on the surface of a sphere because all the points on the _14:21_: equator are _14:23_: 10,000 kilometres away from the pool, _14:26_: obviously. _14:28_: But that means that the _14:32_: and and the circumference of the Earth is 40,000 kilometres. _14:37_: So the _14:40_: there's a conference divided by the diameter _14:43_: is 2, _14:45_: not π. It's two. _14:48_: So even things like the ratio of the circumference of a circle to _14:53_: its diameter is quotes wrong on the surface of a sphere. So you, _14:57_: you, you, you've been looking at it. You're familiar with the _15:01_: game, _15:03_: you are familiar with the non Euclidean geometry. You just _15:05_: haven't been think of it in those terms. _15:10_: So, so want you to you know hold on to that thought. _15:17_: And now here, as is probably fairly obvious and I could be _15:20_: illuminated by by looking at the, the detailed expression for _15:24_: the internal angles and so on. But we're not going to be _15:29_: it's clear that as that radius gets bigger _15:33_: this will that this triangle will get closer. If these _15:38_: lengths, _15:40_: you know an bits of string lengths on the surfaces you _15:43_: don't don't don't get bigger as well as this this radius gets _15:47_: bigger that will become closer and closer to our plane. I mean, _15:51_: as we look around here that the Earth looks flat and locally _15:55_: it's flattish, _15:57_: so the the amount by which _16:01_: the special trigonometry of the surface of this fear differs _16:05_: from plain geometry, _16:08_: It's governed by the size of that radius. _16:10_: So there's a parameter _16:12_: which tells us how much we've gone wrong from Euclidean _16:16_: geometry, _16:19_: and in this case the parameter is obviously the radius. _16:25_: What turns out to be a useful _16:28_: and re rephrasing of that if you like is a thing called the _16:31_: curvature which in this case is just one over the radius. _16:36_: So as the the radius gets bigger the coverage one of the readers _16:40_: gets smaller and the cover should moves to zero. You're _16:43_: arbitrarily large radius the the thing becomes you know non _16:46_: curved and and more like Euclidean _16:49_: geometry. So _16:52_: to summarise that you are familiar with the a non _16:55_: euclidean geometry, _16:57_: it's fairly obvious that there is a a parameter, in this case _17:01_: the radius, which characterises how much that deviates from _17:05_: nuclear geometry. And as the curvature, in this case one over _17:09_: the radius, gets smaller, you end up going back to flat space _17:15_: just implanting those words in your head. _17:20_: OK. _17:23_: And that curvature is, as I mentioned, a property of the _17:26_: space. You you don't know how to be looking at it from outside. _17:30_: You don't have to have a radius for that curvature to be _17:33_: present. _17:35_: So that is a sphere as a surface with constant curvature because _17:40_: the fairly obviously the the radius is the same all over the _17:45_: sphere _17:47_: if you smell like an ellipsoid, so like a like a a squashed _17:54_: and _17:55_: squashed sphere. And the curvature is is fairly obviously _17:59_: different at different points in this. In like a rugby ball, the _18:03_: curvature is less in the middle than it is at the end, so the _18:06_: curvature can vary over _18:08_: over the over the space. So different points in the 2D space _18:11_: at the surface of that of that rugby ball have different _18:14_: curvatures _18:16_: and you can track the as you move around you can you can you _18:19_: can see you can measure the curvature being different, _18:22_: different different points, and discover the way that that that _18:25_: that the the geometry is different at those different _18:28_: points, _18:29_: those of positive curvature. You'd also get spaces which have _18:32_: a negative curvature, _18:35_: and that's not so easily interpretable in terms of _18:39_: radius. But _18:41_: believe the algebra followed through and and you end up with _18:45_: a curvature here, which is conveniently characterised with _18:48_: a negative number. And that also makes sense though _18:52_: because here if you start, if you look at that point in the _18:55_: middle, _18:56_: call that the pool see _18:58_: and look at one of the _19:01_: and and head out to one of