Transcript of gr-l04
==========


_0:07_: Excellent. Lecture 4 and before we get

_0:12_: going in this lecture, we'll finish off.

_0:15_: Part 2 is is the plan and that

_0:17_: shouldn't be too difficult.

_0:19_: The first half of this is going to be a bit

_0:22_: more of a remark through notation I'm afraid.

_0:24_: The second-half should be more examples

_0:26_: and should be a little bit more conceptual,

_0:28_: so basis for the first half,

_0:30_: but it should be over soon

_0:32_: before we get going.

_0:33_: I mentioned a couple of things on the Moodle.

_0:36_: And one of which is that,

_0:39_: oh, that the microphone on.

_0:42_: That, but that's.

_0:46_: I'm putting a button called mute,

_0:47_: mute, mute, OK. One thing is that.

_0:53_: On the middle page there is a link to.

_0:60_: Oh, I can't get to that

_1:01_: piece from here anyway.

_1:05_: A couple of changes of things

_1:07_: as I mentioned in the message.

_1:09_: That said, via Moodle,

_1:10_: there is a podcast of the audio

_1:12_: recordings of the lectures which

_1:13_: should be findable at the end of the

_1:16_: link that was in that modal message.

_1:19_: So that is a bit of an experiment,

_1:22_: both technical pedagogically and technically.

_1:24_: So any comments or thoughts with that?

_1:26_: I am very keen to hear them

_1:30_: other things that are here.

_1:33_: Are though. Let audio podcast.

_1:36_: But we can't get there from this computer.

_1:40_: That thing, they're all notes with HTML.

_1:42_: That's our bundle of the entire

_1:44_: collection of notes format as HTML.

_1:46_: It's sort of readable and might

_1:48_: be useful in some circumstances.

_1:50_: It's a lot less pretty than the PDF version,

_1:52_: but it might be of some interest.

_1:54_: The panel I mentioned a moment.

_1:56_: Now, the stream channel, which,

_1:59_: well, let's not worry about that,

_2:01_: which I think I pointed to you before.

_2:04_: It has a couple of things.

_2:06_: It will have the brief 5 minute overviews

_2:10_: of the various parts will appear there.

_2:13_: Part 2 is there.

_2:15_: Part Three will appear.

_2:16_: Part Four will appear also

_2:18_: as of 10 minutes ago.

_2:20_: It has copies of the first couple of my

_2:26_: lectures from autumn 2020, so they're

_2:29_: the recordings of the zoom lectures.

_2:31_: The the first ones I did so

_2:33_: the little bit Robbie but.

_2:35_: I'll, I'll, I'll,

_2:37_: I thought.

_2:39_: I've been two minds about putting

_2:41_: those up and because.

_2:45_: I was in two minds. I'll put them up.

_2:50_: The deduction is not that.

_2:51_: The that these lectures are are not

_2:53_: important, and I'll put them up

_2:55_: with a bit of a delay in any case,

_2:57_: but there are an extra resource

_2:59_: to use as you think appropriate.

_3:02_: You're all have whole new learning

_3:04_: strategies over the years and

_3:06_: I'm sure you'll be sensible

_3:07_: about these are the imaginative,

_3:09_: so I will put them up there.

_3:11_: Put these two year old lectures up as.

_3:16_: In time. And

_3:21_: I will point you again to the lecture

_3:23_: notes folder in the middle which has the

_3:26_: notes and the screen versions and the

_3:28_: slides of the parts that have completed.

_3:31_: So after today I'll put up

_3:34_: the Part 2 slides PDF.

_3:36_: There's nothing in the slides as

_3:38_: you're as you're aware there isn't

_3:39_: in the notes apart from the sort of

_3:42_: answers to the quick questions things,

_3:43_: but they will appear there in good time.

_3:47_: I think I, yes,

_3:48_: I think I have to date there.

_3:50_: Regarding the padlet,

_3:51_: I see there are some good questions there,

_3:55_: a couple of which I think I've answered.

_3:56_: As I say, if you think I haven't answered,

_3:58_: if I haven't answered that then change the

_4:01_: color back to White and I'll see it again.

_4:04_: And I've just entered to the of resolutions.

_4:07_: I I took compendium version of all

_4:08_: the solutions and the compendium

_4:09_: of of notes on the solutions.

_4:11_: I'll release that either at the end of

_4:12_: the semester or beginning of the next one.

_4:14_: So it's a good time for revision but

_4:17_: appears after after the lectures

_4:18_: and some of the slides.

_4:20_: And I answered these two questions as

_4:23_: they change the color back to white.

_4:25_: If you think that's if you disagree

_4:27_: and I haven't had a look at these

_4:29_: other to quick look look at this one.

_4:34_: Someone asking, I'll post a note on this.

_4:37_: Is there any significance

_4:39_: to this order being swapped?

_4:41_: No, and there isn't,

_4:42_: because the metric is our symmetric tensor.

_4:46_: So in general these two things

_4:49_: would be importantly different.

_4:51_: Because the metric is by

_4:53_: hypothesis asymmetric tensor,

_4:54_: it doesn't matter. OK, so.

_4:58_: The question was we why is the

_5:03_: basis vector being dropped in the

_5:05_: 1st place here and 2nd place here?

