Transcript of a2-l08
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_0:10_: Hello, this is I'm not sure if this is Lecture 7 or Lecture 8,

_0:13_: but anyways, today's lecture

_0:16_: we're going on to talk at the beginning of Chapter

_0:21_: 6, Foot kinematics. Now this is a bit of a a shift from what

_0:25_: we've been doing up to this point, not not conceptually, but

_0:30_: we're we're moving on to the next stage

_0:35_: as I mentioned the Lorentz transformation equations which

_0:39_: we talked about the last two lectures and we found some of

_0:42_: the paradoxes puzzles with them in the last one

_0:46_: are very important and it would pay. It would repay you to go to

_0:50_: be quite diligent about the exercises for

_0:54_: chapter 5 because they will settle down. A lot of your

_0:58_: understanding of the ideas here and there are also nicely

_1:02_: examinable that they're the sort of things that are really just

_1:06_: easy to see. Ohh yeah I've got something about that in the

_1:10_: exam. That's a hint. It's pretty obvious hint, since the only

_1:14_: objective for chapter five was was one.

_1:18_: Anyway, we're now moving on to what we've been talking up to

_1:21_: now about events, and I've been stressing again and again that

_1:25_: events are frame independent things. They have a position and

_1:29_: a time

_1:30_: and that the the story that that the point of the recent

_1:33_: transformation is to talk about the coordinates of that event in

_1:36_: this frame and in that frame which are moving with respect to

_1:39_: each other.

_1:42_: All the only thing that been moving up to this point have

_1:45_: been framed.

_1:46_: But this is a physics course as well as an astronomy course. So

_1:50_: we're interested in things moving. We're interested in in

_1:52_: the description of things moving.

_1:55_: And before we can describe, before we can explain how things

_1:58_: move dynamics, we have to describe how things move

_2:01_: kinematics.

_2:02_: And so this is where we talk about

_2:07_: velocity, we'll talk about acceleration.

_2:10_: And

_2:13_: that brings in talking about four dimensional vectors, 4D

_2:16_: vectors, 4 vectors.

_2:19_: So that's the goal I the plan is to to to to. There's much

_2:23_: relation 8 should come to think of it, because the plan is to do

_2:27_: the next few chapters in

_2:29_: three lectures. So we're ready to move on to GR. Talking about

_2:33_: GR, some some topics in GR in lecture 11, we seem to be on

_2:37_: schedule, well

_2:39_: schedule at this year. So any questions about structure, about

_2:44_: things like supervisions, about things like the product, which I

_2:49_: hadn't actually checked, but which I hope that is, we'll find

_2:53_: your questions there.

_2:55_: Anything else that folks are thinking about, worrying about,

_2:59_: puzzled about,

_3:01_: And it was Mr Tape. OK,

_3:04_: let's go.

_3:08_: So in a way, we're returning to physics here by talk, by talking

_3:11_: about velocity, acceleration and things like that.

_3:15_: So before we can go and talk about those, we're gonna have to

_3:18_: talk about to to, to expand the notion of what you understand

_3:21_: about vectors to to to to to to to discover what the vectors are

_3:24_: in that, in that context. And then we can move on.

_3:28_: So,

_3:29_: and I'll move this to here.

_3:34_: What are we

_3:37_: so really understanding? The concept of all vector is is the

_3:41_: key thing in this whole thing, and that's various things that

_3:45_: you one can do. So this is

_3:48_: and it seems as there were technique have used several

_3:51_: times in this course so far. I'm going to describe from the

_3:55_: you're familiar with in unfamiliar terms because if

_3:57_: those terms, they're going to adapt those terms to use in a

_4:01_: context that you are unfamiliar with. So

_4:04_: this is a vector a in the XY plane,

_4:08_: which are completely familiar.

_4:11_: The vector has components.

_4:14_: There's a basis vector, a unit vector particle on the X axis of

_4:18_: the unit vector pointing along the Y axis. And you can

_4:22_: construct the vector A

_4:24_: with

_4:25_: so much times the

_4:29_: a unit vector on the X axis plus so much with the unit vector

_4:33_: along the Y axis. And you can get the vector E back, and so

_4:37_: those ex and EY are the components of that vector.

_4:41_: OK, nothing exotic there. If you were to change your mind about

_4:45_: the frame and rotate the frame to the frame frame

_4:49_: and the vector doesn't change, the vector doesn't care what

_4:52_: frame you're talking about,

_4:54_: but the components of the vector

_4:57_: would change

_4:58_: and be different in the other frame. So if there's a basis

_5:02_: vector pony X prime axis, a basis vector Polygon that the

_5:06_: the Y prime axis and the same vector E is ex primed

_5:10_: times

_5:11_: the X plus AY primed

_5:14_: times. EY,

_5:16_: nothing exotic there.