the pick a a a radiate head, head _19:06_: head out along it a bit that's that's the radius of of of a of _19:11_: a circle. If you trace that _19:14_: circle around _19:17_: I particular radius, see you pick that one, see and and just _19:20_: follow it round you'll see It's fairly obvious that the length _19:24_: of that will be more _19:26_: than π times the diameter. _19:29_: When the sphere are a a circle was less than PI10 diameter, _19:33_: here a radius. A circle has circumference which is more than _19:37_: π times the diameter. So the geometry on this is also _19:41_: different. _19:42_: It also intrinsic to the to to to the space. It's not an _19:45_: illusion caused by looking at from an infamous site, _19:49_: and it's characterised by a coverage parameter, which in _19:52_: this case is negative, _19:54_: and the the culture of that speed is constant, just like the _19:57_: sphere, except sort of the other way around. _20:02_: How that feeling so far? _20:04_: Any questions? _20:07_: Thank you. _20:11_: So that's _20:13_: spatial, spatial, spatial spaces. _20:17_: But the space that we are interested in _20:20_: is not one of these. Is not a space. So these are, if you _20:23_: like, variance of this sort of space, A space looking like _20:27_: that. The space we're interested in is not that, but instead a _20:31_: space _20:38_: of. Minkowski speaks _20:40_: where the interval, the definition of length, the _20:43_: definition of how far apart two events are. So there's an event, _20:47_: there's an event. They're separated by amount of space and _20:50_: amount of time. How far apart are they? We've learned that _20:54_: it's this quantity _20:56_: that characterises this separation between 22 events, _20:59_: and that's the thing that is our flat space, if you like, in this _21:03_: context. So just as we can start playing games with the different _21:08_: different coefficients in front of of this metric and get things _21:12_: like sphere, spherical trigonometry, _21:16_: we can discover that there are variants of this _21:20_: which are also curved spaces. _21:22_: But there's spaces where one of the dimensions is in time. And _21:26_: in purely mathematical terms the the the key thing here is that - _21:30_: because all of the spaces I was talking about up to this point _21:34_: had all plus signs in there. So that in mathematical terms _21:37_: that's the key difference between that and the the the the _21:41_: sort of Minkowski space times that we've that we've been _21:44_: talking about. _21:52_: Thank you. _21:53_: Moving on. _21:55_: So creature is, is the thing that you've remembered from _21:58_: that. _22:01_: The next thing we have to talk about is _22:03_: that I cheerfully talked up to this point about going going _22:08_: from the centre of, for example, that diagram and heading out to _22:13_: a particular radius. _22:15_: Here I talked to start at the North Pole and heading out in a _22:20_: straight line to the equator. _22:22_: And you know what a straight line is on the surface of a _22:25_: sphere. It's a great circle. You you remember that from last year _22:28_: that that's what a straight line means in a curve speed in a in a _22:31_: curved space. _22:33_: Why a great circle? What's special about great circle that _22:37_: makes it this the the obvious straight line? In that context, _22:41_: the question turns into well, what is a straight line anyway? _22:46_: No, _22:47_: it seems fairly obvious what a straight line is. _22:49_: I go up here to there. I don't wiggle. _22:53_: I I don't do that, _22:56_: OK? I just go from here _22:59_: by the shortest path. _23:01_: So in Euclidean space and indeed on a surface like that, a _23:05_: straight line is the shortest path. _23:08_: So they get if if if. I tried a a bit of string to the table _23:12_: like there and pulled it tight. _23:14_: This string would trace out a straight line _23:19_: between there and here and _23:22_: that's. It's a straight line by definition. That's what we mean _23:26_: by a straight line in _23:29_: in geometrical terms. And similarly, if I tied a bit of _23:32_: string to the North Pole, went down to the equator and pulled, _23:37_: what I would get would be a great circle. _23:40_: OK, _23:43_: but that's not the only way you can define a straight line. _23:46_: Because the other way