_5:06_: There's no significance to that,

_5:08_: OK, it's just.

_5:11_: I want to change my mind to the two cases and

_5:13_: it doesn't matter because Jesus symmetric.

_5:14_: So I'll make a note of that later

_5:17_: and I'll get around to this.

_5:19_: Point. In a moment, right,

_5:23_: we shall proceed.

_5:24_: Are there any questions about either

_5:26_: what I've said there or about burning

_5:29_: questions which haven't been in the

_5:31_: middle about what we've covered last time?

_5:33_: No.

_5:35_: Good.

_5:35_: OK, so the plan now is for me to get rid of.

_5:42_: This and.

_5:48_: Get these again.

_5:54_: And.

_6:00_: I think we've got 2.

_6:06_: But there. Didn't we?

_6:11_: Yes, we got a bit there.

_6:14_: So the. The.

_6:24_: Right, we go to the end of section 226.

_6:27_: So what we discussed last time then,

_6:30_: was your first look at the,

_6:34_: well, the fiddly technology of.

_6:37_: Components and all the things

_6:40_: that algebra you can now do with

_6:42_: the components of 10,001 forms.

_6:44_: And now we're going to go on

_6:48_: to the important question of.

_6:51_: Last time I talked about the.

_6:54_: We talked repeatedly obesity vectors and

_6:57_: the basis one forms that were due to them,

_7:01_: but there's nothing special about

_7:02_: any basis that we pick and that's

_7:05_: the that that goes back to the

_7:07_: principle of covariance that

_7:09_: mentioned in the first lecture.

_7:11_: What that principle is,

_7:12_: is statement that there was nothing

_7:15_: special about any basis that you pick it.

_7:17_: There's nothing special but this

_7:18_: initial frame, or that one,

_7:20_: or or or any or any other.

_7:22_: What that means in turn is is that

_7:24_: if you change your mind about

_7:26_: what is a good basic set of basis

_7:28_: basis vectors or corresponding

_7:30_: to good set of basis one forms.

_7:32_: Then you have to be able to go from 1.

_7:35_: Basis set to another and that's we're

_7:37_: going to talk about in the first half,

_7:39_: I hope all of this lecture.

_7:42_: So what that means is some

_7:46_: more bit more notation and.

_7:49_: What's the best way of writing this?

_7:52_: I'll leave that up there.

_7:55_: New projector document camera.

_7:59_: To that point.

_8:04_: Four which for you?

_8:08_: OK. So we have a basis.

_8:14_: Our vector E could you

_8:16_: is is the late annoying.

_8:18_: Can you read that? OK, OK,

_8:20_: which we're gonna write as a IDE.

_8:24_: Aye with acid 4 the implied

_8:26_: some of the dummy index I.

_8:33_: And the components there are

_8:35_: just the. Set of of numbers.

_8:37_: And would you get when you?

_8:42_: Apply the vector A regarded as

_8:46_: a function. To the one form,

_8:48_: the one of the basis one forms.

_8:54_: OK. Now let's change our mind about

_8:59_: what the basis vectors are and

_9:02_: rather than the basis vectors EI.

_9:05_: Well, state have the beta vectors EI bar.

_9:09_: Now that looks a completely

_9:11_: demented notation.

_9:12_: What, not least because it's slightly

_9:15_: fiddly to write what I've done is I put a.

_9:19_: I I'm your bar over the index. Here.

_9:25_: Now there are other ways of writing of there.

_9:28_: Are there other notations?

_9:29_: Are notional alternatives here and different

_9:31_: books sometimes used to different things,

_9:33_: some are prime.

_9:35_: Some use hat some,

_9:37_: but the general consensus is that the.

_9:41_: The the, the the indicator good over

_9:43_: the index and not over as you might

_9:45_: expect over the basis vector itself.

_9:47_: So we don't, so we don't write.

_9:50_: E primed I, for example.

_9:54_: We don't write that, although that that

_9:56_: might be what you might first guess.

_9:59_: And the reason for this?

_10:00_: Well, I hope becomes slightly clear.

_10:03_: So.

_10:05_: That means that just as we can

_10:07_: write a in terms of the.

_10:10_: Vector EI we can also write.

_10:13_: It in terms of basis vectors.

_10:17_: AI bar EEI bar where there's a again an

_10:21_: applied some over the dummy index I bar.

_10:24_: OK. So that's just as good.

_10:31_: Business as before and just as before the.

_10:35_: The components AI bar are the

_10:38_: vector A applied to Omega I bar,

_10:41_: where the Omega I bar is the are

_10:44_: the basis one forms dual to the

_10:48_: basis vectors EI bar. OK. So I.

_10:52_: All I've done this is isn't is notation.

_10:56_: I'm just saying that this is

_10:57_: what I'd changing your mind

_10:58_: looks like in this notation.

_11:05_: No. And again.

_11:11_: So this is the component of

_11:15_: the vector in the. Alternative.

_11:21_: Vicious. But we can expand a.

_11:25_: Her relic? We can,

_11:26_: for example, write A is AI.

_11:30_: EI. When we go. High bar.

_11:36_: I think that that that light is annoying

_11:39_: and we'll see if it might not be possible.

_11:42_: However, that important to think.

_11:47_: I don't think it's possible

_11:49_: to make that as visible.