_5:18_: The key thing is, the vector is the same in both cases,

_5:23_: right? It's a displacement vector. It's not a position

_5:25_: vector. It's not. How do you get the order into a? It's a a

_5:28_: length and A and a direction in the plane.

_5:33_: Good.

_5:35_: Any questions?

_5:37_: Right. So that that's really the the the terminology that I I'm

_5:43_: using here

_5:45_: and

_5:46_: what we could do then is ask well how do we get if we know

_5:50_: what axe and AY are, how do we work out what ex prime and AY

_5:54_: primes are? And again, that's not. I mean you might not write

_5:58_: this down off the top of your head, but you're not surprised

_6:02_: to see something like that. I trust

_6:05_: the if instead of of the the that's a that are we talked

_6:09_: about a displacement vector does XYZ

_6:13_: in three-dimensional space.

_6:16_: That seemed Victor. That seemed displacement of. How do you get

_6:18_: from here to here?

_6:19_: We have different components in different frame and the related.

_6:22_: By something like that I mean as I say you may not write that

_6:25_: down on top of your head but you're not surprised to see a

_6:28_: you cost sign - cost in that expression

_6:33_: and and and and the the point here is that

_6:38_: that displacement vector I in the previous thing I used a

_6:42_: general vector a but I'm going to talk about displacement

_6:45_: vector as the just the displacement from this point to

_6:49_: this point. So delta X out of Y, delta Z, I'm going to use that

_6:53_: displacement vector as our prototype vector. OK. And by

_6:57_: prototype I mean that anything which looks like that vector,

_7:00_: any which looks at that, we're going to call a vector as well,

_7:05_: right?

_7:10_: And in the

_7:12_: in the lecture notes folder on the middle there's the numbered

_7:16_: chapters. There's also a document called Recipe and

_7:18_: Document called Maths. The document called Maths at the end

_7:21_: of it has a little review of linear algebra and it's 2

_7:24_: paragraphs which just lists basically just a number of words

_7:27_: that I'm going to presume you're familiar with. You've done in

_7:30_: previous years maths, so. So you might have to go back and look

_7:33_: at you, just remember what those words mean. But I'm going to

_7:36_: presume you're familiar with those words,

_7:41_: right?

_7:43_: So much for three vectors.

_7:49_: We saw the displacement vector in

_7:53_: feed in three dimensions dot XYZ.

_7:57_: We can now talk about the displacement vector in 4

_7:60_: dimensions

_8:01_: and and at the moment right now I'm only talking these numbers

_8:04_: haven't haven't got to the vector but yet

_8:08_: delta T, delta Y, delta X and delta Z. And we know from

_8:13_: chapter 5 transformation that they are related to the

_8:21_: displacements. That doesn't frame the delta XYZ frame by an

_8:25_: expression like that. If you multiply that out, multiply that

_8:29_: matrix equation, you'll get the Lorentz transformation equations

_8:33_: from halfway through chapter

_8:37_: 3, and that creates the notes, let's say

_8:41_: 6.2.

_8:43_: And if we

_8:46_: that, that's just a matrix, so the inverse of that matrix is

_8:48_: just the the matrix inverse of that.

_8:54_: Now if we talk about

_8:57_: notice you park that thought

_9:02_: and and think about what a vector in Minkowski space would

_9:06_: look like.

_9:08_: Here, rather than the X&Y axis, we've got X and the T axis

_9:13_: and the X prime and the T prime axis and as usual, rather than

_9:16_: being rotated like that, they've gone like that When you go from

_9:20_: when you when you project from one frame into the other

_9:24_: and you have a vector there, and think of that displacement

_9:28_: vector for now, delta X, delta T

_9:31_: white dotted as well,

_9:34_: that links to points in

_9:37_: space-time.

_9:39_: It links to events.

_9:41_: There's an event to the beginning of the at the end, and

_9:43_: this is the difference between them. There's just exist what

_9:46_: and T in there and that vector we can decide, OK, we're going

_9:49_: to call that vector as well.

_9:52_: Has component A1

_9:55_: when you project it onto the X axis

_9:57_: component. East Naughty.

_9:59_: You project it onto the T axis. Some rather old fashioned

_10:02_: textbooks refer to that as as the Force dimension. We refer to

_10:05_: the zeroth dimension because just a bit tidier,

_10:09_: much more common nowadays. And similarly, if you instead not

_10:12_: project it vertically downwards, but project the vector parallel,

_10:16_: you know, projected parallel to the T frame axis and ending up

_10:20_: on the X frame axis, that's A1A primed one.

_10:24_: So the component of the vector E

_10:27_: and the basis vector pointing along the X prime axis and

_10:30_: similarly project it from a onto the the T frame axis, moving it

_10:34_: parallel to X prime.

