Transcript of a2-l09 ========== _0:11_: This is lecture 9. _0:13_: Remember that there are 10 in special activity for over 5 _0:17_: where I talk about the beginnings of general activity. _0:21_: So the aim is to get this the end of this chapter and Chapter _0:25_: 7 on dynamics done in the next two lectures and we can move _0:29_: straight on. At that point I might drift slightly into _0:33_: slightly intellectual 11, but that's that's not a big problem. _0:37_: If it happens, _0:40_: I don't think there are announcements other than to _0:43_: mention and to remind you that the paddle there exists and is _0:46_: useful. What previous years, I found it very useful by this _0:49_: stage. In previous years, the party has been full of _0:52_: questions, _0:54_: ask questions. I mean, obviously I'm not _0:58_: terror systematic about looking at that every day, but I do aim _1:01_: to look at it fairly regularly and questions that you put there _1:04_: can be answered and other folk can see those excellent _1:07_: questions. What you thought about during the process of _1:10_: mulling things over in your head. After the lectures _1:14_: I shall move on. Are there any questions? _1:19_: We're about where we are _1:21_: and _1:25_: where we are now, _1:28_: emotionally or otherwise. _1:37_: Any questions? _1:39_: OK. Well, you're being you'll, you'll be excellent asking _1:42_: questions in the lecture. So that's good. _1:47_: Where we got to last time _1:50_: was I had introduced the idea of four vectors _1:55_: as the fairly direct analogue of three vectors in ordinary clean _2:01_: space. There's an XYZ and ATT, and the point I was hammering _2:05_: home _2:07_: was that these lectures, these vectors in 4 dimensions have _2:11_: basically the same properties as the vector you're familiar with, _2:16_: except that the definition of length _2:20_: in for these vectors _2:22_: had those extra minus signs. _2:24_: So the the definition, I don't think I have a slide which has _2:28_: that on it. _2:30_: Well yes, I I talked to the the inner product or the scale of _2:33_: product or the dot product for everyone to call it. You know _2:36_: magician would distinguish those various things. We're not going _2:40_: to bother. The dot product of 2 four vectors _2:44_: has this these remaining signs _2:47_: which and and. And the matrix of the ETA which has those lines in _2:52_: it. The diagonal matrix plus minus, minus, minus _2:57_: is. Wild maintenance come from. That's called the metric. It's a _3:01_: constant for special activity it becomes non constant and of _3:05_: physical interest for general activity. I noted that the _3:09_: length of the displacement vector that's just dot X, dot Y _3:13_: delta Z is _3:14_: recovers this invariant interval that I talked about several _3:18_: times in previous lectures. So that's that's interesting and _3:21_: and and and that's that shows where doing something familiar. _3:26_: I talked about the way we did a quick quick question in which I _3:29_: looked at the lengths and inner product of 2 vectors which we _3:33_: discover that those two vectors A&B were orthogonal to each _3:37_: other in the specific sense that their inner product product, _3:40_: scalar product we want to call it _3:43_: is 0. _3:45_: So that so that they are as orthogonal in Minkowski space as _3:49_: two right angles to the right angles would be in three space _3:53_: and they both have different magnitudes. _4:00_: The _4:01_: if A has a squared length, squared magnitude of which is _4:05_: negative, so it's a space lake _4:08_: vector. _4:10_: It points roughly along the X axis. _4:14_: Vector B had a positive squared magnitude. It's time like it _4:17_: points roughly in a sort of time like along the same axis _4:21_: direction. _4:23_: And I went on to talk about _4:26_: the _4:27_: derivatives of the displacement vector _4:31_: and how we can define the _4:35_: relativistic velocity and just the component by component _4:40_: derivative of the position _4:43_: DX0 editor DX1 by detour DX and so on. We're dividing by the _4:48_: Tor, not DT, because DT Tor the is the Lorentz scalar that _4:52_: that's the framing variant thing. _4:56_: The the the time between two events is A-frame dependent _4:59_: thing, so that we can't divide by that is not that that changes _5:02_: from frame to frame, but tall. The proper time between two _5:05_: events is frame invariant, so we can divide by that. It's a _5:08_: scalar under the red transformation. We can do that _5:11_: twice and get an acceleration. _5:14_: So it's nothing