_11:51_: What I could do is put it on that

_11:54_: other one that's probably smarter.

_11:56_: Do both. Not look at camera.

_12:04_: Should be eligible somewhere.

_12:05_: Ah, right, that's the problem.

_12:07_: There's light, OK,

_12:09_: you just have to deal with. So.

_12:16_: We don't know what that is.

_12:17_: We can't just decide what that,

_12:20_: that, that, that basis vector

_12:22_: applied to that basis of one form is.

_12:24_: So we'll see. Instead it is.

_12:27_: It's some set of numbers Lambda I bar I.

_12:35_: And that Lambda, which is a matrix,

_12:37_: is a collection of numbers.

_12:40_: Of and by numbers.

_12:42_: That matrix Lambda is what characterizes the

_12:46_: relationship between one basis and the other.

_12:50_: And it is just this bit of vector EI

_12:53_: applied to the base of one form EI bar.

_12:56_: OK. So that's your transformation

_12:59_: matrix that takes you from 1.

_13:01_: Vicious to the other.

_13:04_: So, so that that.

_13:06_: But it's probably not a.

_13:08_: This is a long way to something

_13:10_: you possibly have thought of

_13:13_: before you know the context.

_13:15_: Umm.

_13:19_: Yeah, we've written the matrix at that. And.

_13:26_: In the same way. The.

_13:31_: And. Let's see the.

_13:34_: The components of our.

_13:38_: One form P will be P.

_13:46_: Applies to E. Ibar. And.

_13:53_: Again we can write P is equal to Pi Omega.

_13:59_: IE I bar which is equal to Lambda I

_14:05_: I bar Pi and you can see this all

_14:09_: sort of works in terms of the of the.

_14:13_: Commission convention because the the

_14:15_: pattern of of of raising load indexes

_14:17_: hangs together and we end up with,

_14:20_: as here, one eye bar in the bottom and

_14:24_: a pair of of of recent Lord eyes on

_14:27_: the on the right hand side. So. Um.

_14:42_: So I will. Go I I sort of want to see.

_14:48_: Have a look through these yourself.

_14:54_: I'm indecision.

_14:60_: Yeah, let's go through this. I'm, I'm.

_15:04_: I'm slightly nervous here because the

_15:06_: it's always very easy to get these

_15:08_: indexes wrong when so doing this as

_15:10_: it were alive in front of people, but.

_15:14_: The next step? Is to. Uh.

_15:24_: Yeah, I do it that way.

_15:30_: We want to look at the.

_15:37_: I don't make this too complicated,

_15:38_: but I don't wanna make it trivial and

_15:41_: let's write down this Ohga. I e.g.

_15:49_: Which is, so this is the,

_15:51_: the basis one for this happening

_15:53_: just in one in one frame and

_15:56_: we'll write that down as delta I.

_15:59_: Edgy, so I'm on 2 minutes

_16:01_: going through the step by step.

_16:03_: I think I will because it's useful

_16:06_: as an illustration of the handle

_16:09_: turning of of of the relevant algebra.

_16:13_: So we have an expression like that,

_16:16_: but we can also write each of those.

_16:20_: And that says and this is what

_16:22_: written down here is just the

_16:24_: thing we decided on last time that

_16:26_: the that the basis one forms are

_16:27_: going to be dual to the basis of

_16:30_: vectors in this very specific sense

_16:33_: that we will one apply to E1 is 1

_16:35_: when we go one apply to any other

_16:37_: basis basis vector is is 0.

_16:39_: So that's that's all that we're

_16:41_: seeing there that's that's the

_16:43_: definition of of of dullness.

_16:44_: Well, then write down these two things

_16:46_: in terms of the expressions we have here.

_16:48_: That's Lambda I.

_16:50_: I bar Omega. High bar.

_16:55_: Applied to Lambda, G Bar G. E.g bar.

_17:03_: I can't obviously write.

_17:06_: I I buy I here because I've already

_17:08_: used up I already in the sum,

_17:10_: so I've gotta pick another dummy index.

_17:13_: Tensor application is linear in its argument,

_17:16_: so we can take that Lambda out.

_17:19_: Get Lambda I I bar. Lambda G by G.

_17:25_: Omega. I bar E. G bar.

_17:30_: But we are also going to presume that the in

_17:36_: the change basis the the the basis one forms.

_17:41_: I bar are also due to the basis vectors

_17:45_: in that basis, so that will be.

_17:49_: Lambda I I bar Lambda G Bar G Delta

_17:55_: I bar G bar and we do that sum.

_17:59_: You end up with Lambda I bar I. And.

_18:08_: Lambda I bar G.

_18:13_: Which is the unit, the identity matrix.

_18:18_: Remember, Delta is the components

_18:20_: of the the matrix with the the

_18:22_: with all ones on the diagonal.

_18:25_: We're just telling you that Lambda

_18:27_: I I I bar a Lambda I bar I.

_18:31_: Are. Matrix inverses.

_18:34_: So this is an example.

_18:37_: I think I earlier said that if you

_18:39_: look at the components of a tensor.

_18:42_: Then they will are always

_18:44_: representable at a matrix.

_18:46_: So you just because you've got an array of

_18:49_: of end by end by whatever whatever numbers.

_18:52_: But there are the.