_10:36_: You end up with an E prime knot,

_10:38_: and the idea

_10:39_: is these are the components of that vector

_10:44_: 4 vector in the two different frames, exactly analogously to

_10:48_: the way we saw the components of the three vector in the two

_10:52_: rotated frames.

_10:54_: OK.

_10:55_: And if that vector E is this delta X, delta Y, delta Z, delta

_11:01_: T vector, then we discover

_11:05_: that the relationship between those components is just that

_11:08_: thing that we are, that it's fairly obvious that there's no

_11:10_: fairly obvious from what we learned about the Lorentz

_11:13_: transformation equation. So this is the point that this is the

_11:16_: bit where we're going from Chapter 5 to chapter six. OK,

_11:19_: the stuff we learned. Chapter 5 we're now applying

_11:22_: but really grasping hold of the of the idea of geometry in

_11:26_: chapter 6.

_11:32_: And again we're going to take the displacement vector in 4

_11:35_: dimensions. There's delta T dot XY&Z are the prototype

_11:39_: displacement vector and so anything that looks like that

_11:43_: and that behaves like that is a vector in 4 dimensions, A4

_11:47_: vector.

_11:48_: OK,

_11:52_: um,

_11:55_: so just as before, an arbitrary vector has components in

_11:58_: different frames. The vector is the same in all the frames, it's

_12:01_: just the numbers that change.

_12:05_: OK, so that's and that that's a key thing that that that that

_12:08_: sounds like a fairly neutral thing to say, but it's actually

_12:11_: a very important thing to see. The victor is the if like the

_12:14_: physical thing we're going to talk about velocities in a bit,

_12:17_: the velocity vector and if something is moving that's a

_12:19_: physical thing that frame independent thing. It's the the

_12:22_: numbers that the components in particular frame

_12:25_: that that you choose that are frame dependent.

_12:31_: Uh, so

_12:34_: we've got that and justice as with the displacement vector.

_12:37_: Therefore for an arbitrary vector A

_12:41_: the components are going to transform like this, so just the

_12:45_: same as the displacement vector but with for an operative vector

_12:49_: components with components A.

_12:53_: So the risk of just banging on with this at too many times.

_12:57_: These are described in the same vector a

_12:60_: but with the numbers a naughty to a three and a not primed to a

_13:03_: three. Prime is being different simply because you're choosing

_13:06_: different frames,

_13:08_: right? So I want to just clear that too many times just so you

_13:12_: that's it doesn't sound as important when I see it first

_13:16_: time as it is.

_13:18_: So

_13:23_: just just really for completeness,

_13:26_: the same thing is true if you draw this in the in the whole

_13:30_: thing, this whole thing in the other way around as it were,

_13:35_: with the

_13:36_: the framed frame being sort of the basic one and the other one

_13:40_: being the one distorted away.

_13:44_: And what we are then going to

_13:49_: how do we introduce them, we come to that.

_13:56_: So we'll we'll, I think we'll do a bit more linear algebra here

_13:60_: and

_14:01_: and uh

_14:04_: how do I?

_14:06_: So now at this point it's slightly I'll just switch both

_14:09_: of the visualizer. There's one or other of these is the one

_14:13_: that gets onto the E360. I don't know which one it is. I'll just

_14:16_: switch both over and

_14:20_: OK, I think, I think, I think this is actually 8.

_14:25_: So

_14:26_: what do we want to write down here?

_14:30_: So I think the standard things if A

_14:34_: and B our four vectors

_14:37_: then

_14:41_: A+B

_14:43_: is also 4 vector and the components of that are C

_14:51_: See naughty will be a naughty plus

_14:56_: be naughty. C1 equals A1 plus B1

_15:01_: and so on, so so so the add in that in in in the expected way.

_15:04_: The way that you're familiar with adding vectors in three

_15:07_: dimensions.

_15:10_: Now the scalar product. If you think back to to to the vector

_15:14_: stuff with vectors you learned. One of the things you learn

_15:17_: about the scalar product is the which measures how how separate

_15:21_: the projection of one right onto onto another. In this context,

_15:26_: the

_15:27_: scalar product

_15:29_: is odd

_15:31_: and the

_15:33_: the scalar product of A&B

_15:36_: is going to be a Naughty. B naughty what? What? What if we

_15:40_: have a dot B? They're both free vectors, you remember? That's a

_15:45_: naughty

_15:46_: being. Naughty plus A1B1 plus A2B2. That's familiar, I trust.

_15:53_: Yeah. Here is minus A1

_15:57_: B 1 -, a two, B 2 -, a

_16:01_: three,

_16:03_: B three,

_16:05_: and that can be written

_16:08_: as

_16:14_: E

_16:16_: Peter B,

_16:17_: where these are to pick up

_16:21_: slipping into a matrix version of this where the the matrix ETA

_16:26_: is

_16:28_: a diagonal matrix 1 -, 1, -, 1, -, 1.