that I can talk about a straight line _23:50_: but _23:53_: search for beer _23:54_: is to say, OK, I'm facing in that direction. We're closing my _23:57_: eyes and just walk in a straight line _23:60_: and where do I end up? _24:02_: So I'm not going to go, I'm not going to turn any corners. I'm _24:05_: just going to start at one place and move, move one, put 1 foot _24:09_: ahead of the other and keep moving. _24:13_: Think as far as I'm concerned right here right now is the _24:16_: straightforward direction, _24:18_: and in this case I trace out the same same path _24:23_: and in that case I treat it the same path. If I start at the _24:27_: equator, _24:29_: face north, and just keep walking, I end up at the North _24:33_: Pole. _24:35_: I don't end up anywhere else. There's nowhere else I can go _24:38_: because because what I I do now I I trace that out is I follow a _24:41_: great circle. _24:44_: So in these two cases, _24:46_: the straight line is both the straight ahead that that that's _24:49_: straight ahead. _24:51_: Street line is the. _24:55_: IT matches the shortest distance between two points. Straight _24:58_: line. _24:60_: But also if I start at the equator _25:04_: and headed South _25:07_: and just kept going, I'll go. I'll go past the South Pole, go _25:10_: up and come to the North Pole. Again, _25:15_: I go that that means I go the, the, the, the. The route I've _25:18_: taken from this point to the North Pole is, at that point the _25:22_: longest route I could possibly take in a straight between those _25:25_: two points of looping around myself or anything. _25:29_: So that means that the straight line in that context is the _25:33_: extremal _25:35_: path between 2 two points. _25:39_: And these two things as I say match up. _25:42_: But the important distinction between them is if I talk about _25:45_: the type of string to the table leg and pulling it tight, _25:49_: I have no sense pre, pre chosen where I'm going to go _25:52_: that that that's intrinsically a a fairly global definition of _25:56_: straight line. Whereas the definition where I say OK, I'm _25:59_: just going to look at my feet and walk straight forward and _26:02_: making sure that my feet move in in what is, as far as I'm _26:05_: concerned right now, a straight line, that's an entirely local _26:10_: definition of straight line _26:12_: and and and and that locality, that localness is key, could be _26:17_: we we want our physics to be local. We don't really. One of _26:21_: the things that Newton didn't like about the law of universal _26:26_: gravitation was that it depended on an element of non locality. _26:30_: It depended on the the sun _26:34_: setting up this structure of gravitational field all through _26:37_: the universe and everything in it. Then, you know, paying _26:40_: attention to that, but but paying attention to a thing that _26:43_: was a very large distance away _26:45_: and we don't really like that. I mean, and Newton didn't like _26:49_: that because Newton wanted a contact theory. In other words, _26:52_: that that the only things that would affect something were the _26:55_: things that were touching it or next to it. _26:58_: And so Newton was disappointed _27:00_: that the theory that that worked was the one that worked _27:05_: and we are still just and we are disappointed about it as well. _27:08_: So we want things that are local only. So the fact that we can _27:10_: define straight line in terms of of, I don't care where I'm _27:13_: going, I'm just looking at paying attention to my feet and _27:15_: walking forward, that's a good thing, _27:18_: the locality _27:20_: important word there. _27:26_: And so _27:31_: there's a quick question which is useful to meditate on in a _27:35_: bit because there's quite a long, a lot of chat we could _27:39_: talk about. And that's just an illustration of the idea that to _27:43_: go from one place to another on, for example, the surface sphere, _27:47_: you can do so by just following a straight line and what you end _27:52_: up tracing out is _27:54_: well as geodesic. _27:56_: It's what it's called the sorry that's the keyword here. The _28:01_: process of doing so, the process of of of just walking forwards _28:06_: media trace out a geodesic and the the the thing that that that _28:10_: usually is the thing that you you, you, you