_18:53_: The converse is not true.

_18:54_: This is an example.

_18:56_: Lambda is an example of our.

_18:59_: An array of numbers,

_19:01_: a matrix which does not have

_19:04_: a corresponding tensor.

_19:05_: Because there isn't a geometrical object

_19:08_: which those lambdas are the components of.

_19:12_: OK, that's a key thing.

_19:15_: So the the the matrix of components of.

_19:19_: Or the the the vector of the

_19:21_: the the column of of components

_19:23_: of a vector or one form or the

_19:26_: matrix of components of a tensor

_19:28_: are not just a random matrix,

_19:30_: they are linked to the geometrical

_19:32_: object single geometrical object

_19:34_: which is frame independent.

_19:35_: But Lambda is not a tensor and that's

_19:39_: why just just parenthetically we write

_19:41_: the indexes just one above the other.

_19:44_: If you remember when we write the

_19:47_: the matrix the components of our.

_19:50_: Are two taken frontier.

_19:51_: We carefully staggered them

_19:53_: because they're referring to

_19:55_: different arguments of the tensor.

_19:57_: In this case, this isn't attention.

_19:59_: There's no need to stagger the indexes.

_20:00_: And then we write the one above the other.

_20:03_: And in this for something like that,

_20:05_: you can sort of start to see

_20:08_: why the bar goes over the.

_20:10_: Index rather than the rather the vector.

_20:15_: The vector,

_20:16_: because it lets us keep track

_20:18_: of all of which way round.

_20:20_: Or which we up if you're like

_20:24_: the the matrix Lambda is.

_20:26_: And that,

_20:27_: but you couldn't do if if if the

_20:29_: dictation were elsewhere the question, sorry.

_20:33_: 2nd.

_20:36_: This should be I, bar J,

_20:37_: bar J yeah, yeah, this one,

_20:40_: that's I I bar I bar G because we have.

_20:45_: Summed over the of the J bar.

_20:49_: So that is if we do the sum over J bar.

_20:54_: This term here will be 0.

_20:58_: Except where G bar is equal to I bar.

_21:01_: So the only time that survived out of

_21:03_: that out of that sum is the term where

_21:05_: G bar is equal to I is equal to I bar.

_21:08_: So that's how we get that.

_21:09_: Thank you. Yes, so it's important.

_21:11_: So part of the point of of of me writing

_21:14_: this out long hand is to go through exactly

_21:16_: that the step by step quite slowly. So.

_21:19_: So yes we are doing that sum over J bar

_21:22_: and that's why that deals with there

_21:24_: you know changes that G bar into an I bar.

_21:27_: So we're left with a sum I bar I.

_21:31_: Do W indexes were left with

_21:33_: an I raised eye a, lowered G?

_21:35_: Just as a relieved I Lord G.

_21:38_: So everything matches. OK.

_21:41_: If the the the indexes you got on one side

_21:45_: do much indexes you got on the other side,

_21:48_: you have done it wrong.

_21:50_: OK, OK, what you've done wrong,

_21:52_: but you've done something wrong.

_21:53_: So that's always a check.

_21:55_: Can be at every step.

_21:57_: Through this calculation, we could check.

_22:01_: Does the does the have one

_22:03_: raised 11 lower G duplicated.

_22:05_: I bar Dublin G bar.

_22:06_: That's good,

_22:07_: that line is good and so on.

_22:09_: So you can check each line against

_22:11_: that with that Santa check.

_22:15_: Umm.

_22:18_: And there's.

_22:25_: Right and and. I'm not going to go

_22:29_: back and forth through the the notes,

_22:31_: but by going through a similar sort of

_22:34_: calculation you can discover that this.

_22:37_: Lambda matrix is transformation matrix.

_22:39_: It doesn't just transform components,

_22:42_: it also ends up being the.

_22:47_: How you turn?

_22:54_: 1. Basis. Into another. And Omega.

_23:02_: I bar equal to Lambda I bar I.

_23:08_: Now you may think, Oh my God,

_23:10_: that's an awful lot of things to memorize.

_23:12_: You really don't need to memorize

_23:14_: anything because once you once you've

_23:16_: got the idea that there's a an,

_23:17_: in this case an E Lambda and a knee.

_23:21_: There's only one way the indexes can go.

_23:25_: So you don't have to memorize this,

_23:27_: you just have to know that you know

_23:30_: the general idea and and and the

_23:32_: index is fit in it only one way.

_23:33_: So there's no memorization here.

_23:36_: But there's a a useful exercise.

_23:41_: It might be.

_23:44_: What exercise 214 to 217,

_23:47_: which invites you to form a I I

_23:49_: just a table of all of these things.

_23:52_: It's not terribly exciting,

_23:53_: excited, but it it helps you to

_23:55_: drill that a little bit more.

_23:57_: So that is useful.

_23:59_: And this sort of thing also is why

_24:01_: I think it's useful to use bars

_24:03_: rather than hats or or primes,

_24:06_: because it's just slightly easier

_24:08_: to write them neatly, I find,

_24:11_: rather than having dashes or or or

_24:14_: circumflex or all over the place,

_24:16_: and then later on we start introducing

_24:18_: punctuation to this notation,

_24:19_: things would get a bit hairy

_24:22_: unless we end up with a very neat

_24:24_: handwriting at the end of this course.