complicated, just it is just the component _5:17_: component derivative. _5:20_: And I walked through the the brief calculation that shows _5:24_: that this is another another manifestation of time violation. _5:29_: The the the rate of change of of of the time coordinate with _5:34_: proper time _5:38_: is _5:39_: non. It is not one that they don't vary in the same in the _5:43_: same way. _5:44_: I committed at that a little a little more. I also described _5:48_: mentioned the little notation for velocity rather than _5:56_: the sanitation. Where the the 0 component of of the velocity U0 _6:02_: is gamma, _6:04_: the ex important is gamma VX, gamma vy, gamma VZ and so on and _6:09_: write that in that particular way and I showed you. I hope I _6:14_: convinced you. I hope that the dot product of you with itself _6:20_: the the length of the vector or the velocity vector was _6:24_: over 1. _6:25_: And because that's true in one frame and the nice easy frame to _6:29_: calculate, which is the case for is not moving, so the velocity _6:33_: vector points along the T axis, we can easily calculate what U _6:36_: is in that frame. And since the frame independent because _6:40_: there's reports all frame independent, that's true always. _6:43_: So the velocity vector is strange _6:46_: in that its length is always one _6:49_: that unexpectedly _6:52_: so. The _6:54_: the point being that as you move, as something moves at _6:57_: rustic speed, its velocity vector doesn't get longer, _7:01_: doesn't change length, you just change direction in the _7:04_: Minkowski space, _7:09_: right? So that's where we got to last time. Where we're going to _7:13_: go on is _7:15_: and _7:19_: coming up appropriately. _7:21_: OK, _7:26_: I mentioned that you know it was equal to DX naughty by D Tau or _7:33_: more generally that you MU is DX UB yd _7:40_: reminding you that the this index MU the Greek ones go over _7:46_: 0123 Latin one goal 123. _7:51_: So we have the velocity, _7:54_: how do we get the the next thing we do? Look at the acceleration, _7:59_: the acceleration vector. That's straight forward. So a naughty _8:03_: is going to be _8:05_: the _8:06_: you not by detour _8:09_: before, _8:11_: which is _8:14_: the tea by detour _8:17_: at the you not by _8:21_: DT. _8:30_: I'm gonna write this. _8:40_: Which _8:42_: yes and remembering that you you not _8:47_: but you is equal to gamma. One VU naughty is equal to gamma. _8:54_: So this is going to be gamma _8:57_: the gamma by DT _9:01_: gamma gamma dot. _9:05_: I'm not doing anything complicated that that looks very _9:07_: fiddly, but I'm not doing anything more complicated than _9:10_: just _9:12_: between rule for differentiation. _9:15_: Already you can see this is looking a little bit a little _9:19_: little messier than the than the. I've lost you _9:24_: vector _9:26_: the special components AI. That's DU _9:32_: UI by D Tau _9:35_: equals DT by D Tau _9:40_: DUI by _9:42_: ET, _9:44_: which is Gamma _9:49_: D by DT _9:52_: Gamma _9:53_: VI. _9:55_: So the the the the the 8th component of you is just Gamma _9:59_: VI _10:01_: from the thing on the right. _10:03_: Differentiate that _10:05_: and we get _10:06_: gamma. _10:08_: Gamma dot _10:10_: VI plus gamma _10:12_: would be _10:14_: I dot _10:16_: which is Gamma _10:18_: Gamma dot _10:20_: the I plus gamma _10:22_: EI. Defining the _10:29_: EI as the just straightforward derivative with respect to the _10:33_: claim coordinate of T and you can see that looks a bit _10:36_: messier. _10:38_: The the relationship between position, velocity and speed is _10:41_: nice and simple. In Newtonian mechanics, it ends up being a _10:44_: bit messier _10:46_: in this case, but it's just because there's a gamma in there _10:50_: making things up. _10:55_: So OK, that's that's messy. We don't want we don't want to do _10:59_: calculations with that as much to the extent we can avoid it. _11:04_: But _11:06_: a point worth making is that _11:09_: there is no _11:11_: there are no frame in which a particle is not. If a particle _11:14_: is accelerating _11:16_: and there's no frame in which it's not accelerate, _11:20_: the particles moving at a constant speed and there's a _11:22_: frame in which it is stationary, _11:23_: there's no freedom which is not accelerating. _11:26_: It's exaggerating one after the other. _11:30_: Admit that it's complicated. _11:33_: If a particle is moving to constant speed, then there is a _11:36_: frame in which that particle is not moving _11:39_: the the frame that's cool moving with the particle. _11:42_: A