trace out. When _28:14_: you do that process, _28:16_: it had also _28:19_: a line of extremal length. In _28:23_: geometry is with this signature it's it's minimal length. In _28:28_: terms of this signature, it's maximal length because you if _28:33_: you think back to chapter 5 and the mention of the _28:38_: the twins paradox and the McCarthy diagram where one of _28:42_: the two of the two twins heads off to the some distance and _28:45_: comes back, I mentioned in passing. I didn't make a big _28:49_: deal of it at the time. I mentioned in passing that the _28:53_: the reason that one way of thinking about the reason why _28:57_: Odyssey is staying at home _28:59_: go and going to the the the the the rendezvous point. By what _29:03_: looks like the straight line along the the the T axis, _29:08_: Odysseus is taking the longest route between those two points. _29:11_: Whereas Penelope, who's taking their dog leg root, _29:15_: is taking a shorter route. Although it looks longer on the _29:19_: diagram _29:20_: Minkowski space, it's the shorter route. So every every _29:24_: deviation you take from this, the apparently straight line _29:31_: and _29:33_: root _29:34_: decreases the length and that's that's that's rather strange, _29:39_: but that's why we talk about duties being a lane of extremal _29:42_: length rather than minimal lens. _29:46_: So at this point we now have all of the _29:51_: the, the, the, _29:53_: the key ideas to go and talk about general activity and we're _29:56_: very close to having to getting a bit an insight about about GR. _30:01_: First of all, last week we talked about inertial frames _30:06_: and I I said that the the definition of inertial frames in _30:10_: general activity is basically the same as in special _30:13_: relativity but with an important generalisation. _30:17_: Which is an inertial frame is what you are in when you are in _30:21_: free fall _30:23_: and freefall means way out in space, or it means and this was _30:27_: a physical insight that Einstein had. Or freefall is moving _30:32_: entirely under the influence of gravity. So if you jump up and _30:36_: down _30:37_: then you are in freefall False. You're doing that and you are _30:40_: and and and the quarters tax. You are the coordinate of an _30:44_: inertial frame _30:45_: because in that frame _30:47_: Newton's laws work. While you are falling, _30:50_: you can let something go and it will fall with you. That was _30:54_: what Galileo discovered, dropping things from from the _30:57_: tower of Peters. Allegedly, that everything falls accelerates the _31:01_: same rate. _31:03_: If I get something, a nudge while Foster was falling, you're _31:07_: in this falling lift shaft. You get something, a nudge, and it _31:10_: would move at a constant speed in your frame until it hit _31:14_: something. So nuisance laws work in our national frame, and _31:18_: that's an important link. From the physics before to the _31:21_: physics of relativity _31:23_: to national fame are important. _31:26_: I talked about the equivalence principle _31:30_: and I gave 2 versions of it. _31:32_: One would depend on the notion of. If I think I have those _31:38_: three version, I have two versions, _31:42_: you know, come back to that one. One which was talking about _31:45_: uniform motion and uniform gravitational field. _31:49_: One which generalised that to say that all free falling _31:53_: inertial frames were equivalent, _31:56_: completely equivalent for the performance of all physical _31:59_: experiments. And from that basis we directly deduced that light _32:02_: must curve the gravitational field. _32:05_: Who national frames equivalence principle. _32:09_: I've talked of curvature _32:11_: in this lecture and the the notion that you can characterise _32:16_: the geometry _32:17_: of a space _32:20_: by making measurements internal to it and using things like the _32:23_: metric to characterise that geometry and to build your _32:27_: calculations on top of. And I, and I haven't gone into the _32:30_: details of that _32:32_: other than, you know, very sketchily the last time. And _32:37_: I've talked about geodesics as the way you in a sense discover _32:42_: what the space that you are in is like. So as I walk forward in _32:47_: a flat space, I discover I explore the space along that _32:51_: line. As