Transcript of gr-l07
==========


_0:10_: Hello everyone and welcome to lecture 7.

_0:13_: I'm fairly sure we are still in good time.

_0:18_: Although we can't hang about

_0:19_: with the rest of the section,

_0:20_: the plan is to is to cover.

_0:27_: If we spend 9 lectures 910 and

_0:30_: 11 on part three, that's good.

_0:32_: So I my hope is to deal with

_0:35_: the rest of of this chapter,

_0:37_: this part in this lecture and and the next.

_0:41_: So unfortunately we are both this is,

_0:46_: this is Scotland, we have sunlight.

_0:49_: Known again from awkward angles and

_0:51_: we don't seem to have blinds on

_0:53_: these on these windows unfortunately,

_0:55_: so you may have to squint but look cool.

_1:00_: It's autumn where we got to last time

_1:05_: when we finished off this section 322.

_1:08_: And these are the things we covered there.

_1:11_: We talked about the essentially we

_1:14_: talked about the covariant derivative in

_1:16_: flat space and how we could define that

_1:19_: and the covariant derivative in flat

_1:22_: space is the answer to the question,

_1:24_: how does this field vary?

_1:28_: As you move across the space,

_1:31_: given that the space is flat and

_1:33_: so the change in the.

_1:36_: We calculate the change in the.

_1:38_: Um.

_1:39_: In the field has to take account of

_1:42_: the fact that the basis vectors will

_1:46_: be changing across will potentially be

_1:48_: changing across the space for the example,

_1:51_: being the motor the motivating example.

_1:54_: Being the vectors the basis vectors in

_1:59_: in the plane in the basis of spherical

_2:01_: Polaris or of of plane polarized.

_2:03_: Where as you as you're aware the

_2:05_: the the the direction of the of the

_2:08_: radial basis vector changes and the

_2:11_: direction and size of the tangential

_2:13_: basis vector changes as you move

_2:15_: across the space.

_2:17_: And we were able to deal with

_2:19_: that with the current derivative.

_2:21_: I said that for every vector V vector

_2:26_: vector field V there's a tensor.

_2:29_: With one rank higher, called Nabla V.

_2:32_: The components of which tell you

_2:34_: how the vector changes.

_2:36_: The vector field changes as

_2:38_: you move across the space.

_2:39_: And we were able to find the

_2:44_: components of that tensor.

_2:47_: With this rather strange notation,

_2:48_: I think that's the last,

_2:50_: basically the last bit

_2:51_: of notational annoyance.

_2:53_: Are we subjecting you to this

_2:56_: VI semi colon J?

_2:57_: Which is the iconology,

_2:59_: which is just the straightforward

_3:01_: derivative of the the.

_3:02_: That's DV,

_3:03_: the derivative of the ith component of

_3:06_: the of the vector with respect to X J

_3:09_: + a term involving this Christoffel

_3:12_: symbol called the social connection,

_3:14_: which is basically the thing that encodes.

_3:18_: The way in which the basis

_3:19_: vectors change across the space.

_3:24_: And I said that though I didn't

_3:27_: go into Labor line detail,

_3:28_: there are corresponding expressions for

_3:30_: the covariant derivative of function,

_3:32_: which turns out to be just the.

_3:35_: Derivative. That the gradient operator

_3:38_: we learned about a few lectures back.

_3:42_: And there's corresponding expression for

_3:44_: the great derivative of one form and

_3:47_: associated ones for higher rank tensors,

_3:49_: which we're not going to go

_3:51_: into because the the key idea,

_3:53_: there's nothing really new there.

_3:55_: The key idea is in this expression here,

_3:58_: and that's what we're actually

_4:00_: we're most often use.

_4:02_: So that's what we got last time.

_4:04_: What we're going to do now is 1 bit

_4:07_: of extra calculation and then move

_4:09_: on to the apparently much more exotic

_4:12_: question of how do we do the same thing.

_4:15_: In a case where the space is curved so

_4:18_: so the basis vectors are changing in a

_4:22_: curved space rather just a flat space.

_4:24_: And we'll discover it actually less

_4:26_: less hard than you might anticipate.

_4:29_: But unlikely. Rather annoying.

_4:30_: I think I'll have to put in some

_4:33_: sort of request for blinds to be.

_4:35_: Fixed in this room.

_4:38_: OK. Any questions about that?

_4:41_: We've got to OK, that was just a revision.

_4:45_: But Papa. OK, I'll go back to.

_4:52_: This.

_4:55_: Now what would you do now is work

_4:57_: out how to differentiate the metric,

_5:00_: and that will be illuminating.

_5:02_: In a way. It's illuminating because

_5:04_: it shows a calculation happening,

_5:06_: and it'll be illuminating

_5:06_: because the result is 1.

_5:08_: We'll briefly use later on.

_5:11_: Is that impossible to see?

_5:15_: Really hard, OK?

_5:19_: OK.

_5:23_: So we have a. So we have a vector field V.

_5:29_: Now as you will recall the the

_5:31_: the metric gives us a way of

_5:33_: associating with any vector.