_16:33_: Now I've done several things in those couple of lines.

_16:38_: The first thing is, I've just pulled this definition of scale

_16:41_: part out of the hat. Really, you know, minus A1B1 and so on. And

_16:46_: you think those might seem completely arbitrary. But if you

_16:49_: think back to the definition of the scale of the invariant

_16:52_: interval squared,

_16:53_: that's fundamental for those maintains come from and we'll

_16:55_: see the link again in a moment. So, So that's a bit of a I've

_16:58_: stopped that out of the hat. We shouldn't be too much of a

_17:00_: surprise.

_17:02_: The next line is just that that that same thing written in.

_17:08_: In metric terms

_17:10_: we have introduced this 4 by 4 diagonal matrix ETA

_17:15_: where the central the diagonal 1 -, 1 mitral -1 and everything

_17:19_: else zero. And you can and if you start it a bit you'll see

_17:23_: that if you form that matrix product with that and the and

_17:27_: the components of the matrix E then you'll get that term back

_17:31_: again. Why do I do this other than to make it look

_17:34_: complicated? I don't do it to make it look complicated. I do

_17:38_: it because this matrix eater

_17:40_: is terrifically important.

_17:43_: If you remember in the in the

_17:49_: discussion of the scale of product in in in 3D space, the

_17:53_: scale of production 2 vectors is.

_17:56_: It tells you it includes information about how long the

_17:59_: vectors are and what the angle is between them and so on. It's

_18:02_: AB Cos Theta as you recall.

_18:04_: If you take the the the the scale of product of a vector

_18:06_: with itself,

_18:08_: that gives you the length of the vector.

_18:10_: If you recall, because of course cost zero is 1 and a a a Cos

_18:14_: Theta is going to be a ^2. So the the the the dot product. The

_18:18_: scalar product of a vector with itself gives the length of a

_18:21_: vector.

_18:23_: And here if I ask what is the

_18:26_: a scalar product of of the vector E with itself,

_18:32_: that's going to be a not a naught minus A1, A 1 -, a two A

_18:40_: 2 -, a three A3.

_18:43_: And if we talk about the

_18:47_: the displacement vector R

_18:50_: which is equal to Delta T, delta Y&X, delta Y

_18:56_: doctor Z. So that's just the displacement from this event to

_18:60_: this event

_19:01_: and ask what is

_19:03_: our dot

_19:05_: dot R?

_19:07_: We see that it's just a t ^2 minus Delta X ^2 minus Delta y

_19:11_: ^2 minus

_19:13_: Doctor Seth Green.

_19:16_: So we've recover the invariant interval

_19:19_: from this definition of the scalar product.

_19:21_: That's. So it's it's not really pulled out of a hat.

_19:24_: Was not pulled out of the hat. But it's entirely reasonable

_19:27_: because we end up with the right answer

_19:29_: that's telling us that if that displacement vector doesn't re

_19:33_: goes from this event of that event,

_19:36_: the length of that vector is the invariant interval between those

_19:41_: the the the delta squared that we found in two chapters ago. So

_19:47_: that's why this

_19:49_: matrix here meta is important because it recovers this

_19:55_: and that's why that matrix there

_19:58_: is encoding the definition of length in this space.

_20:03_: A hand up.

_20:05_: What does that say?

_20:07_: After

_20:09_: or diag.

_20:18_: Lots of tea,

_20:19_: yeah.

_20:24_: So what do you mean

_20:25_: to the

_20:27_: what you transpose? Etab

_20:30_: yeah yeah sorry. He transpose yes because you've both got a

_20:33_: column vectors. So so if you think of the the matrix product

_20:37_: of the row vector matrix, column vector.

_20:42_: So this is the in the same way that the

_20:47_: scalar product of

_20:50_: a free vector a with itself gives you a square plus a

_20:53_: naughty squared.

_20:55_: So that should be so that that that that would be a dot B

_21:02_: equals A1B1

_21:04_: plus A2B2 plus A

_21:08_: BB3 as well.

_21:10_: So there the scalar product of a vector with itself would be a 1

_21:15_: ^2 plus A 2 ^2 + a ^2 or X ^2 + y ^2 + Z ^2, which gives the

_21:19_: length of a vector code Pythagoras theorem

_21:23_: when you

_21:25_: the scale.

_21:30_: Or why the minus signs? Why the minus signs here? Yeah, that's

_21:35_: that's the bit I pulled out of a hat. But it makes sense because

_21:39_: we end up if we do the same thing with the displacement

_21:43_: vector dot R which is this separation between two events.

_21:48_: We get this thing here

_21:50_: which you recognise

_21:53_: at the invariant interval.

_21:55_: So those minus signs are there so that we get the right answer.

_21:60_: Right answer here.