particle is accelerating, then there is no frame in which it is _11:45_: not accelerating. _11:46_: It's just accelerating from a different. _11:49_: But _11:50_: at any point there is a frame in which the particle, though _11:54_: accelerating, is briefly at rest. _11:56_: And if you if for example, if you throw something up, _11:60_: then all the way through its motion it's accelerating, but at _12:03_: the very top of its motion it's it's vertical speed is 0, so _12:07_: it's more entirely at rest, _12:10_: but it was accelerating. _12:12_: OK, _12:13_: that's called the momentarily cool moving reference frame or _12:16_: instantaneously cool moving reference. _12:19_: I'll come back to that. Yeah. It's it's it's what do you, you _12:23_: know, lodge that thought in your head. And now in that frame. _12:28_: The _12:29_: velocity acceleration vectors are nice and simple _12:32_: in that frame. _12:39_: The moment you can move reference frame, _12:42_: the _12:45_: velocity is _12:48_: nice and simple. It's directed entirely along the T axis. It's _12:53_: the special component of. Its velocity is 0. It's. It's _12:57_: temporarily at rest _12:59_: in that frame. Also _13:02_: the _13:03_: you look at look at this. The velocity is 0 and gamma is equal _13:09_: to 1 _13:12_: in in that frame. _13:14_: So that means the acceleration is 0. Velocity is 0 _13:20_: and the gamma is _13:25_: is 1 _13:28_: so that the _13:30_: acceleration in that frame is just is _13:37_: camera 1 and gamma dot is _13:41_: 2nd _13:43_: gamma dot. It turns out continuous factor of of of TV. _13:48_: So dot in that reference stream is 0 _13:51_: and V0 gamma dots 0, so gamma is 1. _13:57_: So in that moment actually that's more convenient in that _13:59_: more entire local movement. The point. The point is, in that _14:02_: moment helical movement reference frame, the _14:05_: international symbol, _14:07_: the velocity has zero spatial component and gamma is 1. The _14:11_: acceleration _14:14_: of the gamma, gamma dot which tends to be 0 in that frame and _14:18_: justice acceleration. But look what we found in that frame. _14:24_: You A _14:27_: is equal to 0, _14:28_: it's 1 * 0 minus _14:31_: zero times dot east. _14:34_: So in that frame _14:36_: you and A are orthogonal. _14:39_: But _14:40_: that sort of recent variant is just a number, _14:44_: so it's not frame dependent. _14:46_: So it's true in every frame. _14:48_: So in any frame, _14:51_: the velocity and its aeration are _14:56_: orthogonal. _14:59_: And that's that's that's a technique that _15:02_: this is another variant of choose the right coordinates, _15:06_: which you you you, you. Well, we've drummed into you again and _15:09_: again over the course of your physical mathematical career. _15:12_: Pick the right coordinates and and problems are simple _15:17_: or problems can be simple and if you have an invariance then the _15:21_: answer you get in that simple frame is the answer you get and _15:25_: what more generally. And that sort of makes sense, because if _15:29_: you _15:30_: remember that the velocity vector is always _15:35_: the same length, you don't use always one _15:37_: and change the velocity correspond to changing the _15:40_: direction of that vector in Minkowski space, _15:43_: then something is accelerating. What's happening is not it's _15:46_: being. _15:47_: This vector is being extended, it's just being _15:50_: moved, _15:51_: and that sort of makes sense that the acceleration acting on _15:55_: the velocity would rotate it if you like, rather than extend it. _16:01_: That's not a detailed mathematical _16:04_: a code, but it it intended to to make this feel a little less _16:07_: strange, _16:11_: right? Also in that frame, _16:16_: we can look at _16:18_: what _16:20_: the length of the of the acceleration vector is. _16:24_: It's just the timing component squared minus the space _16:27_: component squared 0 ^2 -, a ^2. _16:34_: So the acceleration vector _16:37_: is space lake. It's negative. A space like vector _16:42_: the length of which is referred is called the proper _16:45_: acceleration, _16:47_: and that's true in this realm, and therefore it's true in every _16:51_: stream. Even though in in another frame, _16:54_: the components of U would change in that other frame, the _16:57_: components of a would change in that other frame, but in such a _16:60_: way that the _17:02_: the dot product of the two would have been the same. The _17:04_: components of the acceleration 4 vector would be different in a _17:07_: different frame, _17:08_: but the