_5:35_: I want phone. In the very

_5:39_: straightforward fashion. One form is.

_5:45_: Um.

_5:48_: We. We.

_5:52_: Metric G.

_5:57_: With. This vector what are the holes?

_6:01_: One of the slots filled in with the vector

_6:04_: and leaves another vector shaped hole.

_6:06_: The result is different one form,

_6:08_: so that's the, that's the one

_6:11_: form corresponding which is sort

_6:13_: of dual to the vector V and it's

_6:15_: mediated by the the metric.

_6:17_: And as we saw that had a has the effect.

_6:22_: In component form of saying that

_6:24_: the components of this one form

_6:29_: VIAGIG. Fiji. Where?

_6:33_: The component of the metric

_6:36_: of the metric vector.

_6:38_: Are these the components of

_6:39_: the corresponding one form?

_6:40_: Are these now?

_6:42_: What happens if we differentiate?

_6:45_: This one form.

_6:48_: What we'll do is we'll

_6:49_: differentiate the left hand side,

_6:50_: we'll differentiate the right

_6:51_: hand side and see what we get.

_6:56_: Now this is our geometrical equation.

_7:00_: In other words,

_7:01_: it's a coordinate independent equation.

_7:03_: There's no, there's no,

_7:04_: although we can talk about the

_7:06_: coordinates of this equation.

_7:08_: This by itself is a coordinate

_7:10_: independent equation.

_7:11_: So we can calculate with it in any.

_7:14_: Coordinate system we're like.

_7:16_: So we pick a coordinate system in

_7:18_: which the calculation is easy.

_7:20_: Of course you pick,

_7:21_: and we're not doing it here.

_7:23_: Here, anything that you haven't

_7:24_: been taught to do previous

_7:26_: stages in your physics education,

_7:27_: you pick coordinates so that

_7:29_: the calculation is easy.

_7:30_: Here the the coordinates we pick are

_7:32_: going to be Cartesian coordinates,

_7:35_: so we're going to calculate with

_7:37_: this using Cartesian coordinates

_7:38_: and in Cartesian coordinates.

_7:39_: The special thing with Cartesian

_7:41_: coordinates is that the basis

_7:44_: vectors are constant.

_7:45_: Across the whole space.

_7:47_: That's when we recorded creating coordinates.

_7:50_: The X&Y basis vectors are are what

_7:53_: they are across the whole space,

_7:56_: and what that means is that the

_7:59_: christophel symbols for Cartesian

_8:00_: coordinates are all zero.

_8:02_: The Christoffel symbols pick up

_8:04_: the change in the basis vectors

_8:05_: as you move around the space,

_8:07_: so the basis vectors don't change.

_8:08_: The Christoffel symbols are zero,

_8:10_: so conveying differentiation in

_8:13_: these coordinates is trivial.

_8:18_: Um. What this means is that.

_8:21_: Uh. And in other words.

_8:27_: In these coordinates.

_8:29_: Equivalent derivative of the vector V.

_8:33_: Is just. DVI by DXJ. E.

_8:40_: Aye without the corresponding term

_8:46_: which involves differentiation of the.

_8:49_: Basis vectors because

_8:50_: that derivative is 0. OK.

_8:56_: So so you can either think

_8:58_: of this as being the.

_9:02_: The the the the expression for the

_9:04_: current derivative with the covenant

_9:05_: with the Christoffel symbol zero.

_9:07_: Or you can think of as just

_9:10_: libras rule with the second term

_9:12_: which would be IDE IDE I DXG.

_9:14_: Disputing because DVD I DXG.

_9:33_: Because that's zero in Cartesian coordinates.

_9:39_: If we even had, you didn't have that

_9:42_: Shadow Cross. It would be better.

_9:46_: And what that means?

_9:48_: Is that if we ask what is the?

_9:51_: And what happens if we apply?

_9:55_: That vector. THG.

_10:01_: To the. Metric metric. That's.

_10:09_: By two and DVI by DX JEI.

_10:17_: And because the metric is a tensor,

_10:20_: it's linear in uterus arguments,

_10:22_: so that's a DVI by DX JPG.

_10:34_: OK.

_10:37_: Now. This. Thing here.

_10:44_: Is. And. If you think of it.

_10:50_: Well, it has the property

_10:53_: that it that is dual. To the.

_10:57_: Basis vectors EI. Because

_11:03_: GEI. EJ. Is equal to well delta.

_11:10_: IG's if you could do one

_11:12_: when the statement and.

_11:14_: 00 otherwise that that's

_11:15_: we know that to begin with.

_11:16_: In other words,

_11:20_: GEI.

_11:23_: Is equal to.

_11:26_: One of the beaches one forms.

_11:33_: In other words. This expression

_11:37_: here. Um is. DVI by DXJ.

_11:46_: Who written this? Yes.

_11:49_: Omega I is DVI by DX J Omega I.

_11:60_: Summed over I.

_12:02_: Something over I because this is.

_12:07_: The two eyes are both raised to the

_12:10_: instantiation convention doesn't apply,

_12:12_: so I've got to explicitly say

_12:14_: that what we're summing here.

