Transcript of gr-l08
==========


_0:09_: Well, this is exciting.

_0:10_: A whole new, a new, whole new place.

_0:13_: Are you all. I'm not going to

_0:14_: be galloping around the place,

_0:16_: so I think it's really important that

_0:18_: you can see me and see a screen so

_0:21_: we adjust yourself as appropriate.

_0:24_: This is lecture 8 and is basically going

_0:27_: to be the last of the of of the the,

_0:31_: the, the maths bits.

_0:33_: We're going to finish off Part 3 today

_0:36_: and move on to physics next time.

_0:39_: Hooray.

_0:40_: And now there's quite a lot to get

_0:42_: through and this is quite an intricate one.

_0:44_: I don't think it's terribly deep,

_0:46_: but it is quite intricate,

_0:47_: so I propose to get on with

_0:50_: things without too much I do.

_0:54_: I can't think of any announcements,

_0:56_: but are there any questions to do with?

_0:58_: Stuff.

_1:01_: I thought they were in here from now on.

_1:04_: To have left other room behind and can you

_1:08_: hear me OK at that at the back end there?

_1:12_: No, I'm seeing thumbs up in a sort of

_1:15_: middle I I should aim to project suitably.

_1:18_: I think that I think this is

_1:19_: actually a designed lecture theatre.

_1:20_: So I think this will probably have

_1:23_: substantially better acoustics

_1:24_: than the rooms we've been in,

_1:25_: than the room we've been in so far.

_1:28_: Actually the poor astronomy too,

_1:30_: because they've been going from room to

_1:31_: room to room somebody with a days notice.

_1:33_: So you have, you know,

_1:35_: whatever the acoustics of,

_1:37_: of OF446 or 466, it could have been worse.

_1:42_: OK, where we got to last time?

_1:45_: Was talking about. Quite often.

_1:53_: We defined differentiation.

_1:57_: And discovered that the.

_2:02_: With this definition of differentiation, the.

_2:08_: The metric tensor had the property that

_2:10_: it's convenient derivative was zero,

_2:12_: and I mentioned that partly

_2:14_: because it's interesting,

_2:16_: partly because it justifies the our

_2:19_: ascription of the idea of the metric

_2:23_: providing a length which we draw true

_2:26_: for granted if that property that makes

_2:28_: that a reasonable thing to do to to take

_2:31_: the metric to be a definition of length,

_2:33_: and also because working that out was a a

_2:36_: useful way of using some of the mathematical.

_2:38_: Technology we had built up up to that point.

_2:43_: So the a couple of different payoffs,

_2:45_: you're distinct payoffs for that?

_2:50_: For that, now, the next thing

_2:52_: we talk about is a crucial idea,

_2:55_: the notion of geodesics.

_2:57_: A geodesic, as we'll discover,

_2:59_: is the next best is not the next best thing.

_3:01_: It is what it is.

_3:02_: A straight line is in a flat space,

_3:04_: and the thing that most natural corresponds

_3:07_: to a straight line in a curved space is

_3:10_: we have to talk about what autism is.

_3:12_: And to do that we have to 1st we

_3:14_: have to go back to the idea of what

_3:17_: a parallel transport was.

_3:18_: And at what happens if you parallel

_3:21_: transport a vector along a curve?

_3:24_: So imagine those.

_3:28_: Let's take first of all.

_3:32_: A vector field which is.

_3:34_: That's what's at each point in space.

_3:37_: There is a vector.

_3:38_: These both these solid arrows

_3:41_: defined that there's there's a AV,

_3:43_: let's say at each point in the space.

_3:46_: Now let's have a curve.

_3:48_: A curve there.

_3:49_: And that curve will have a a a

_3:52_: function Lambda of T or whatever it is.

_3:56_: So as we advertise T,

_3:58_: we move through the space along

_3:59_: the path in a particular way,

_4:01_: parameterized by the particular

_4:02_: functional form of Lambda that we choose.

_4:05_: Remember these things be a path,

_4:06_: a set of points,

_4:08_: and a curve at one of the most

_4:10_: multiple possible ways you could

_4:13_: parameterize that set of of points

_4:15_: as a function of a parameter T.