_24:26_: Well, you're very neat handwriting

_24:28_: over a few letters your your queues

_24:30_: might look terrible, but your eyes,

_24:32_: G's and keys will be perfect.

_24:34_: Umm.

_24:37_: And and this this is these are

_24:42_: both generalizable, it turns out,

_24:44_: so that something like TI bar.

_24:47_: AG Bar Key bar is equal to Lambda I bar

_24:52_: I Lambda G bar G Lambda K key bar T.

_24:59_: IG. OK. And and again,

_25:03_: I didn't have to remember.

_25:04_: I didn't memorize anything there.

_25:05_: I just remembered the pattern.

_25:07_: There's one Lambda per index,

_25:08_: and there's only one way

_25:10_: that the index is fit in.

_25:14_: Umm.

_25:20_: But, but boom, there's a few other remarks

_25:22_: which in at the end of that section which

_25:25_: I don't think it's useful to belabor.

_25:27_: But in the in the last half hour I

_25:30_: will going to move on a bit but first.

_25:33_: What questions have you about that?

_25:38_: I question the. See.

_25:43_: To the.

_25:50_: Here.

_25:53_: Yeah.

_25:56_: Yes, because I think that's a very good

_26:01_: point that's useful to highlight that

_26:03_: these because they're matrices and

_26:05_: because that's just a matrix of numbers,

_26:08_: these are all numbers. They're all,

_26:10_: they're all things on the real line,

_26:12_: so you can swap them over on all you

_26:15_: like and because the the the sum.

_26:19_: So. So one of the advantages of this

_26:21_: component notation is that it means you

_26:23_: can swap things arbitrarily, because

_26:24_: they're just numbers and numbers commute.

_26:26_: So when we later come on to talk

_26:29_: about differentiating things.

_26:30_: Differential operators don't commute

_26:32_: with numbers, but numbers do.

_26:34_: So is that what you meant?

_26:37_: You. Yeah, yeah.

_26:38_: So yes, you could if you wanted

_26:40_: to write those another order.

_26:41_: Weird, but you're allowed.

_26:43_: OK, your question there.

_26:45_: Swap them around without having

_26:47_: to to change the index.

_26:49_: Indeed, absolutely so I could if I wanted

_26:51_: to if I was really confused myself, right?

_26:55_: T like Lambda I bar.

_26:60_: Key Lambda G bar M.

_27:07_: Sorry. If we bar Lambda K.

_27:14_: Bar and Lambda. And.

_27:19_: G Yeah, G. Cheap bar equals T.

_27:27_: KM. Gee or something?

_27:30_: Now that would be perverse,

_27:32_: but but there's nothing stopping

_27:36_: me doing that because all that I've

_27:38_: done there is I've changed is.

_27:40_: I've changed my mind about the dummy indexes.

_27:43_: And to a stupid thing.

_27:45_: But I'm an allowed allowable thing, OK?

_27:50_: So, so I've just to to check one dummy.

_27:54_: Dummy, yes.

_27:54_: So I've I've still got a IIRG bar.

_27:59_: So I've I've I've given

_27:60_: myself too much leeway there,

_28:01_: so sorry, that should be G bar.

_28:04_: And that's that has to be a key bar.

_28:07_: So there's a a matching.

_28:10_: I bought gbar keybar.

_28:13_: But

_28:16_: KM&G in this case I don't mean

_28:18_: indexes, so so so disappear.

_28:22_: Please.

_28:26_: Yeah.

_28:29_: And second position then we

_28:31_: would have to put K bar.

_28:37_: No. So if it's what those

_28:40_: dreaming no cause, because still.

_28:43_: The, the, the, the the sum would be.

_28:46_: OK, let's write that down.

_28:47_: Like it's just just just just show it.

_28:51_: So if we wrote a Lambda G bar

_28:55_: G Lambda I bar I Lambda key

_29:01_: bar tigg key that you mean?

_29:05_: Yeah. Yeah. And the the again,

_29:08_: it's still just a sum over.

_29:12_: I G&K and because they're just numbers.

_29:17_: And each element in that sum,

_29:20_: all that will have happened in the

_29:22_: there's quite long sum that is an end

_29:24_: by end by end sum is that the the

_29:26_: real numbers in the product of each of

_29:29_: those terms will be in a different order.

_29:31_: So. So yes,

_29:33_: these lambdas are just numbers.

_29:36_: How to decide the order of the T?

_29:40_: Order of the right size,

_29:43_: up or down, right.

_29:46_: In both these cases, this is a 2.

_29:50_: There's this appears to be a A21 tensor.

_29:53_: There's 10s, it's a tensor

_29:54_: which takes I've just you.

_29:55_: I just picked a a rank to to illustrate this.

_29:58_: So it's a A21 tensor.

_29:60_: So there will be a two ways and one Lord.

_30:04_: Indexes. So once,

_30:05_: but once you've got and and and it's,

_30:08_: it's a 21 tensor here, it's a two.

_30:11_: It's the same true one tensor here.

_30:13_: So it'll have the same pattern of indexes.

_30:16_: And once we've got that pattern,

_30:18_: then the pattern of of lambdas

_30:21_: inside in this transformation follow.