_12:17_: OK, so that looks rather

_12:19_: strange expression. OK.

_12:22_: Now let's look at that.

_12:25_: That's what we've done by.

_12:28_: Especially differentiating

_12:28_: the right hand side of this.

_12:32_: Of of the situation here.

_12:35_: If we now ask how, what happens if

_12:36_: we differentiate the left hand side?

_12:40_: Then what we get. Um.

_12:47_: Is. The level that you.

_12:55_: 32. V tilde. So we're looking at this again.

_13:01_: No difference in the left hand side there.

_13:06_: That's differentiating.

_13:09_: The. I what we got what we got I

_13:14_: because V is our our one form so it

_13:17_: will have some components in the

_13:20_: one form in the one form basis.

_13:27_: Since the. Basis vectors.

_13:29_: Are constant in this basis in

_13:32_: the in this coordinate system.

_13:34_: Then the one form is a constant

_13:36_: in this coordinate system,

_13:37_: so the one form is also.

_13:39_: Do not vary as we move across the space,

_13:44_: so again this ends up being DVI.

_13:48_: By the XG. Omega. Aye with, with.

_13:54_: No D Omega by the XJ term.

_13:59_: Because the basis vectors are.

_14:01_: Constant.

_14:06_: But. In these coordinates. The.

_14:13_: Special thing is that.

_14:15_: The components. No.

_14:20_: I think I have somewhere

_14:23_: explained why this is obvious.

_14:26_: I'm you know I'm have to be

_14:31_: recalled but have thought but

_14:33_: the in these coordinates.

_14:35_: The components of base of

_14:37_: vectors in one forms are this

_14:40_: are equal in these coordinates.

_14:48_: Only in creating coordinates

_14:49_: and what that means is this.

_14:52_: And this. Are equal, so DVI.

_14:56_: Raised by the XG and DVI

_14:59_: lowered by the XG are are equal

_15:02_: so this is equal to this.

_15:05_: And what that tells us.

_15:09_: Is that they are equal in these

_15:12_: in this coordinate system.

_15:14_: But if they're equal as components.

_15:17_: In in one coordinate system.

_15:20_: Then they are equal as tensors

_15:22_: in all coordinate systems.

_15:23_: So two things are equal. In.

_15:26_: One basis. Component by component,

_15:29_: and that's telling you that they

_15:30_: point in the same direction.

_15:32_: They are the same vectors.

_15:34_: And so you've gone from doing

_15:36_: the calculation in a nice easy.

_15:38_: And coordinate system Cartesian

_15:40_: vector Cartesian coordinates.

_15:42_: But it would draw geometrical conclusion

_15:44_: that these two things are equal as vectors.

_15:47_: But that means that they're equal,

_15:49_: independent of the coordinate system.

_15:51_: So we've picked a nice coordinate

_15:53_: system to make the calculation easy.

_15:55_: But still ended up with our

_15:59_: geometrical result which is that the.

_16:02_: Um.

_16:03_: This.

_16:09_: Um. Derivative of the?

_16:15_: Of this one form V. Is equal to.

_16:20_: Um, this thing here? She.

_16:29_: As a geometrical result.

_16:33_: So.

_16:36_: Next right.

_16:41_: 10.2 here. Look of it.

_16:43_: So where do we go next from there?

_16:48_: The um.

_16:56_: The component form of this

_16:57_: is this expression here.

_17:01_: And.

_17:07_: VI is equal to

_17:12_: GIJVG.

_17:14_: And the. Component form of. This expression.

_17:22_: Umm.

_17:25_: Yep. Is VI semi colon G? Is equal to G. IG.

_17:37_: VK. To him I key. They're calling G.

_17:46_: And you may have to, you know,

_17:48_: see that a bit to reassure

_17:49_: yourself that that's the case.

_17:53_: So.

_17:57_: This isn't trivial. We know that

_18:00_: there is a given this tensor V VK

_18:03_: semi colon G we know there's some. A

_18:09_: tensor, which is what you get

_18:13_: when you lower the indexes.

_18:16_: What this what worked out is the chance

_18:18_: that you get when you lower the top

_18:21_: index in that covariant derivative.

_18:23_: The chance you get is this covariant

_18:26_: derivative of the corresponding one form.

_18:28_: OK. That's good.

_18:32_: So what we didn't do is we didn't get that

_18:35_: expression there by differentiating that.

_18:38_: It looks like this expression

_18:40_: is just the derivative of that,

_18:42_: but it's not.

_18:42_: We got that by a different route

_18:44_: this calculation here.

_18:47_: So what do we get?

_18:49_: Who would differentiate this?

_18:52_: That's just life.

_18:53_: That's the rule, really.

_18:54_: So V. I semi colon.

_19:00_: Ugg. Is equal to

_19:05_: GIK semi colon JVJ. Plus.

_19:11_: GIKVK. semi colon. Gee.

_19:14_: And that's just liveness rule.

_19:18_: Like this rule where you you

_19:20_: you differentiate a product

_19:21_: by differentiating one you

_19:23_: learned about in school.