components would change systematically in a way that the _17:12_: dot product would remain at this value. That's what it means, _17:15_: that the dot product is frame invariant. _17:18_: The components of the vectors change the dot product, doesn't _17:22_: it? Our family variant quantity. The components are frame _17:25_: dependent. The dot product is framed, independent _17:29_: and her very important distinction to have in mind. _17:35_: Um, _17:38_: and and and. I'm not going to go through the details, but but _17:42_: if we have two velocity vectors then _17:46_: we can find a quantity like that and go through the details if _17:49_: you're interested. It's not terribly important, but if you _17:52_: look at the notes _17:54_: and the point here is that _17:57_: this _17:58_: these expressions for the acceleration for Vector are _18:02_: looking messy. _18:04_: Messi, isn't good _18:06_: looking messy. But if we instead fall back on the geometrical _18:10_: properties of these vectors, _18:13_: we can find these we clearly very important relations really _18:17_: quite promptly. We didn't have to do a lot of fiddly _18:20_: differentiation here. We didn't have to explicitly change frames _18:23_: here. We just had to be smart, do the calculation in the right _18:27_: frame, _18:28_: get the answer very straightforwardly _18:30_: and we've, we've discovered a couple of very important, _18:34_: clearly very important properties of these two vectors _18:37_: just from the geometry, _18:40_: OK. I think that's quite a good example of where the geometry, _18:44_: our general approach, has power. _18:50_: Umm, _18:54_: on the stage do we have? _18:56_: That's what I've just _18:57_: soon _18:59_: quick question. _19:02_: If I threw a ball vertically into the air at the top of its _19:05_: path, what is it? What is the instantaneously cool moving _19:07_: reference frame? _19:10_: Who is it with the family room? _19:14_: Who would see it with a free moving upwards with the same _19:16_: speed as I threw the ball? _19:19_: Who's? Who's? Who's able to frame moving downwards with the _19:22_: balls terminal velocity? _19:25_: Who had put the hands up yet? _19:28_: OK, then we'll start again. Guess. I mean, I don't care if _19:30_: you get wildly because in the moment you're gonna talk to each _19:33_: other and explain why? Why? Why? Why why you, right. Who said the _19:36_: frame of the room? _19:38_: Who said it was a frame of upwards with same speed _19:42_: free moving downward of the terminal velocity? _19:45_: I still have to eat all those hands. If if you have to guess, _19:48_: I don't mind. I just want to see all those hands up. I want to _19:51_: make some sort of commitment at some point. _19:54_: Who would see with the frame of The _19:56_: Who has stayed with the free movie upwards? Who say with the _19:59_: frame moving downwards? _20:00_: That's enough. You talk to your neighbours. Tell them why you're _20:03_: right. _20:05_: Look what it should be. _20:26_: Well, _20:44_: OK, having I thought. I remember. I say again, I I don't _20:47_: keep track of who answers what. I don't care. All I'm all I'm _20:51_: interested is aggregate numbers to see if I'm going about the _20:55_: right speed. _20:57_: Who's here with the name of the room? _20:60_: Who is it with this free move upwards with the speed of the _21:02_: ball? _21:03_: Who is it with the free moving downwards the terminal velocity. _21:06_: OK. It's the fame of the room and and and and and and and I _21:10_: think I'm going to high speed three of the room because at _21:14_: that at the point where the balls at the top of its of of _21:17_: its trajectory or the pens at the top of its trajectory there _21:23_: the frame in which it's not which is not moving is the frame _21:26_: of the room. So at the very end, for instantaneously at the top _21:30_: of its of its of its trajectory is not moving vertically. _21:35_: OK, _21:36_: it's that's the frame. So the frame which is not moving _21:39_: vertically is instantaneously called moving frame and that _21:42_: frame is the frame of the room. So does that make sense? _21:47_: OK. _21:48_: And _21:50_: I mean in every other instant the ball is moving with respect _21:53_: to the room. Although all the all the pain or whatever is _21:56_: moved respects the room, but only that that that instant is _21:58_: the is instantaneously _22:02_: key points _22:05_: right now we're going to go on to an exciting different thing. _22:14_: So that's all very nice and somewhat abstract. _22:19_: Ohh, I'll, I'll just mention