_30:27_: The last line, that's right.

_30:29_: It is last line below.

_30:32_: So it is 21 tensor.

_30:35_: It's so true. 21 tensor, yes.

_30:37_: The two up and one down, one down.

_30:40_: Yeah, yeah. You have to show up all three.

_30:43_: Or can you just swap high bar to I

_30:46_: and then just give J bar and J bar?

_30:48_: Yes you would have to because the.

_30:53_: What what this set of numbers is?

_30:56_: Is the components of that tensor

_30:58_: in that transformed basis.

_31:00_: So. In each, so it is. So

_31:09_: TIGI bar G Bar K bar is equal to T Omega.

_31:13_: I bar. When we get a cheap bar.

_31:19_: Ek bar and it wouldn't make any sense,

_31:21_: really. To to, yeah,

_31:25_: if you were to to drop in one forms

_31:31_: and vectors from two different bases,

_31:34_: you get a number, but it wouldn't mean any.

_31:37_: They are just. The different forms

_31:40_: of the tensor to write down.

_31:43_: These. This is one form in Japan.

_31:47_: The below is another form,

_31:51_: so this tends to this tensor.

_31:53_: Here is A21 tensor, yes?

_31:55_: So it's the same.

_31:59_: The same with yes,

_32:01_: it seemed interesting.

_32:02_: It's just the truth.

_32:04_: It seemed unsure. Yeah,

_32:08_: just inform the same tension,

_32:09_: OK, and one better move on.

_32:12_: But one last question there.

_32:17_: Yep.

_32:20_: It equals the components of tensors, yeah,

_32:22_: so equals the components of a tensor, yes.

_32:26_: So and so each of these is a matrix.

_32:31_: That's a matrix which is the the

_32:33_: the matrix components of a tensor.

_32:35_: And similarly that's a metric which

_32:37_: is the component of tender and so,

_32:40_: so this is our.

_32:42_: One tensor is equal to N by N by N.

_32:47_: Terms which are a number times

_32:49_: the components of a tensor.

_32:52_: Is that what you were asking on the next,

_32:55_: but you just got the three members?

_32:59_: So that that's not that.

_33:01_: That's not supposed to be.

_33:03_: Or is it? Yeah, yes.

_33:06_: So not the not equal sign there,

_33:08_: but spotted well someone,

_33:10_: someone is paying close attention.

_33:12_: OK. That is the all of the components,

_33:20_: all the components gymnastics

_33:22_: that we were introducing.

_33:24_: So the last bit is slightly less.

_33:28_: These were heavy and just a

_33:30_: few more examples of BC's and

_33:32_: transformations and spaces.

_33:35_: So the first example of a space

_33:37_: in which we can talk these

_33:40_: things is flat Cartesian space.

_33:42_: Now it's flat in the sense that

_33:44_: I think do I define that here?

_33:47_: I think there's.

_33:51_: Just flight, welcome back to

_33:52_: flat in the morning. And you're

_33:53_: familiar with flat Cartesian space.

_33:56_: Flight Euclidean space if we were flat

_33:59_: Euclidean space with a Cartesian basis.

_34:02_: So Euclidean space is the space

_34:04_: we're familiar with, but if we're,

_34:07_: Pythagoras theorem works.

_34:08_: The Cartesian basis is the X&Y basis,

_34:12_: so so things like are given.

_34:15_: The basis vectors are.

_34:19_: Orthogonal to each other.

_34:21_: No, we and the unit and the other

_34:23_: unit length although right now

_34:25_: because well up to the point where

_34:28_: we define a a metric on that space.

_34:30_: We can't talk about length or about angles.

_34:33_: But you know you you have

_34:34_: a metric in your head.

_34:35_: Pythagoras theorem is the

_34:36_: definition of a metric. It is.

_34:38_: It's how you turn direction.

_34:41_: Directions into into distances.

_34:45_: So in Floaties,

_34:48_: because he's in space, we have.

_34:52_: Umm.

_34:58_: EX. 1 E y = e two and this is

_35:05_: a good point to say that I

_35:08_: will sometimes swap between.

_35:10_: And numbering the basis vectors

_35:12_: and giving them, you know,

_35:14_: I'll say more pneumonic. Remove.

_35:16_: There we can sort of sum over these things.

_35:19_: We can't remove these,

_35:20_: but if if we're different specific things,

_35:22_: it's useful to write things like that, so.

_35:26_: Our vector in fact you clean space

_35:29_: or in any thing A1E1 plus A2E2 and

_35:34_: which we can say right as a ***

_35:39_: just like the informal notation.

_35:43_: And that's something you learned

_35:44_: about in secondary school?

_35:47_: So that's there's nothing exotic there.

_35:50_: So this is a very,

_35:51_: a very long way to come back to.

_35:53_: Some of you learn about school,

_35:55_: but there's nothing,

_35:56_: there's nothing extra at this point.

_35:58_: Um. So what are the?

_36:03_: The one forms in this space.

_36:06_: Well, there's no,

_36:07_: as we said earlier that we that we there's

_36:10_: no constraint about what the one forms are,

_36:13_: but we can choose them so that

_36:16_: the contract to form the direct

_36:19_: delta function to the so the basis,

_36:22_: the basis one form #1 contracted

_36:26_: with basis vector one is 1.