_22:03_: In other words, that matrix ETA,

_22:06_: now four matrix heater push a diagonal is telling us about the

_22:11_: geometry of Minkowski space. It's all encoded in that matrix.

_22:15_: That matrix is called the metric

_22:18_: metric, as an metro and as in metric system. But it means

_22:22_: measurement. The metric. It's what defines distance in this

_22:26_: space and in special activity. It's always that

_22:31_: in general relativity,

_22:33_: where generally comes in

_22:35_: or is that we're we're not talking about special activity

_22:38_: anymore. We're talking about things which are moving and are

_22:41_: accelerating, or which are moving on the influence of

_22:44_: gravity. And the effect of that, it turns out,

_22:48_: is the metric. In a curved space-time is curved because the

_22:52_: metric is different from that diagonal.

_22:56_: So this is the first time you've you you'll you'll see this that

_22:60_: that that that that this method but that stays with the whole

_23:03_: study of relativity all the way up to the

_23:08_: the key part of general activity. The Einstein's

_23:10_: equations for general activity are basically what is the metric

_23:13_: given that there are matters here and here, that that, that,

_23:16_: that, that, that, that's that's what solving Einstein

_23:18_: ingredients means.

_23:20_: But

_23:21_: in special activity it all with that, because the geometry of

_23:25_: Minkowski space of

_23:28_: of speaking, moving, moving at constant speed is nice and

_23:31_: simple.

_23:33_: I said it's always that sometimes it will be different

_23:37_: from that because some textbooks choose

_23:40_: this to be minus, plus, plus, plus,

_23:44_: and that and both, in which case the metric is minus, diagonal,

_23:48_: minus, plus, plus, plus, and both are reasonable.

_23:53_: I think that there's one that's more common in the ecology, one

_23:56_: that's more common in particle physics. I don't know which ones

_23:60_: which, but there's just our conventions in different areas.

_24:03_: So if you're looking at another textbook on relativity,

_24:07_: be careful and justice check what the signs in this

_24:10_: expression are or in this expression here are, because

_24:13_: they might be exactly the other way around

_24:17_: and that caused a couple of things elsewhere to change. But

_24:19_: it's not fundamentally. It's not fundamentally important

_24:25_: now to keep a good place to pause briefly have any questions

_24:28_: about

_24:31_: or happy with that or at least

_24:35_: unhappy about it. To happy extent

_24:38_: content question. Ohh, excellent. Well look, other

_24:41_: questions could change the metric to be -1. Yeah. Does that

_24:44_: mean that

_24:46_: like the way

_24:47_: derive the invariant interval forward that will change the

_24:50_: settings?

_24:53_: It would be because there will be another - somewhere else in

_24:58_: that so. So at one point in that definition.

_25:04_: I can't remember exactly how it would be different, but there

_25:07_: will be another minor saying somewhere in in in chapter 4

_25:12_: and I like this way of doing it for this class because this way

_25:18_: proper time is a positive.

_25:21_: So so if if two things are

_25:26_: two events which at the same place and and and and the simply

_25:30_: in different times. In other words, with is a watch observer

_25:34_: at both events that invent interval between those is the

_25:37_: proper time property means and it's positive

_25:41_: it. With this definition, which is. I think it's nice, yeah, but

_25:44_: but but but yes, so there's an arbitrariness in here. We all

_25:48_: have to be consistent, but it is basically arbitrary.

_25:53_: OK,

_25:55_: umm

_25:60_: and.

_26:04_: I think I have

_26:08_: alright. Yes,

_26:10_: today.

_26:15_: OK.

_26:16_: Great question.

_26:17_: And

_26:21_: the history of vectors

_26:23_: what's the put hands up here but have I think I'll give you I'll

_26:27_: come to think what's E dot B and what's the magnitude of and the

_26:31_: magnitude of just in your in your heads what's the

_26:35_: have opposed and talk to your neighbour and work out what it

_26:38_: would be is alright

_26:41_: right

_26:43_: yeah

_26:45_: quick quick

_26:46_: you know we.

_27:21_: OK,

_27:23_: having discussed that until over that a bit,

_27:26_: any books

_27:33_: and and any what I want, I want to show what a dot B would be

_27:38_: zero. Yeah, one 1 * 2 - 2 * 1

_27:44_: a. What's the length of a?

_27:49_: What are you there?

_27:53_: No, no, no Route 5.

_27:57_: It would be 1 ^2 -, 2 ^2.

_28:01_: The remaining three would be the but the length squared of that

_28:05_: and similarly

_28:08_: the other one would be 2 ^2 -, 1 ^2.

_28:13_: So what that shows is that those two vectors A&B 12002100 are

_28:18_: orthogonal to each other.