that. _22:22_: Yep. _22:24_: And in in your experience of Newtonian mechanics, you're _22:27_: you're you're you're familiar with acceleration and force and _22:30_: so on, being things that you use all the time to solve problems _22:34_: and so on. They turn out not to be slightly unexpectedly they _22:37_: turn out not to be terribly useful in relativity context, _22:41_: partly because they're a bit messy like that. _22:45_: But there is a sort of constant acceleration equations for _22:49_: interactivity, which I quote from the notes just so you've _22:52_: seen them. But they turn out not to be as as useful because in _22:57_: relative to context we do the sort of puzzle we have to solve, _23:01_: like how fast is this electron moving in the accelerator? How _23:05_: fast is this accretion disc moving around _23:09_: a black hole? They tend not to be constant acceleration type _23:12_: problems, so that that that that tends not to be an important _23:15_: technique. But they do exist and they're in the notes, _23:19_: OK. And we're moving on _23:22_: applications of this. _23:24_: I've mentioned the _23:26_: velocity acceleration for vectors and I said we can derive _23:30_: them fairly straightforwardly just differentiating the _23:33_: displacement vector component by component. _23:36_: This is straight forward, but I I have just admitted they're not _23:39_: terrifically useful too for talking problem. _23:42_: But there is another important four vector which we can look at _23:47_: quite closely and discover _23:50_: useful things immediately. _23:54_: We're going to do is. _23:59_: So imagine _24:05_: ah _24:06_: ohh it's a wave fronts. _24:09_: But let's talk about these being realistic. The in water or _24:12_: whatever right now would be completely generic, but these _24:15_: are waves off _24:16_: the way France _24:19_: and there's a _24:25_: wavelength. Of course _24:27_: there's a _24:30_: a direction, _24:31_: a special direction, and that's _24:35_: out of LMN type vector. Go write that in. _24:44_: Nothing special with that of that. At that age, _24:48_: I would imagine 2 successive events _24:52_: on this wave front. _24:54_: And then _24:56_: a little while later when the we've moved on a bit _25:01_: on _25:02_: an event _25:04_: on another wave front _25:08_: and we're going to add what are the displacement _25:12_: between those two events. _25:16_: We could call that Delta R _25:20_: and that Delta R _25:23_: it's going to be Delta T, Delta X. _25:26_: But why _25:28_: Doctor Z in the expected fashion? _25:32_: OK, _25:39_: no. In the team, there's way front is moving at a speed You _25:49_: I've lost to you. _25:50_: So in a time delta T _25:58_: each wave front will have moved forward _26:01_: a distance. _26:03_: You just _26:05_: so the length _26:08_: between _26:10_: one and two _26:14_: is going to be. _26:18_: You _26:20_: Delta T plus _26:22_: in Lambda with _26:25_: any of the number of away fronts we're talking about. _26:28_: So we're building up this construction step by step. _26:39_: But that _26:41_: special separation _26:43_: between 1:00 and 2:00 is also _26:47_: just _26:48_: and dot dot R _26:53_: which is equal to L delta T _26:56_: plus m.m dot X + M dot Y + N _27:03_: doctor Z. _27:06_: Keep that with that. Just the, the, the, the, the length. Or _27:11_: two different ways of of getting the length of the spatial _27:15_: distance between these two events, and we rearrange that a _27:18_: bit. We get _27:20_: you just a T minus L X -, m, Delta Y minus N, Delta, Z _27:29_: equals minus _27:31_: and Lambda. And you from the pattern of saying you could _27:34_: probably see where this is going _27:41_: and we can divide through by Lambda _27:43_: and write F is equal to U for Lambda that F dot at t -, L over _27:50_: Lambda, delta X -, m over Lambda, delta Y minus N over _27:56_: Lambda Z. You're going to make N _27:60_: and we can write that _28:02_: as _28:04_: L _28:07_: dot R where delta R is this thing here _28:11_: before _28:13_: and L squeeze it in. The bottom here _28:16_: is _28:17_: F, _28:19_: L over Lambda, _28:21_: over Lambda, and over Lambda. _28:25_: So if I the point is all right, *****. _28:32_: If I define _28:34_: this thing L in this way, _28:38_: then and and recall the definition of delta R as above, _28:42_: then L to R is equal to this thing here. _28:47_: OK, that's _28:49_: suggestive. _28:51_: This looks a bit like a a bit like a vector. Is it a vector? _28:54_: Or is it just some some numbers in a row _29:01_: and _29:07_: I'll look at? Let's look at that, just