_19:24_: It's the I think,

_19:26_: I think it's all generally.

_19:31_: So if.

_19:36_: So this we obtained by this argument.

_19:38_: This we obtained by differentiating that.

_19:42_: If these should be equal.

_19:46_: That term is the same.

_19:48_: And this term must be 0.

_19:50_: So what we have done is

_19:53_: discover that this that well

_19:55_: this term is going to be 0,

_19:57_: so this term must be 0.

_20:00_: In other words.

_20:05_: GIK, semi colon G. Is equal to 0.

_20:11_: It could be a derivative of the metric.

_20:14_: 0. What does that mean?

_20:18_: If we ask ourselves what? And.

_20:26_: Consider the.

_20:30_: And. Inner product AB. That's GIGA.

_20:40_: BG. And if we differentiate that, so ask.

_20:46_: With the driver of that of that number.

_20:49_: Libraries. Really. Again. GI. G.

_20:56_: So the the the case component of that.

_21:01_: Semi colon key AIBJ difference in that

_21:05_: 1 + g I. GA semi colon K. PG plus G.

_21:16_: IGE. IB. G. key. But if this is 0.

_21:26_: Then that tells us.

_21:28_: That as we move around to the,

_21:30_: the as we move around the space. The.

_21:39_: The way that the inner product varies.

_21:43_: Is purely due to the way that

_21:46_: the to the derivatives of A&B.

_21:49_: Is not due to the inner product is,

_21:53_: which you feel like is the.

_21:59_: The size it describes the the the

_22:01_: the size of this of this object.

_22:03_: So it is a dot A for example,

_22:05_: it would be describing the

_22:07_: side side of the of the vector.

_22:10_: What they're telling us is that as that

_22:13_: varies as you move around the space.

_22:16_: The change in that is

_22:17_: purely just changes in the.

_22:19_: The vector and not changes in the

_22:22_: coordinate in the underlying coordinates.

_22:24_: So something.

_22:25_: So in other words this is telling us.

_22:27_: This is in a sense what gives us

_22:30_: license to think of the metric as

_22:31_: being a measure of the size of the

_22:33_: of of a vector at different points.

_22:35_: Because this is telling us that when

_22:37_: the the the metric between 8:00 or E

_22:39_: changes as you move around is not just

_22:41_: an artifact which of the coordinates

_22:43_: changing under you it's it's it's the,

_22:46_: the, the.

_22:47_: The, the, the,

_22:48_: the vector,

_22:49_: the vector field changing rather than the

_22:51_: an artifact of the coordinate change.

_22:53_: So this is.

_22:54_: This in a sense is what gives us

_22:56_: license to talk about G as a metric as as,

_22:58_: as, as a length,

_22:59_: we and and also via the the dot

_23:01_: product as a way of talking about

_23:03_: the angle between two things.

_23:05_: To that angle is at a well defined thing.

_23:08_: The direct question in the sense

_23:10_: that you say that the derivative

_23:12_: of the number is not.

_23:15_: The derivative of somebody?

_23:16_: No, it's not. Yes, so.

_23:20_: So say that again.

_23:21_: We know that a dot B yeah wouldn't

_23:24_: it be vectors is a number yes.

_23:26_: And then you differentiate the number

_23:28_: but you didn't get 0 differentiated it.

_23:32_: Yes so that it was derivative of a

_23:36_: that would be the length of the vector.

_23:38_: So I as you move around the the space

_23:40_: in different parts of the space this

_23:43_: electric fuel see might not only change

_23:46_: in direction might change the length.

_23:48_: So the derivative of of a dot A

_23:52_: would be the rate the derivative

_23:54_: of the length of that vector.

_23:59_: So it went from here to here.

_24:03_: Then that derivative would be that

_24:05_: that that length will be changing.

_24:07_: And what this is telling you is that

_24:08_: that's because the vector has changed.

_24:09_: Length is not just a thing

_24:11_: about the coordinates.

_24:16_: So that's an important thing and we'll

_24:17_: come and and that I think also shows

_24:20_: that there are two things that's.

_24:23_: That was three things.

_24:25_: It shows that what I said about the vector,

_24:27_: the metric being useful being a length,

_24:31_: the fact that the covariant

_24:33_: derivative of the metric is 0,

_24:36_: and we will pick up later.

_24:37_: But there's also shows the way that

_24:40_: you can do calculations in this.

_24:42_: Sort of context by doing things

_24:45_: like picking the right coordinates.

_24:47_: And this sort of trick that if

_24:50_: you can come up with a geometrical

_24:53_: result in coordinates.

_24:55_: That these two vectors were equal,

_24:57_: then that geometrical result

_24:59_: is coordinate independent.

_25:01_: Because it's no longer just a

_25:04_: coordinate dependent number.

_25:09_: OK Umm any more to say about that?

_25:14_: Ohh yeah yeah the last thing which I

_25:16_: won't work out is I won't go through

_25:18_: the steps for because it's just

_25:20_: rather tedious is but it's it's it's

_25:22_: important but but the derivation is

_25:24_: not is not particularly interesting is

_25:27_: that you can work out that the before.