_28:21_: Now look orthogonal to each other as you as you visualise

_28:25_: them on on the on the plane. But to the extent that orthogonal

_28:29_: means that the scale of product is 0, those two vectors are

_28:32_: orthogonal to each other

_28:35_: and and and So in the geometry of of of Minkowski space, those

_28:38_: two vectors are are at right angles. We don't know.

_28:41_: Orthogonal right angles is the wrong word to use. They're

_28:44_: orthogonal.

_28:46_: The length of V, the length squared of east is negative. So

_28:49_: picking up the terminology from the end of chapter four, we call

_28:52_: that a space like vector because its length is length squared is

_28:56_: negative

_28:57_: be B is positive. It's something that is pointing along roughly

_29:03_: along the time axis. We call that a time like vector.

_29:10_: OK, so that's about terminology, which is just picking up the

_29:14_: terminology you used a couple of chapters ago.

_29:18_: The other thing I will mention, because oddly enough I don't

_29:22_: have it the notes is that

_29:27_: and this matrix

_29:33_: Gamma, Gamma V

_29:36_: Gamma V Gamma

_29:42_: that we used to transform from one stream to another. We

_29:48_: universally live with that Lambda

_29:50_: and and I I'll I'll use that term happily from no one

_29:55_: negative orientation in space.

_29:59_: Ohh, right. And orientation to the negative sign given

_30:01_: orientation of space

_30:04_: right. There's a mathematical question. I think this is a an

_30:08_: orientable space in the in the sense that if you move around

_30:12_: it, it's not a movie type of space. Is what you meant

_30:18_: and

_30:20_: it it it's all right. I think you both what you meant. Does he

_30:25_: give an orientation to space? I think sort of in the sense that

_30:30_: if you remember the

_30:32_: the that diagram with the light cone that indicates there's a

_30:37_: direction in. It's based in the sense that there's a direction

_30:41_: into the future

_30:43_: if you like. The things along the time axis stay along the

_30:47_: time axis of any frame that you move into. So they are always in

_30:50_: that direction. So that's a sort of directionality to space.

_30:54_: Think about orientation is a of a space is something that is

_30:57_: done in topology. I don't know, which I think is probably not

_31:01_: the question you're asking, but I think this is an oriental

_31:04_: space. But not. I'll come back in a moment, Is that does that

_31:08_: sort of make sense?

_31:11_: Yeah. See, you're already asked to write up and. Would you write

_31:15_: it as that matrix or would you write it as like the

_31:18_: gamma minus gamma? Or does it just depend on the situation

_31:22_: that you've got? Alright, and I think it depends on the

_31:26_: situation because there this is is going from a metric space to

_31:30_: another one in standard configuration. I think that you

_31:33_: know those movements positive V

_31:36_: so the inverse matrix to that.

_31:39_: If you if you were to to look at that and and and invert it then

_31:43_: you would get a a - Here and here we would make sense because

_31:46_: from the point of view of the prime frame the unprimed frame

_31:49_: is moving in the negative direction. So you can you can

_31:52_: get that inverse either by thinking of that way, that way

_31:56_: or minus V or you can take the matrix inverse of that and

_31:59_: discover that ministry.

_32:01_: So that transformation matrix Lambda

_32:04_: is uh.

_32:08_: Well, I I will use that notation to refer to it. And it has the

_32:11_: property that if we,

_32:16_: uh,

_32:18_: yeah,

_32:21_: this, you know, motivated. But if we were to

_32:27_: you, you'll have done matrix transformations at some point.

_32:30_: You know how to get from this vector to that vector using

_32:33_: transformation matrix?

_32:35_: Possibly, possibly not

_32:38_: Have the property that if you

_32:40_: to the matrix product of

_32:43_: the this diagonal matrix ETA and transform that matrix matrix ETA

_32:49_: into

_32:50_: the other frame, what you get is

_32:55_: eta

_32:57_: through the this metric transformed into the other frame

_33:02_: transformed into itself. In other words, the metrics, the

_33:05_: metric is frame independent. In other words, the would be an

_33:10_: interval is framed dependent. In other words, the separation

_33:14_: between two events at either end of a displacement vector is

_33:18_: Freeman variant that's not dependent on the even though the

_33:22_: components of those vectors will change. That's also very

_33:26_: important point. My question there,

_33:32_: well the the the gamma, the gamma is are the gamma that's

_33:36_: one and one. The gammas are just the Gammas that we let of

_33:45_: Yeah, that's gonna be, that's gonna be as well, yeah. Sorry,

_33:49_: 00.

_33:52_: So it's it's block diagonal

_33:55_: and you know this bit diagonal. So the wine, the this is just

_33:60_: that me scribbling out the matrix that's in green 63-B.

_34:06_: For some reason I didn't seem to have pulled out that that, that.

_34:10_: That's the point. We were label that as Lambda.

_34:15_: Good. We're stepping through this,

_34:20_: so

_34:22_: there's a lot going on in this section.