to help to suggest _29:11_: that a bit. _29:18_: And _29:20_: images of water waves move along the X axis at speed 10 metres a _29:24_: second with wavelength 5 metres. _29:29_: What's its frequency? For Vector? _29:32_: We're going to, well, let. Let's all just stare at that for a _29:35_: moment and then I'll get you to talk to each other. And _29:41_: what I'll do is I'll put the other thing on one _29:48_: the other thing _29:53_: you're looking at this. _29:56_: So what's the frequency for vector of that in that _29:60_: situation? _30:06_: And _30:19_: OK, _30:21_: who would say it was the first one? _30:23_: Hoosier was the second one. _30:26_: Who's the third one? _30:28_: OK, _30:29_: I'm just going to. I don't trust my own arithmetics, I'm just _30:32_: going straight to answer. _30:35_: Move along the X axis so So N is equal to 1. _30:40_: The frequency is 2/2 Hertz, 10 metres second divided by 5 _30:45_: metres _30:46_: and so that means that the _30:50_: you you over Lambda _30:53_: the you know the this _30:56_: first component over Lambda in the 5th so it would be two 5th _31:01_: 00. I mean _31:02_: don't worry if you got that I I got I was doing that but doing _31:05_: that with you I got it wrong as well because I've got my _31:08_: arithmetic rubbish. But the point is, there's nothing _31:11_: particularly exotic about this frequency 4 vector. _31:17_: It's just there's not really obvious thing. _31:24_: No. So so where we got to _31:26_: is _31:30_: is this. I'm going to write that down again. _31:33_: L is equal to F&L over Lambda, M over Lambda, N over _31:40_: Lambda and delta R is equal to delta T, Delta X, Delta Y, Delta _31:47_: Z _31:49_: and we worked out the L... R is equal to minus N and that which _31:54_: is just a number. _31:57_: But look back to the beginning of this chapter and one of the _32:02_: things that I showed you _32:05_: was that the components of a vector _32:08_: in one frame _32:10_: turn into _32:12_: component of the same vector in another frame _32:15_: when the when the actually upon by the transformation matrix big _32:19_: Lambda. _32:22_: And so the components of delta R, _32:28_: delta X, Delta Y, delta Z _32:34_: in terms of the components of the same vector in another frame _32:41_: are going to be _32:45_: add primed plus _32:49_: gamma, delta X primed plus V delta T prime primed _32:57_: data. Why prime data _32:60_: their frame? Now we know that simply because we R is our _33:04_: prototype _33:05_: full vector displacement vector is a prototype 4 vector and we _33:09_: deduced that its components transformed in this particular _33:13_: way. _33:14_: I'm falling behind time here. I'm talking too much _33:20_: now. We can substitute that into this expression here _33:24_: and they would rearrange it _33:27_: and we end up with _33:31_: dot dot R _33:33_: equal to _33:34_: camera _33:36_: and _33:37_: F -, V one over Lambda, _33:41_: but elbow Lander _33:43_: camera _33:46_: over Lambda minus _33:49_: PDF. _33:51_: Aim of a Lambda _33:53_: and Orlando. _33:58_: Daughter _34:00_: Print, _34:03_: so all of them there is. Substitute that back into L dot _34:07_: r = -, n and we get the same thing. Is this expression here _34:10_: after a bit of rearrangement. I encourage you to go through the _34:14_: steps of that just to reassure yourself I'm not pulling a fast _34:17_: one. _34:19_: But what we've got here _34:22_: is the _34:25_: the L primed that we would get if we took L prime equal to _34:29_: Lambda L where Lambda is the transformation matrix _34:34_: that gets us this _34:36_: out of that _34:37_: is L primed dot _34:41_: R print. _34:45_: So just to repeat, what I'm doing there is if we took the _34:52_: components of L up at the top there and acted upon them with _34:56_: the same matrix that get got is that that that transformed delta _35:00_: R what we would get is L primed with these components here. _35:05_: So what we discovered _35:07_: is that this vector L _35:10_: would find at the top, _35:13_: transformed in the scene. We have our four vector. _35:17_: It behaves the same way as the displacement vector. It's a four _35:20_: vector in other words. _35:23_: OK, now is that not sort of inevitable? _35:27_: I seem to have been trivial here. _35:30_: No I'm not. Because if I were to say I don't know what what S _35:34_: doing in that in that in in that in that definition of. Well, OK, _35:39_: let it seems pointless. Let's just put define L because I want _35:44_: to _35:45_: to be 0 over Lambda, M over Lambda over Lambda. Why _35:49_: shouldn't I do