_25:30_: We were introduced the Christoffel

_25:33_: symbols at the beginning of near

_25:35_: the beginning of this chapter.

_25:37_: I showed how you could work them

_25:40_: out by working out the.

_25:42_: By explicitly working out the way that

_25:46_: the basis vectors in plain Pollers moved,

_25:49_: you're changed as you moved around the space.

_25:52_: I then said, oh,

_25:53_: and these coefficients are called

_25:55_: the Christoffel symbols.

_25:56_: Match these two equations up and

_25:58_: you can work out what the symbols

_26:01_: are for playing pollers.

_26:02_: They're just the components of the of

_26:04_: the of the expression that we got.

_26:06_: So I'll just, I'll just jump back just.

_26:10_: Think about for blank looks there.

_26:13_: Yeah.

_26:16_: And we saw that.

_26:23_: It's not nobody terribly easy there

_26:25_: but but you'll see that in your

_26:27_: notes that this is the way that the

_26:30_: basis vectors of pain pollers change

_26:32_: over the over the over the plane,

_26:34_: and we can with engine identify that

_26:38_: this 1 / r is the Christoffel symbol

_26:43_: gamma R Theta Theta. That one's gamma.

_26:47_: Theta, R, Theta and so on.

_26:49_: So we can just match those up

_26:51_: and discover what the shuffle

_26:53_: symbols are by that process.

_26:55_: But it's also possible to do this rather

_26:58_: more mechanically and discover that the

_27:01_: and I will copy this down because I

_27:03_: don't want to necessarily get it wrong.

_27:06_: This equation says 36 that the the.

_27:11_: Ijk the IJK is at half.

_27:19_: Gil.

_27:21_: GLK. comma key plus.

_27:27_: GKL. G. Minus. GGK.

_27:37_: So that given the derivatives of the metric.

_27:41_: You can just turn the handle and.

_27:45_: To churn out the values of

_27:48_: the Christoffel symbols.

_27:49_: And I'm not going to ask you to remember.

_27:52_: Memorize that.

_27:52_: But you probably will end up memorizing it,

_27:54_: given that you do enough of the exercises.

_27:57_: And that's a nice. Obvious.

_27:60_: I mean, I'm making your promises here,

_28:01_: but that is a nice sort of.

_28:03_: It's a dull but quite a a nicely

_28:06_: contained exam question to

_28:08_: get you to turn that handle in

_28:11_: and and avoid falling asleep.

_28:13_: It's not pedagogically terribly interesting,

_28:16_: but the same as far as the Examiner

_28:18_: and Peggy were interested in

_28:19_: getting some that will,

_28:20_: that will.

_28:23_: Produce our result.

_28:24_: So I I heartily encourage you to look

_28:26_: at the exercises which are covering

_28:28_: that sort of that sort of thing.

_28:30_: And there's lots of them in

_28:32_: the IT would refer to lunch,

_28:33_: but there's several of them in

_28:36_: the exercises for this part.

_28:38_: And it just gives us practice.

_28:41_: Um, so before we go to the next bit,

_28:44_: we're making good time here.

_28:45_: I'm. I'm. It's so much easier

_28:48_: to keep keep your time face

_28:50_: to face than it is in zoom.

_28:52_: It's so much nicer.

_28:55_: Because I can see you go or

_28:56_: and and and and I can see

_28:58_: you and you can see smile.

_28:59_: It's just I can work out.

_29:01_: If things are are.

_29:03_: Keep keeping up on it after

_29:05_: being neurotic and any

_29:07_: questions before go on.

_29:11_: No, OK.

_29:15_: Um.

_29:29_: OK, so that's the thing.

_29:32_: I just wrote down the turn

_29:34_: the handle expression for the

_29:35_: truffle symbols in terms of

_29:36_: the derivatives of the metric.

_29:41_: And those are the key points from

_29:45_: this section. The key thing the

_29:48_: metric tensor this tensor that we've.

_29:51_: That, I said, was somewhat arbitrary.

_29:54_: But because of its of of the way in

_29:57_: which we use the the the the metric,

_29:59_: well, it is arbitrary.

_30:02_: In the sense that.

_30:04_: You pick a metric tensor,

_30:05_: and you pick the shape of the

_30:07_: of of the species describing.

_30:08_: But in mathematical terms

_30:10_: it's somewhat arbitrary.

_30:11_: But because the tensor is what we

_30:13_: the metric tensor is what we use to

_30:16_: map to turn vectors into one forms.

_30:18_: It because of that.

_30:21_: The problem it has the property.

_30:23_: That the, the, the, the.

_30:26_: The convent derivative of the metric

_30:28_: metric is 0 and which means it's

_30:30_: legitimate to use it as a measure of length.

_30:33_: And then this expression here,

_30:35_: which I omit simply because it's

_30:38_: tedious to actually calculate

_30:40_: and it's not interesting.

_30:42_: But the details are on shoots.

_30:43_: Or think in Carol's most you know,

_30:45_: most most fat GR textbooks will

_30:47_: have a derivation of that if you're

_30:49_: interested or don't believe me, OK.