_4:18_: And as we learned at the beginning of,

_4:20_: at the beginning of this part.

_4:22_: If you have that structure.

_4:25_: Then you can define a vector

_4:29_: as the directional derivative.

_4:31_: Or the derivative as you.

_4:34_: As you vary the curve parameter, let's say T.

_4:39_: So that you can define the

_4:40_: notion of tangent vectors.

_4:41_: And this is a curve where the have

_4:45_: just tangent vectors which vary

_4:47_: as the as the curve moves about.

_4:50_: Now it's clear.

_4:51_: But as you go along this curve.

_4:56_: The vector field. Changes.

_4:60_: Very obviously.

_5:02_: And so,

_5:03_: so these.

_5:06_: Or or or.

_5:09_: More specifically,

_5:10_: if you parallel transport this

_5:12_: initial vector along this curve,

_5:14_: then it's then you you deviate from the

_5:18_: direction that you you you you you start off.

_5:22_: The curve, if you like, turns a corner.

_5:24_: That curve bends.

_5:26_: It bends in the sense that. And.

_5:32_: Um, do I have a diagram of this?

_5:39_: It bands in the sense that.

_5:42_: If you had a curve.

_5:45_: Like a little of that and you.

_5:49_: Parallel transported

_5:50_: along a curve like that.

_5:59_: Could I lock the focus of that?

_6:04_: What you focus.

_6:08_: Which books are good?

_6:10_: So this curve bends in the

_6:13_: sense it deviates from the. The.

_6:17_: Set of parallel transported digital

_6:20_: vectors here with this curve doesn't,

_6:23_: so we already have a sort of intrinsic

_6:26_: notion of what a straight line is.

_6:28_: Just from that picture,

_6:29_: if what we're doing is we are

_6:32_: parallel transporting this

_6:34_: tangent vector along the curve,

_6:37_: and it stays tangent to the

_6:39_: curve in that nice straight

_6:40_: line parallel transporting this

_6:42_: tangent vector along the curve,

_6:44_: it does not remain tangent to the curve.

_6:48_: That curve bends in a in an

_6:51_: entirely coordinate free way.

_6:53_: It depends on our definition of the

_6:54_: Korean derivative and that's and

_6:56_: that's of the parallel transport,

_6:57_: but it but given that it's a,

_6:60_: it's a bending curve.

_7:04_: And a curve which.

_7:07_: Doesn't do that like this.

_7:09_: It's a curve, which when you.

_7:15_: And parallel transport to

_7:17_: that curve along the curve.

_7:19_: So that's the derivative of the

_7:23_: that that that this tensor.

_7:26_: Is the derivative of the.

_7:29_: Field. Of. You.

_7:36_: Along the. Direction of the curve

_7:39_: if that C is parallel. Then that.

_7:45_: Directional derivative will remain zero.

_7:48_: That means a couple of goals to see PIN but.

_7:52_: What I've written down there is

_7:55_: the sort of mathematical version

_7:57_: of this idea of the vector,

_7:60_: not the parallel transport.

_8:02_: The tangent vector being parallel

_8:03_: transported in such a way that

_8:05_: it remains a tangent vector.

_8:08_: Right. No, that's very pretty.

_8:13_: It it it's. Four symbols,

_8:15_: and it means that, and I'm saying I'm

_8:17_: telling you it means a huge amount,

_8:18_: but it's not something you can very

_8:20_: straightforwardly calculate with.

_8:21_: So how do you calculate with that?

_8:22_: How do you find a curve which

_8:25_: satisfies that equation?

_8:28_: And here we get to safely integrate that.

_8:31_: That microphone is annoying me.

_8:37_: But you know we have.

_8:41_: Turning it down, OK. And.

_8:46_: OK, so I'm going to have to refer to this,

_8:50_: so don't make any index mistakes.

_8:59_: We remember that you is equal

_9:02_: to something like UJEJ.

_9:05_: The usual fashion. So the.

_9:12_: Just going to check.