that? Why should I do that? Because if I do the _35:53_: same process here and operate on that with the transformation _35:57_: matrix L Lambda rather _35:60_: what I get _36:01_: is what we'll have a a non 0 component _36:04_: in the in the time slot it'll have changed its form. And _36:09_: moreover this property of the dot product which does R _36:13_: wouldn't remain frame invariant. _36:16_: In other words, it's only when we include that F at non zero _36:20_: with frequency in the frequency 4 vector _36:24_: that we get these correct properties of a vector. So this _36:27_: is behaving like a four vector is the point _36:31_: by by his behaviour. Shall you know it? _36:33_: OK. And that's called the frequency for vector _36:38_: and it's an important thing. _36:41_: Uh, no. Next thing is _36:47_: imagine we had our _36:52_: away free _36:54_: moving right before _36:56_: and in our _36:59_: a prime frame is moving at speed at at at an angle Theta with _37:03_: respect to the X prime axis _37:06_: in that frame. Then _37:09_: the _37:11_: 4 vector would be _37:12_: it. Frequency in that frame _37:17_: cost you frame over Lambda. _37:20_: Something strange over Lambda that's going to be its _37:25_: the the four vector describing the the motion in the prime _37:28_: frame when it's moving with an angle Theta primed to the X _37:31_: axis. So it's moving in that direction, but like that, like _37:34_: that. _37:35_: OK. _37:37_: What then _37:39_: would that look like in the unframed frame? In the frame _37:42_: with respect to which that frame is moving? _37:46_: Uh, _37:47_: I'm _37:50_: that L is equal to _37:52_: inverse transformation. _37:56_: Hell _37:58_: transformation on _37:59_: L primed _38:01_: equals _38:03_: what equals, _38:05_: but that _38:06_: it also the case that in _38:09_: this _38:10_: frame, in the frame with respect to which the frame is moving, _38:15_: the the same wave train is going to have our frequency, _38:18_: it's going to be moving at an angle. _38:22_: So these are all primed as well _38:25_: if we move at an angle Theta not Theta primed _38:29_: over Lambda _38:30_: 0. _38:32_: And we can compare these two expressions for L component by _38:36_: component _38:38_: to get _38:41_: the frequency _38:42_: in the _38:44_: thank you thank you. The frequency in the prime frame as _38:48_: our function _38:51_: frequency in the unframed frame as a function of the frequency _38:54_: in the frame frame. _38:56_: This factory Gamma _38:59_: 1 + V _39:03_: Cos Theta over _39:05_: the frequency of the wave train in that frame, _39:09_: and a similar expression for the the change in the angle. In _39:14_: other words, simply by virtue of _39:17_: the of regarding the four vector as our the the frequency 4 _39:21_: vector as a four vector, _39:25_: that tells us we're allowed to relate the components of that _39:29_: frequency of that vector in the prime frame L prime _39:34_: and deduce the bit of missed out the... Deduce what the frequency _39:38_: pulling point of that same vector R in the unprimed frame _39:43_: and compare them to what we decide that we're going to call _39:47_: those components in this frame and so get a way of transforming _39:51_: the frequency from one frame to another. _39:57_: And then we're we're, we're getting towards the, the, the. _40:03_: At the point here, _40:06_: umm, _40:12_: and because I'm running out of time, I'm not going to go _40:15_: through the the the, the blow by blow details, but we can _40:18_: and discover that the the length of the of the four vector _40:23_: is _40:27_: OK _40:29_: Yeah _40:31_: and what? What do I miss out? _40:41_: OK. I'll I'll, I'll I'll skip a bit about and _40:47_: talk about the important case _40:52_: where the the the the way free in question is _40:56_: a light wave moving at the speed of light. _40:59_: OK, so the velocity U is equal to to one and the bitter missed _41:06_: out. It's a question. _41:14_: ITV Cos Theta primed over U Primed _41:18_: so but but that you is the is the speed of the of the wave _41:21_: train. But we're going to but what we're interested in really _41:23_: is the special case where the wave train is moving. The speed _41:26_: of light is a is a light wave, _41:28_: and in that case _41:30_: the _41:31_: this expression changes. It simplifies to. _41:37_: Is equal to _41:39_: if primed gamma 1 plus _41:42_: the Cos, Theta _41:45_: and Costa _41:48_: equals. _41:52_: You're playing plus V _41:54_: over 1 plus _41:57_: Cos Theta _41:58_: praying where V is the open question. Is it gamma times that _42:03_: or is it gamma off that? Ohh, gamma 10 times that? Yes. Good _42:07_: point. Yes. So it's just gamma of V where V is the relative _42:12_: speed between the two frames. Thank you. Yes, _42:16_: and and that is a nice way. There are other ways of doing _42:20_: it. If you look at Carol and Astley, the recommended book _42:24_: they have, they're a different way of deriving this, which I _42:29_: think is is much messier. But this way we have deduced this _42:33_: relativistic Doppler shift directly from the requirement _42:37_: that the the frequency for vector is a full vector. _42:42_: So it's it. So the first equation we wrote in this in _42:45_: this in this chapter the beginning of chapter 8 about the _42:48_: transformation of vectors from one frame to another. _42:51_: It's what lets this pop out. _42:55_: And you are familiar with the Doppler shift of for sound _42:58_: waves. You're familiar with Doppler shift for light waves, _43:02_: for example. You can tell how fast a Galaxy is is moving back _43:06_: at the Doppler shift from the light emitted from things on the _43:09_: edge of the Galaxy, and so on. And that there's a bit, These _43:13_: are the staples of observational astronomy. _43:17_: That Doppler shift is just the non realistic limit of this _43:20_: expression here. _43:25_: With the, the, the, the, the feature in question being a nice _43:29_: straightforward line of sight. Zero 100 degrees, whatever you _43:33_: want. _43:34_: And so I'm gonna write the remaining 5 minutes _43:39_: having _43:41_: rushed there. And this is this is the double shift. It's it's a _43:46_: very important and and directly observable _43:50_: feature _43:52_: of the change of the change of frequency and wavelength and and _43:56_: and the direction of light we move from one frame to to to to _43:60_: another. It's directly observable. _44:03_: You can see that book you you you learned in in in _44:08_: last year in astronomy last year, how to use the Doppler _44:11_: shift to _44:14_: measure the speeds of things at astronomical distances and the _44:18_: fact that the direction changes. The direction of Starlight _44:22_: changes depending on what direction the Earth is moving _44:26_: in. _44:27_: Because these changes depending on on on on on whether the Earth _44:30_: is moving in that part of it, Robert, or on this part of its _44:33_: orbit, _44:34_: The change of direction is not much that that that. That _44:37_: particular aberration is not much, but it was detected in the _44:41_: 18th century. _44:43_: People were were were able to make precise enough measurements _44:47_: of the procession of astrometric measurements of the direction of _44:52_: stars at different times of the year to detect that there was a _44:56_: an anomaly over the course of the year which with this rustic _44:59_: aberration _45:03_: and you can detect this in a laboratory, it's nice. It's very _45:07_: easy to detect the non relativistic Doppler shift that _45:10_: that that that the longitudinal Doppler shift where where Theta _45:14_: is 0 or or π. They used to take that thing just changed speed _45:18_: change colour very obviously. It's much less less easy to _45:21_: detect the transverse Doppler shift because as something is _45:25_: moving past you _45:27_: perfect you perpendicular state. There's no there's there's zero _45:30_: classical Doppler shift there because it's not moving moving _45:33_: towards you. _45:35_: So the only Doppler shift you get is because of time dilation. _45:39_: If it _45:41_: effectively could have obtained _45:44_: and that's a very small effect _45:46_: and it's very hard to measure. _45:49_: But it it was detected in 1932, I think the two Eyes and _45:54_: Stillwell at Bell Labs. In fact there were _45:59_: there were industrial researchers and they managed to _46:03_: detect this effect. And that was one of the one of the early _46:07_: straightforward observational _46:10_: confirmations of of lots of defects in in in a laboratory. _46:14_: So this is very, very practical, very, very, very _46:17_: straightforward, detectable and an immediate consequence of _46:21_: relativity in the laboratory. _46:25_: And I am going to stop there because I've run out of time. I _46:28_: had hoped to take on to Chapter 9 in this in this lecture, and I _46:32_: think it would be ambitious to you all the way through. Chapter _46:37_: Definitely. I think we're basically get all the way _46:40_: through that next time, so I think we will drift eat into _46:44_: lecture 11 a bit, but I aim to get through Chapter 7 fairly