_30:54_: Next.

_30:58_: Umm.

_31:02_: How old are you?

_31:07_: Right. What we now want to do, as I said,

_31:11_: we've now talked a bit about the

_31:14_: convenient derivative in flat space.

_31:17_: Not necessarily Cartesian

_31:19_: coordinates, but flat space.

_31:22_: Meaning one where Euclid's

_31:24_: geometry works and so on.

_31:26_: Or when Minkowski geometry works,

_31:28_: so Minkowski space is also a flat space.

_31:31_: And I I'm I'm here giving a rather

_31:33_: hand waving definition of flat spaces.

_31:36_: We will later discover what you know,

_31:37_: I'm more more precise

_31:38_: definition of our flat pieces,

_31:39_: but I hope you have a a fairly intuitive

_31:42_: notion what a flat space is at this point.

_31:44_: So what we want to do is also

_31:47_: differentiate things in non flat spaces,

_31:49_: for example on the surface of a sphere or.

_31:53_: In the cosmos or something.

_31:55_: So these are not flat spaces.

_31:56_: The nucleus, the parallel axiom nucleus,

_31:60_: doesn't work on a sphere,

_32:01_: and it doesn't work in the cosmos.

_32:05_: So how do we?

_32:06_: Port what we have learned

_32:08_: about conveying differentiation

_32:09_: in flat space into covering

_32:11_: differentiation on a curved space.

_32:14_: What we do is we jump up and down.

_32:18_: Because I said that it's easy. Well,

_32:22_: it's important to pick your coordinates.

_32:24_: If you do a calculation in the

_32:26_: right coordinates, it's easy.

_32:27_: So what you do is you've picked up well.

_32:31_: So, so so how do you change coordinates?

_32:33_: You change coordinates by picking

_32:34_: a Lambda that, that, that,

_32:36_: that this coordinate transformation matrix

_32:38_: that we learned about back in Part 2.

_32:40_: And you have 4 by 4.

_32:42_: By the case of,

_32:43_: you have N by N numbers there,

_32:46_: and as long as the matrix is invertible,

_32:49_: and as long as.

_32:53_: Uh. It's not singular.

_32:55_: And I think the same thing.

_32:57_: And then you have a number of

_32:59_: degrees of freedom so you can pick.

_33:02_: A transformation from whatever

_33:04_: chords you're starting off with.

_33:06_: On your sphere or whatever into

_33:09_: a flat into flat coordinates.

_33:13_: And you can do better.

_33:14_: You can you can pick coordinates

_33:17_: where not only is the.

_33:21_: Is the metric in that new?

_33:26_: Coordinate system diagonal

_33:28_: meaning it's it's flat,

_33:30_: but the derivatives of those of

_33:33_: those components are also zero.

_33:35_: In other words, you can pick coordinates.

_33:39_: In which the metric? Is.

_33:43_: The metric of special activity,

_33:45_: that's the minus plus,

_33:46_: plus plus or plus, minus, minus, minus,

_33:48_: depending what your convention is.

_33:52_: Plus some of which is a

_33:54_: second order in the. Um.

_33:60_: In in, in, in in the in

_34:02_: the component functions.

_34:03_: And that's very easy to do.

_34:04_: You just do that.

_34:07_: You're in freefall, your metric diagonal.

_34:11_: Diagonalizing your personal metric is easy.

_34:13_: You do it every time you jump up and down,

_34:14_: given you jump up and down with the

_34:16_: late more often than once a day,

_34:18_: which is very important to do.

_34:20_: Um. And what that means is we

_34:24_: can start to talk about um.

_34:27_: So I I said we can't do this.

_34:30_: In a moment I'm going to explain

_34:31_: why that's useful to do.

_34:34_: Because now we're going to talk

_34:36_: about differentiation and why

_34:38_: it's hard in a curved space.

_34:41_: Ohh, that, that's, this is called

_34:43_: the local flatness theorem.

_34:44_: The theorem that you can do this is

_34:45_: called the local flatness theorem.

_34:47_: And the the the frame that you get,

_34:49_: the coordinate system that you get when you

_34:51_: do this is called the local and national

_34:52_: frame because it's a it's a natural frame.

_34:54_: It's in freefall.

_34:55_: It corresponds to the natural

_34:57_: frames of special relativity.

_34:58_: So we know what that is.

_34:60_: We the local national frame of

_35:02_: freefall is something we understand,

_35:04_: and it's the freefall frame that

_35:06_: we're talking about back in part one.

_35:08_: OK, so it's a good, nice,

_35:09_: it's a nice good frame.

_35:10_: And this in this sense is one

_35:12_: reason why we made a fuss about the

_35:14_: local national frame in part one,

_35:16_: because it is the framework

_35:18_: which has these nice properties.

_35:19_: That it's it's just special relativity.

_35:25_: No. You know how to differentiate things.

_35:29_: You learned that in school?

_35:33_: That I trust that looks completely familiar,

_35:36_: right? What's happening here? Is that you?