_36:29_: And the zero contracted with other ones.

_36:31_: And what do the components that look like?

_36:33_: They look exactly the same as the vectors.

_36:36_: So in.

_36:37_: Flat clean space the one forms.

_36:40_: When you turn the handle and find with

_36:42_: root like look exactly like the vectors.

_36:44_: You can't tell them apart and that is why

_36:46_: you never had to learn about them before.

_36:49_: Because in the sense you've always been,

_36:51_: you're dealing with one form.

_36:52_: In fact, you could in space,

_36:54_: but you didn't know it because they looked

_36:56_: at it indistinguishable from vectors.

_36:58_: So if if you want to think of it that way,

_37:01_: you could say that row vectors

_37:03_: in the example you used earlier

_37:05_: are are the one forms of flat.

_37:07_: You could use space,

_37:08_: so you'll be using them all the time,

_37:09_: but you never had to had to care.

_37:13_: Similarly if you.

_37:17_: If you have continued mechanics

_37:18_: in in previous years,

_37:19_: you learned at the National

_37:21_: Center or the strain tensor,

_37:23_: you never had to worry about raise lowered

_37:27_: indexes because the didn't matter.

_37:29_: There's no difference between the reason

_37:31_: Lord indexes if you like in Euclidean space,

_37:33_: the the components the one are the same,

_37:35_: the transformation between them is just the

_37:37_: right delta rather than the more complicated.

_37:39_: Not the right delta,

_37:41_: the chronicle delta.

_37:43_: And.

_37:45_: OK,

_37:45_: let mustn't get bogged down.

_37:50_: And our metric in this space is

_37:53_: just G has components. GIJ equals.

_38:02_: You know that that's.

_38:04_: Here there there's no sums

_38:06_: in the nose imply summations.

_38:08_: Here our metric in.

_38:11_: The for the space is just.

_38:17_: The 1001. You're like, which ends up with

_38:22_: when we apply that to. And. A vector.

_38:30_: We get GIG.

_38:36_: AIAG. Picking up what we did last time.

_38:40_: And if we do those sums. Actually.

_38:45_: I = 1, I equals and so on.

_38:47_: We get a 1E1 plus E2E2 which

_38:53_: is equal to a 1 ^2 plus.

_38:57_: You 2 squared, which is Pythagoras's theorem.

_39:01_: So deciding that this is a

_39:04_: defining metric this way.

_39:07_: Is equivalent to seeing the Pythagorean

_39:10_: Theorem works and the and the.

_39:12_: The length squared of this vector

_39:14_: A is just it's X component squared

_39:17_: because it's white component square.

_39:25_: Um. Ohh yeah. And and and what and and

_39:29_: it's is is this fact that means that

_39:31_: when you raise the index or lower the

_39:34_: index of of of a we we we raise the

_39:36_: indexes of a a vector in this space.

_39:38_: What you get is. The same number.

_39:41_: In other words, vectors and and

_39:44_: one forms have equal components.

_39:46_: In space. And the other points are.

_39:54_: I'll go through the polar coordinates,

_39:56_: but fairly quickly just

_39:57_: because I want just to to.

_39:59_: I'll draw your attention to that section.

_40:01_: It is the same set of ideas,

_40:04_: but with another case that you're

_40:07_: familiar with Paul coordinates,

_40:09_: but which is slightly less

_40:11_: trivial than the the Euclidean

_40:13_: space with Cartesian coordinates.

_40:15_: And you can again discover what the.

_40:20_: The components of this transformation matrix.

_40:24_: We might even have that in our.

_40:30_: And that's.

_40:32_: OK, I'll go through this very quickly.

_40:34_: And so the, the, the basis,

_40:37_: the basis vectors of polar coordinates.

_40:41_: Are just a transformation away from the

_40:43_: basis vectors of Cartesian coordinates.

_40:45_: So say E1 and E2 are the

_40:48_: the X&Y basis vectors.

_40:50_: They're familiar with the.

_40:53_: Basis vector in polar coordinates

_40:55_: is that very obvious transformation

_40:58_: away from that, the basis.

_40:59_: The basis vector, the tangential one,

_41:01_: is a similar one.

_41:04_: That are there is not

_41:05_: what you've seen before.

_41:07_: Usually when you've seen this

_41:09_: transformation written down before

_41:10_: in what implicitly they're scaled

_41:12_: so that that are is not there.

_41:16_: And that is what makes the one you're

_41:19_: more familiar with, the basis vector,

_41:22_: the basis vectors for polar coordinates.

_41:24_: It makes them unit vectors.

_41:27_: So these are not unit vectors.

_41:29_: These are these are natural and

_41:31_: different sense, so that's not a typo.

_41:33_: That is the sort of natural thing.

_41:35_: It's context, and nobody would into that.

_41:37_: But the point is that that here,

_41:39_: this is a concrete example of

_41:41_: a change of basis. And.

_41:43_: Boom, boom, boom, boom.

_41:45_: I've components of the the Lambda matrix,

_41:48_: the transformation matrix Lambda.

_41:50_: So that that's.

_41:51_: So that's just.

_41:53_: The transmission matrix Lambda applied

_41:56_: to the pair of basis vectors E1 and E2.