_9:16_: And I'm going to remind ourselves that this.

_9:23_: Tensor. Remember that this nabla

_9:25_: you is a tensor A11 tensor,

_9:28_: in other words, something which takes.

_9:31_: Are. Vector shaped argument.

_9:35_: And one form shaped argument,

_9:37_: but we write it in an odd way.

_9:39_: So that the.

_9:43_: So the vector.

_9:47_: Argument. We rate down.

_9:49_: And here for for notation like for

_9:51_: reasons of notational convenience.

_9:54_: There's nothing deep there,

_9:55_: it's just handy.

_9:57_: So that which means that this.

_10:03_: Expression here nabla U.

_10:07_: You is really just nabla you with. The.

_10:16_: One of the. With the vector

_10:20_: argument prefilled in.

_10:21_: But since that's the argument to a tensor,

_10:24_: it's linear in that argument, so nabla.

_10:28_: You you will be equal to.

_10:31_: Nabla. You KEK. You.

_10:39_: Which that being linear in that argument,

_10:42_: that pops out to be now you. OK. And?

_10:46_: Nabla and remember he K you and remember

_10:49_: the notation shortcut that rather than

_10:53_: writing this double subscript here,

_10:56_: we we we we end up writing that as you.

_10:58_: Just. I think that's you, I said.

_11:02_: Sorry, that's you. You K now blah.

_11:07_: Key. You. Equals 0.

_11:10_: So, so this is just a sequence of.

_11:13_: There's just a bit of notational

_11:15_: trickery happening here.

_11:16_: There's nothing terribly deep. OK.

_11:22_: But we have an expression for this.

_11:24_: This expression here we we we,

_11:26_: we, we, we, we saw what that was

_11:29_: in the last lecture but one.

_11:33_: To that expression there.

_11:36_: And the danger?

_11:36_: So she makes sure we get the equations right.

_11:39_: It's going to be I.

_11:44_: Yeah. Great, you. G Nabla, G.

_11:49_: You will be equal to. Eugene GUI.

_11:57_: J. The. Aye. OK. Just recalling the

_12:05_: component version of the covenant

_12:07_: derivative that we saw last time.

_12:11_: And. That means we would be equal to

_12:14_: recalling the expansion of that UG.

_12:19_: Uh, you I comma. Gee.

_12:26_: Plus UG. You. Key comma

_12:36_: IGK.

_12:38_: Equals 0.

_12:41_: So again, we're just using the expression

_12:44_: for this this set of components that

_12:47_: were on last time, no time before last.

_12:52_: No. And. Well, good.

_12:55_: Well, haven't said any very

_12:56_: much about the, the, the, the,

_12:58_: the the curve that we're that this

_12:60_: curve we're talking about here.

_13:01_: We were rather that this curve here,

_13:04_: but it'll be this curve here.

_13:08_: Will be. Lambda T. For some,

_13:13_: for some suitable universe suitable

_13:16_: carefully some suitable curve Lambda.

_13:20_: And what we learned before is

_13:23_: that this tangent vector is.

_13:27_: The operator. DDT that will end

_13:29_: at the beginning of this part.

_13:31_: That was our definition of vectors

_13:34_: in this context. And so if we then.

_13:41_: Which means that this this UG.

_13:44_: The JTH component of that vector.

_13:50_: Is going to be you applied to. The XG. That.

_14:00_: The gradient of one of the of the.

_14:05_: That's the basis.

_14:08_: One form corresponding to

_14:10_: the basis vectors without

_14:11_: the component basis vectors.

_14:14_: Which equals DXG by DT.

_14:22_: Just the derivative of

_14:24_: that component along the.

_14:27_: So so so this is this is

_14:30_: fairly directly saying.

_14:32_: Given a curve.

_14:36_: You know that, that, that,

_14:37_: that, that line there, the.

_14:39_: The of the fact is just how much that.

_14:42_: The, the, the, the X or the Y

_14:43_: or the whatever component varies

_14:45_: as you move along that curve.

_14:50_: And that means that looking at

_14:52_: this other term, you I comma G.