_35:40_: Take up your difference your function.

_35:43_: You ask what's the difference

_35:44_: between the function at the point X,

_35:46_: the function a little bit

_35:47_: further along the opposite.

_35:49_: Divide that by how far you've gone

_35:51_: and take the limit as that goes to 0.

_35:55_: Now that's nice and well defined.

_35:58_: But it depends. On. A minus sign.

_36:04_: And it depends on division.

_36:06_: In other words, that depends on

_36:08_: you being able to say what a

_36:10_: function at this point minus a

_36:12_: function at different point means.

_36:14_: It depends on it being possible

_36:16_: to divide that by a number.

_36:18_: And neither of those things have we

_36:21_: got yet when we're differentiating our

_36:23_: vector field as we move across a space.

_36:26_: So what we have to do is define

_36:30_: what subtraction means for our

_36:32_: vector moving around space,

_36:34_: and define what this divided

_36:36_: by each could correspond to.

_36:40_: So that's what we're doing.

_36:42_: OK, it's just that those two

_36:44_: steps are a little bit hard.

_36:45_: What the the second step is quite easy

_36:47_: once you've got the first step right.

_36:52_: And I know there are there are multiple ways

_36:54_: of talking about the derivative of our.

_36:59_: Of of a vector.

_36:60_: And the we were going to do it

_37:02_: is the the Cuban derivative,

_37:04_: in particular the derivative with

_37:06_: the metric connection so-called.

_37:08_: There are things called leader of this,

_37:10_: there are things called flow derivatives,

_37:11_: are things called 2 forms, and so on.

_37:14_: So there's more than one way of doing this,

_37:16_: but this is the way that's useful in in

_37:19_: GR most useful, most immediately useful,

_37:21_: most commonly useful in GR.

_37:24_: And it relies on the notion

_37:26_: of parallel transport.

_37:28_: And parallel transport.

_37:31_: Age where? I take a vector.

_37:35_: And I see. May I borrow this?

_37:40_: Another vector is parallel to it.

_37:42_: If we put them nearby together,

_37:44_: the fairly obvious parallel

_37:46_: mean that's parallel.

_37:47_: Those two vectors are parallel.

_37:48_: Thank you.

_37:50_: Now if we see that red parallel there.

_37:53_: And to there, and to there,

_37:55_: and to there, and to there,

_37:56_: to there and to there.

_37:57_: Incrementally we can infinitesimally.

_37:59_: Rather, we can end up with a

_38:02_: definition thereby of how you

_38:04_: transport a vector from here.

_38:06_: To a collector somewhere else which is

_38:10_: parallel in the sense I think it is

_38:13_: parallel at each infinitesimal separation.

_38:16_: And what that means?

_38:19_: Is that?

_38:20_: So something like this.

_38:22_: So we can move a vector along a path,

_38:25_: keeping it parallel at each point,

_38:26_: and we end up with these two

_38:28_: vectors being parallel even though

_38:30_: they're separated.

_38:31_: And that matters because.

_38:36_: If you remember when we defined

_38:38_: vectors on the manifold.

_38:41_: I said that the death the definition

_38:44_: of the tangent vector in the manifold.

_38:47_: The tangent vector space on the

_38:49_: manifold was in a space which was

_38:52_: attached to the manifold at one point.

_38:54_: And that space? That's the tangent plane.

_38:57_: TP at M is the tangent plane of the

_38:60_: of the manifold M at the point P.

_39:02_: And the tangent plane TQM which is

_39:05_: tangent plane over the manifold am at

_39:07_: the point Q has nothing to do with this.

_39:10_: Which are different species,

_39:12_: so you can't subtract that vector

_39:15_: there from that vector there.

_39:17_: Which is what we want to do. Super.

_39:20_: What we can do is take a vector here.

_39:24_: And parallel transported along this

_39:28_: curve Lambda. Until we get back,

_39:30_: until we get it into for some Lambda

_39:32_: that goes through both points until

_39:34_: we get it into the same space here.

_39:38_: At that point there are two

_39:40_: vectors in the same space.

_39:41_: So we can subtract.

_39:45_: So we've done the first part of

_39:47_: what we wanted to do with our

_39:49_: definition, differentiation.

_39:50_: We've discovered a way of subtracting

_39:54_: a vector here. Of tracting.

_39:59_: The vector here from the vector here,

_40:01_: which is a little bit further along the path.

_40:05_: And we get a vector as the answer.

_40:08_: We know how far we've had to transport it,

_40:10_: what the difference is in in T so

_40:12_: we've got our H in the bottom line.

_40:14_: We can divide that that vector there by

_40:18_: the amount we've had to do to do this.

_40:21_: At that point we are talking

_40:23_: about the derivative. And and.

_40:29_: So that means that we have are able to

_40:32_: to find the the two steps involved in

_40:35_: defining your derivative in a curved space.

_40:38_: OK, because this is only depends on

_40:41_: the presence of this. Curve Lambda.

_40:48_: No. You know, we haven't said very much here.

_40:52_: Because what we have been vague about.