I take a walk in a straight line in this local _32:55_: sense on the surface of a sphere, I discovered I I go to _32:60_: to both poles and and and and. _33:02_: You want the notion of a straight line. A geodesic _33:07_: in space is very important, _33:12_: so _33:14_: we can now talk about gravity. _33:17_: So we've been talking about geometry at Raptor now, but now _33:19_: we're about to talk about gravity. _33:23_: And the question that gravity is about is about answering is if _33:28_: you are _33:29_: near a gravitating object _33:32_: such as the sun, _33:34_: how do you move? What? What what, what, what constrains your _33:37_: emotion? How can you predict where you're gonna move that? _33:41_: That's the question that a a theory of gravity is trying to _33:44_: answer. They're trying to answer for, for example, the Earth and _33:48_: the sun or the moon could run the Earth or or or a ball being _33:51_: thrown from one place to another. But in all cases, the _33:54_: question is where does this ball move? And the answer is, well, _33:58_: let's let's use gravity to work it out _34:03_: what we can. That then I'm going to jump forward a little bit, _34:11_: is a third version of the equivalence principle, _34:16_: which is what we do before we do that. _34:21_: We knew _34:23_: how things move. We know we're Newton told us how things move. _34:27_: And I've, I've referred to that multiple times. So it said, if _34:31_: you use the three laws, if you just leave something it don't _34:35_: touch, it doesn't move. _34:37_: No that's not true under gravity. But if we're inertial _34:40_: frame that is true, you just leave something and it's falling _34:43_: with you just if it is. And if you if something is moving the _34:46_: straight line, it carried on moving in that straight line _34:48_: until it hits something. _34:50_: OK, _34:52_: now the context of special activity. _34:54_: The space we're talking about _34:56_: is not just the XY&Z over lift cabin, but it's the space _35:00_: of because it's Mankowski space. _35:04_: So _35:06_: if we take a bit of a jump and say, well, what does that _35:12_: Newton's law of principle tell us about how things move _35:14_: Minkowski space, It says, well, perhaps they move in a straight _35:17_: line _35:19_: and in a straight line. And in Minkowski space the simplest _35:22_: motion _35:23_: is not to move at all, _35:25_: just standing where you are in your frame and you move in a _35:29_: straight line along the same axis. _35:34_: And _35:36_: what if that is true more universally? _35:40_: And the third version of the College Principle says exactly _35:44_: that. It says that the that _35:47_: any physical law _35:49_: that can be expressed geometrically in special _35:52_: activity and exactly the same form in a locally inertial frame _35:55_: of a curved space-time. _35:57_: There's another of those versions of the course principle _35:60_: for all the words mean some are important. _36:03_: So a physical law _36:05_: we're talking about a a statement of how the universe _36:09_: works. We're not a mathematical things, _36:12_: we're expressing the physical and mathematical form. But we _36:15_: are seeing something which could be, which could be _36:18_: mathematically otherwise. But in our universe, it's true. _36:21_: That's what we mean by physical law _36:24_: that can be expressed geometrically in Sr _36:28_: And what I said there about when you're standing still, you're _36:31_: moving in a straight line through Makovsky space. That's _36:34_: in a sense of physical, a geometrical statement of of how _36:38_: things move. _36:39_: If you if you're form momentum is pointing in this direction, _36:44_: for example along the team member that prime axis that's _36:48_: going to carry on doing so. _36:51_: And that's a physical law that's that's in its in the version of _36:54_: distance laws in expressing Minkowski space. And This is why _36:58_: we're making such a fuss about geometry when talking about _37:02_: special activity in the 1st 10 lectures of this course. _37:07_: And the equivalent of said that has the same form _37:11_: when expressed in a locally inertial frame, of course _37:13_: space-time. _37:14_: In other words, if you are in our _37:17_: if we could have a reference frame _37:20_: and it's an inertial reference frame. That