_34:24_: I've seemed to just

_34:29_: see vectors exist

_34:31_: and this may each exists and that it's important, but that's

_34:35_: quite a lot. Yeah, there's a lot going on in this section because

_34:41_: one could talk about the maths in the section quite a lot

_34:45_: because it's talking about symmetry, it's talking about

_34:48_: invariance, very important things in maths, and some of the

_34:52_: maths underlie generativity. So we could go on about this for

_34:56_: quite a long time. So this is a bit of mathematical technology

_35:00_: and sort of dumping on you, which you'll get familiar with

_35:04_: in the electorate to come. But it is a weirdly crucial in the

_35:08_: sense that if you, if you study more of relativity, you stay

_35:12_: with those things again and again and again, indefinitely.

_35:16_: More or less.

_35:19_: Good,

_35:21_: right? Sorry,

_35:24_: I think that there is not much more to say about that. Good.

_35:29_: OK, so we have therefore the displacement vector.

_35:36_: Delta R is delta T, Delta X, Delta Y.

_35:42_: What is it?

_35:44_: And we also have thing to have to play with.

_35:48_: Well, I think we have to play with. We have the proper time

_35:52_: and remember the proper time is the time between two events in

_35:56_: A-frame in which those two events are the same place. In

_35:59_: other words, in other words, where the the spatial separation

_36:02_: between those two events is 0,

_36:06_: So what we can. So if

_36:09_: that displacement vector got R is like that,

_36:13_: we can talk about

_36:14_: Dr which is an infinitesimal version of that

_36:19_: the

_36:21_: Deez

_36:23_: and we'll rate that just by that we as DX

_36:27_: mute

_36:28_: but rotation here. So

_36:31_: in the same way that you you, you might see a a vector with

_36:36_: components 123 whatever. This impressive vector has components

_36:42_: and so,

_36:45_: and I'll write that as

_36:47_: DX0DX1DX2DX3.

_36:54_: And the convention is that when I use a Greek index here,

_37:00_: I'm I'm I'm using it to refer to index 0123. When I use a Latin

_37:05_: index through GK, wherever I'm using to refer to indexes 123.

_37:09_: So it's just the spatial indexes. That's a standard

_37:13_: convention. So we have this,

_37:16_: if it's impossible displacement, we can divide it

_37:22_: yeah

_37:25_: by the

_37:26_: the proper time detour

_37:29_: and get

_37:30_: DX0 by D Tau,

_37:33_: the X1 by

_37:36_: tall the X tube ID tall DX, D by D Tau,

_37:44_: right?

_37:46_: And we define that to be the

_37:52_: good,

_37:58_: the

_37:60_: the, the, the four velocity.

_38:02_: You

_38:04_: so remember that if R is a vector, then so will DRB. It has

_38:09_: the same properties. Tower is just a number, so D tall. We're

_38:14_: just a number. So those four things will behave as a vector

_38:18_: still. So therefore this thing behaves as the prototype dispute

_38:23_: vector does. Therefore it's a vector.

_38:27_: OK.

_38:31_: And we can do the same thing with

_38:34_: the for mentum

_38:36_: for acceleration rather

_38:38_: D2

_38:40_: ohh by D

_38:42_: tall squared

_38:44_: which is equal to

_38:46_: D2X naughty by D2 squared,

_38:50_: et cetera.

_38:52_: And we will write things like you MU is equal to DX MU

_38:59_: but he told me.

_39:01_: But that's just I can walk communication for the thing at

_39:04_: the top.

_39:08_: OK,

_39:11_: so let's look at the components in a bit more detail.

_39:19_: Yeah, this is what I'm going to follow my notes quite closely,

_39:24_: taking the art with the infinitesimal displacement of a

_39:27_: particle. So just moving through space and time

_39:31_: in A-frame in which the particle is not moving. So in the

_39:36_: particles rest frame,

_39:38_: the

_39:41_: just it it's in. In that frame, it's physical displacement.

_39:45_: Spatial displacement will be 0, so DX will be 0

_39:49_: and it's so the length of that discrete vector will be

_39:59_: Dr Dot Dr equals.

_40:03_: That are tough squared because that's just the the length of

_40:08_: the displacement in time in our frame in which the particle has

_40:12_: no displacement in space S that's do you DT squared

_40:17_: minus

_40:18_: the X ^2 where the equity is 0 and that the definition of the

_40:21_: property.

_40:23_: So that means that the thing of this particle that has a

_40:28_: displacement in physical displacement,

_40:32_: that displacement in I think will be the TD t ^2 minus DX

_40:38_: squared. And I'm going to just now confine this to movement in

_40:44_: the X direction and to press Y&Z that DT squared minus DX

_40:50_: squared

_40:52_: will be equal to D Tau squared.