_42:01_: And I'll leave you for that section

_42:05_: slightly more slowly in a moment.

_42:08_: And it's worth and and we can point out that.

_42:15_: Um.

_42:18_: I'll let you look at that after I

_42:20_: put the slides up in a moment and

_42:22_: the the metric of polar coordinates.

_42:32_: G is equal to 100. R ^2.

_42:38_: Which I mentioned just to show that it's

_42:41_: not the the the diagonal unit matrix of

_42:44_: the metric of in Cartesian coordinates.

_42:47_: That's again, that's all in the notes, so.

_42:51_: And. I'm going to skip over.

_42:58_: I'm going to skip over taking 233

_43:00_: because although it's not false,

_43:01_: it's it's a potential little

_43:04_: confusing and go to another very

_43:07_: important special example where the.

_43:16_: It's Minkowski space where the metric.

_43:20_: And you do the traditional thing of

_43:23_: rating the components in mikovsky

_43:25_: space with Greek letters. And.

_43:29_: The metric is also traditionally

_43:31_: referred with water rather

_43:33_: than G is the diagonal matrix.

_43:35_: And. Made 1111.

_43:42_: Umm.

_43:45_: And each a here is a matrix with

_43:49_: a particular constant components.

_43:50_: It's not, it's not the,

_43:51_: they're not component of a tensor.

_43:56_: The vectors in this space are a.

_44:02_: A equals to a.

_44:06_: MUEMU. And we can write the.

_44:14_: Really. I. And the the. Umm.

_44:21_: Yeah, the metric applied to

_44:25_: two vectors GABG mu nu. AM UB.

_44:30_: New which will be equal to.

_44:37_: MUBMU, which were equal

_44:39_: to minus. Oh, thank you.

_44:44_: 0B0 plus A1B1 plus A2B2 plus A3B3 where

_44:52_: I am sticking with the convention

_44:55_: that in Minkowski space the basis

_44:58_: vectors are numbered 0123. The indexes.

_45:04_: Index is run over 0123 and there's

_45:07_: it's a four dimensional space.

_45:09_: And So what you what we have got there?

_45:13_: Is the inner product in Minkowski space

_45:16_: the inner product of special relativity

_45:18_: that you may recall from the last time

_45:21_: that you studied special relativity?

_45:23_: So this is a prompt,

_45:24_: a hint to perhaps drift back to

_45:27_: those notes from two years ago

_45:30_: and remind yourself a little bit

_45:32_: of what was the question.

_45:35_: Using it.

_45:37_: Than in.

_45:43_: Because it's arbitrary. The signature. Why?

_45:51_: In second year we use the metric

_45:53_: which was plus, minus, minus, minus.

_45:57_: And if you make that the signature,

_45:60_: then the signature of 1 +,

_46:01_: -, -, 1 -, 1 -, 1 -, 2,

_46:03_: the signature it will for because it will

_46:06_: always be either minus two or minus or +2,

_46:09_: depending on your on your convention

_46:11_: and a number of other equations

_46:12_: you change in turn now.

_46:16_: I prefer when you talk with special

_46:18_: activity to use the signature minus two,

_46:21_: because then the interval

_46:23_: is the same as proper time.

_46:26_: It is more conventional in GR.

_46:31_: And to use this, the signature the

_46:34_: opposite way around so that um. The the.

_46:39_: And spatial sector has the plus,

_46:42_: plus, plus.

_46:42_: So and that's really just a matter

_46:44_: of taste some extent sort of taste

_46:46_: and tradition you know we so it

_46:48_: would perfectly reasonable to

_46:50_: introduce special activity with with

_46:51_: that metric but it is arbitrary we

_46:54_: were undergoing nothing changes.

_46:57_: So with in 30 seconds I've questioned yes.

_47:02_: The maintenance.

_47:03_: Or diagonal, so that that's the

_47:06_: expression diagonal mobile.

_47:07_: So it's a matrix which which is is 0

_47:10_: except along the along that diagonal.

_47:13_: OK. And? Um.

_47:17_: Blah blah and the transformation matrix,

_47:21_: which takes you from one basis

_47:24_: vector with that basis vectors in.

_47:26_: Because his space.

_47:28_: To another basis vector.

_47:31_: Instead of vectors in Minkowski space

_47:34_: is this transformation matrix here,

_47:36_: which you may be familiar with

_47:38_: as Lorentz transformation.

_47:39_: So all the right transformation is

_47:42_: is just how you get from one basis

_47:44_: basis set attached to the station

_47:46_: platform to another basis set

_47:48_: attached to a moving a moving object,

_47:50_: a moving train.

_47:52_: That's what I transformation is.

_47:55_: It's a basis,

_47:55_: a change of basis and the and the

_47:57_: point of all of this positivity and

_47:59_: yeah is that the base is the physics

_48:01_: doesn't change when you do that.

_48:05_: And that in fairly decent time is is us.

_48:10_: There's a few extra marks in in in in

_48:12_: section 2.4 we talk about coordinates,

_48:14_: bases, and just clear bits of terminology.

_48:17_: It would be good to have a look

_48:18_: at that section just to get

_48:20_: your head straight around those.

_48:21_: But that's I've done with with Part 2.

_48:23_: So we'll go into Part 3 next time,

_48:25_: which is next Wednesday.