_40:55_: Is. What this? What we mean by parallel,

_41:02_: I mean what I said was I hope persuasive

_41:04_: that thing to parallel if they are, you know,

_41:07_: if when they're nearby they have the,

_41:09_: they're obviously parallel.

_41:11_: But that is, that's that's that statement.

_41:14_: Has an obviously in it,

_41:16_: so it's not really a mathematical statement.

_41:18_: But we can be precise about what

_41:20_: we mean by parallel by saying, OK,

_41:23_: two things are parallel if in the

_41:26_: local national frame. They are.

_41:28_: Have the same coordinates, so two to so,

_41:31_: so 2 vectors in the local national frame.

_41:34_: Are parallel if the other thing components.

_41:37_: That gives us a definition of parallelism,

_41:39_: a place definition of parallelism.

_41:41_: Which is enough to let us define this.

_41:47_: This. The the the difference.

_41:51_: You let us go from here to here,

_41:53_: and that, and and.

_41:56_: That means that we end up defining.

_41:59_: Great differentiation. In.

_42:03_: And. In the coordinate system

_42:06_: of the local national frame.

_42:07_: But the local metal frame is flat.

_42:12_: And that means that we know how to do that.

_42:15_: So we've gone from defining this

_42:19_: in the recovery space to doing the

_42:24_: same parallel transport thing in.

_42:26_: Our flat space. Which we can do.

_42:30_: So we just have to import what we learned

_42:33_: in the previous section to this case.

_42:39_: And that means that the.

_42:43_: 100 hundred freeze this,

_42:46_: and that means that. Well,

_42:49_: that means we're actually finished.

_42:51_: Because we don't have to do

_42:53_: anything else. We can just.

_42:57_: Do I have a slide for that?

_43:02_: The the the definition of the the the.

_43:05_: Greater TV? In. Approach piece

_43:10_: is again just fee I semi colon G.

_43:16_: Umm.

_43:19_: Now, I've I haven't gone through

_43:21_: every step in that whole derivation

_43:24_: because there's more than one way of

_43:26_: of you creeping up on that conclusion.

_43:29_: But I think that I hope

_43:31_: that the the point is clear.

_43:34_: First of all, the parallel transport is

_43:35_: key and in order to give a definition

_43:37_: of what we mean by parallel transport,

_43:38_: we can use the local national frame.

_43:40_: Those two steps are the steps that

_43:42_: allow us to instance step back to the

_43:45_: flat space that we understand already.

_43:51_: But I think it's important to do you know

_43:53_: that you appreciate where those what

_43:54_: what the inputs to that conclusion are.

_43:59_: Um. And and I'm not gonna go through it, but.

_44:10_: You know there are two

_44:13_: further remarks there that.

_44:15_: In the local national frame.

_44:18_: The is it. It's flat,

_44:20_: so the console symbols are zero,

_44:24_: so in the local national frame.

_44:26_: The item Icon G is just VI comma G.

_44:33_: In the local initial. But that's.

_44:41_: That's true for anything,

_44:43_: so it's true for IgG semi colon key.

_44:48_: Well, in this case be IgG IgG comma. Key.

_44:56_: Which because of the what we worked out.

_45:04_: Yep, which because of the definition of the.

_45:11_: Local of the local flatness theorem.

_45:13_: We were able to see not only that

_45:15_: that the vector that the metric

_45:17_: in the new coordinates was zero,

_45:18_: but the derivatives of the metric

_45:20_: in these coordinates was also zero.

_45:22_: So we discovered that this is 0.

_45:26_: In the in the local national frame.

_45:30_: But. We've got tensor equation here

_45:34_: we've got GIIG semi colon K = 0.

_45:39_: That's true as a component calculation.

_45:42_: But seeing this tensor,

_45:44_: the remainder of this tensor is 0.

_45:48_: Is. A geometrical statement.

_45:51_: So although we worked out in these

_45:53_: coordinates the nice easy coordinates,

_45:55_: it's true in general.

_45:57_: So if again worked out.

_45:59_: That the.

_46:01_: And that GGK.

_46:06_: Key is 0. In.

_46:11_: In any coordinates.

_46:14_: So this is one of two comical,

_46:17_: cynical rules.

_46:18_: The other one is insensible.

_46:20_: Interesting. This is a mathematical trick.

_46:23_: We've done the calculation in A-frame,

_46:25_: in which the calculation is easy.

_46:27_: Discovered that by doing so we

_46:30_: have a geometrical statement.

_46:32_: Realize that that is therefore true

_46:34_: in not a coordinate dependent thing,

_46:36_: but our geometrical thing,

_46:37_: and so we're able to go from that

_46:39_: in one frame a special frame,

_46:41_: to the same statement in any frame

_46:43_: as a geometrical statement.

_46:49_: Umm. Right. In the last

_46:52_: couple of moments, I'll just.

_46:58_: No, but it's more sensible if

_47:00_: we start thinking 2.4 next time,

_47:02_: and I'll aim to get through the the

_47:04_: the remainder of this part in the

_47:06_: next lecture which we like to eat,

_47:07_: and that will give us lectures