_40:57_: No. If we

_41:00_: and

_41:04_: divide both sides of that by three t ^2, we get

_41:09_: You talk squared by DT. Squared is equal to 1 -, X ^2 by D

_41:16_: t ^2

_41:18_: and and notation is getting a bit confusing here

_41:22_: and we just clarify it with that,

_41:26_: so that that. But that's equal to 1 minus DX by DT squared, and

_41:33_: any pure mathematicians can look away just now which is

_41:42_: 1 -, v ^2,

_41:44_: which is one over

_41:46_: gamma squared.

_41:49_: Or in other words, DT

_41:54_: by D Tau

_41:57_: is equal to gamma.

_42:00_: Or in other words, the rate of change at which the clock

_42:06_: I wish the the time coordinate in the frame of which the

_42:09_: particle is moving

_42:11_: with respect to the

_42:13_: time coordinate on the particles own clock

_42:18_: is equal to the time dilation factor. So that's another

_42:20_: expression of time dilation there.

_42:23_: Quite abstract one, but that's that's just time relation

_42:26_: happening there again.

_42:28_: So the difference between the coordinate time

_42:31_: in what I think was particles moving and the coordinate time

_42:33_: in A-frame of which the particle is not moving

_42:36_: is that

_42:39_: um

_42:42_: And. And what that tells us, turning the handle a bit, is

_42:46_: that you're not,

_42:48_: which is good. X naughty by D tall, which is DT by D Tau is

_42:55_: equal to gamma

_42:58_: and uh

_43:02_: you I

_43:04_: which is DX I. By

_43:08_: the tour

_43:09_: it ends up being gamma

_43:12_: VI and you see I've switched I'm I'm consistent with this

_43:17_: notation I've I've used MU as the index for something which

_43:22_: goes from zero to to 30123, and I've used I Latin index for the

_43:27_: spatial bits. With index goes 123 and so on.

_43:33_: So those are the components of the velocity vector

_43:37_: expect activity for a particle moving at speed V.

_43:41_: There's the components in our frame of of of the philosophy

_43:44_: vector of a particle which is moving with respect to us at

_43:48_: speed V

_43:50_: Umm.

_43:53_: And we could also write that out

_43:56_: as

_43:58_: gamma gamma VX,

_44:01_: Gamma BY gamma

_44:05_: VZ,

_44:10_: which is gamma 1V XVY,

_44:16_: he said.

_44:17_: And it looks like slangy. We will also write that as gamma

_44:20_: one

_44:24_: an underlying.

_44:26_: So this line here is just rewriting this line

_44:31_: make it look like a vector.

_44:33_: This line is just taking the gamma out. The common factor

_44:37_: gamma just number out, and this is just a slightly slangy

_44:40_: rewriting of that to indicate that this is still a four vector

_44:44_: where the three special vectors of written as a vector.

_44:49_: OK, that's good. No, we're making

_44:53_: you were going to OK, I'm going to get drink. One more thing

_44:57_: done

_44:58_: and what we stop our pick up next time in this in this is a a

_45:04_: good bit. In the frame in which

_45:08_: the particle is not moving,

_45:10_: the velocity is 0,

_45:13_: obviously.

_45:14_: So in that frame

_45:16_: you will be equal to gamma will be one, because if V is zero, we

_45:21_: won

_45:23_: zero

_45:24_: and in that frame

_45:26_: the length of the velocity vector,

_45:29_: he says. It's simple. It's one, It's one thing one.

_45:32_: So you don't. You

_45:35_: is equal to 1, so the length of the velocity vector in the frame

_45:38_: of which particle is moving is 1.

_45:41_: But remember that the

_45:45_: length of a vector,

_45:46_: the and the metric are framing variant. And what that means is

_45:51_: that the length of the of the velocity vector

_45:55_: it's invariant

_45:58_: and with the same in all frames. So in any other frame it was

_46:01_: part of moving, the length of the velocity vector in that

_46:04_: frame with those components in that frame will also be one,

_46:08_: so the velocity vector. So you're used to that in 3D space.

_46:12_: Used to the idea that something was moving faster. Has a bigger

_46:16_: velocity vector. It's the velocity vector is longer, V is

_46:19_: longer.

_46:20_: In this context we see that's not the case. The velocity

_46:23_: vector is always the same length,

_46:26_: but in the frame in which the particle is not moving, the

_46:29_: velocity vector is pointing directly along the T axis.

_46:33_: In another frame,

_46:35_: the velocity vector is pointing in a different direction. It's

_46:37_: the same length

_46:38_: and and in these terms, but it's pointing a different direction.

_46:42_: So when you see something moving at roughly 6 speed,

_46:46_: what you're seeing is it's

_46:49_: velocity vector, foreshortened if you like, and it ends up

_46:52_: looking a different shape,

_46:54_: but it's still the same length

_46:57_: and that's a natural place to stop. It will pick up that field