is, you're moving _37:22_: only under the influence of gravity, so we are in space or _37:25_: you're free fall. _37:27_: Then _37:28_: the you can't tell _37:31_: that you are. You can't tell which of those possibilities it _37:36_: is, _37:37_: even when _37:39_: the space-time you're falling through _37:41_: is curved, _37:44_: even when the space-time fallen through is curved. And that's _37:48_: why we're talking about _37:51_: this. This local definition of _37:54_: what geodesic is is important _37:57_: because it's seeing that you don't have to worry about how _37:60_: the space around you is curved. _38:02_: It could be round the twist completely, _38:05_: but the physical law that we're talking about here is 1, which _38:09_: says you your velocity vector stays the same _38:14_: in _38:15_: the local and national frame. The nation that's local to you, _38:18_: that's around you, _38:20_: That's small enough that you can approximate it as being special _38:23_: activity. _38:26_: That's why the locality is important. _38:28_: What this says doesn't happen, _38:31_: or what this says does not happen is that _38:36_: the the the the the the statement that you move along _38:39_: the time axis of your local inertial frame. That doesn't get _38:43_: any complications from the amount of curvature _38:46_: in the space you're you're in. So you can imagine _38:49_: that Newton's laws had an extra term in them which was to do _38:53_: with how covered the space was, which just happened to be 0 in _38:58_: the the, the, the, the, the physics we're familiar with _39:02_: that could that. There's nothing mathematically wrong with that. _39:05_: That's a perfectly mathematically _39:08_: good guess to make. _39:10_: This says that doesn't have. _39:13_: There's no extra terms because the space you're moving through _39:16_: is is weirdly curved. _39:18_: OK. _39:21_: So that that is the physical content of that. But that seems _39:24_: to phase _39:26_: um, _39:29_: I'm I'm, I'm minimal thing to be confused about. _39:32_: This thing that you might have thought might have happened _39:34_: doesn't. _39:35_: But it's actually terrifically important _39:38_: because it's seeing. What that means is that if you understand _39:41_: how things move in special relativity, and we do, then you _39:44_: understand how things move in general activity as well. _39:49_: That's the bridge from special activity _39:52_: to general activity. This statement, the fact there's _39:54_: nothing it not any more complicated as long as you're _39:57_: concerned only with your local inertial frame, your your your _40:01_: local frame, and a locally inertial frame. _40:04_: But what's most important? _40:10_: And _40:12_: the fact that _40:15_: this _40:18_: diagram here tells us how things move, they move in a geodesic _40:24_: along the local time axis _40:27_: tells us how things move in a curved space. Time things move _40:31_: along a geodesic of the space. _40:34_: They could use principles. Let's let's make that jump. _40:37_: In other words, at this point you have understood half of _40:40_: general relativity, _40:44_: which is that _40:48_: space-time tells matter how to move. _40:52_: So space-time is curved. Why? It's curved will come to in a _40:55_: moment, _40:56_: but it's curved _40:59_: and we move along geodesics in that space. When we're in _41:02_: freefall, we move along geodesics in that space, _41:06_: and so the space curves we will. _41:10_: You go with it _41:11_: and that's our path through space-time, through time and _41:14_: space _41:15_: and that you well that was have seen videos where people put _41:19_: weights on on rubber sheets and so on. _41:22_: What happening there is is that the, the, the, the, the the _41:25_: weight undersheet makes the initially flat rubber sheet into _41:29_: a curved space _41:30_: and then with the with some rules are marvel across that the _41:34_: marble is instead of going in a straight line it's you know it's _41:37_: not being being acted on by any breaths of breaths of air. I _41:40_: mean, it's going as far as concerned, the straight line, _41:43_: but because it's going in a straight line through a curved _41:46_: space, it does things like orbiting _41:49_: and that is the equivalence principle _41:52_: that once you've worked out how the space is curved, you've