_14:56_: Is just D by DX J. And.

_15:07_: DXI. ID T.

_15:14_: And uh. That means.

_15:21_: That. This expression here. Is. DX.

_15:31_: JPITT. And.

_15:41_: DX by DX.

_15:45_: G. Yeah. DX I by DT plus gamma I JK.

_15:57_: DX J by DT. We just completed

_16:03_: the XK by DT equals 0.

_16:08_: And that um. Combination.

_16:14_: There is incredibly deep ODT.

_16:17_: DX. I by DT plus gamma

_16:25_: IGKEDXJ by DT DX K by DT equals 0.

_16:33_: So we have turned this.

_16:38_: Elegant, but rather.

_16:40_: Impenetrable equation,

_16:42_: which we jumped out from considering what

_16:45_: the what a straight line consists of.

_16:48_: We've turned that into a second order

_16:52_: differential equation in the functions

_16:54_: in the coordinate functions X of of T.

_16:60_: And that's a second order

_17:02_: of differential equation.

_17:03_: So from the theory of 2nd

_17:05_: order differential equations,

_17:06_: you can discover that that

_17:08_: will have a solution.

_17:09_: And that solution is the the an

_17:14_: expression for the path followed

_17:16_: by that geodesic or by by the

_17:19_: by our our path which which

_17:21_: satisfies this property.

_17:23_: In the coordinates X of TXI of T.

_17:34_: I don't hear you worried faces

_17:37_: particularly, but OK. Umm.

_17:43_: And a bit bit bit bit a remark.

_17:45_: I, I, I, I'd, I'd said that this curve

_17:49_: Lambda was a suitable curve and it's not

_17:53_: an arbitrary curve that will do that.

_17:57_: Subset of curves which will form

_18:00_: suitable solutions to that,

_18:02_: but once you have a solution Lambda of T.

_18:09_: If Lambda of T. And.

_18:15_: If Lambda 2 is a geodesic then.

_18:19_: That means that Lambda of a T + B. Is.

_18:28_: It's very hard to write to that

_18:31_: person's angle. And that is.

_18:34_: Up and our parameter T which has that

_18:37_: property is known as an affine parameter.

_18:40_: And what that affine parameter

_18:41_: is doing it well what we've seen

_18:44_: that it is affine saying that

_18:45_: this this set of rescaling of that

_18:48_: parameter what that's that's.

_18:51_: Telling you is that you can rescale and

_18:55_: and shift the curve the parameters of

_18:58_: your geodesics more or less at will.

_19:02_: Which makes sense because if

_19:04_: you measure time in seconds.

_19:06_: And and and you have a a duty

_19:09_: that goes through that that that

_19:10_: describes the fall of a of a ball.

_19:13_: You can also whatever that that

_19:15_: that equation of motion is.

_19:16_: You can also rescale it and talk about it

_19:18_: not in in seconds since since midnight,

_19:21_: but in ours since 2:00 o'clock for example.

_19:24_: So you can change the,

_19:25_: you can rescale the the the parameter,

_19:28_: in this case time,

_19:30_: and you can shift the origin of arbitrarily.

_19:34_: And and that is and.

_19:36_: And one way of seeing that well

_19:38_: known we have think of talking about

_19:40_: that is that affine parameters are

_19:42_: defined so that motion looks simple.

_19:43_: If you were to decide that.

_19:46_: Part of the talk about seconds,

_19:48_: I'm going to write down my geodesic

_19:51_: equation in terms of second squared.

_19:53_: You silly,

_19:54_: because that would make the your

_19:55_: duties a really complicated.

_19:57_: Your expression for the parabola,

_19:60_: it would make you the parameter

_20:02_: rewritten in terms of of of

_20:04_: second squared from midnight.

_20:06_: It's going to be a mess.

_20:08_: Motion looks simple in the when

_20:12_: you're using the when you pick up a.

_20:16_: A form for the that your geodesic which.

_20:21_: It's not an affine parameter.

_20:22_: I'm saying to go around circles here,

_20:24_: but but there there's a.

_20:26_: There are more upper stigmatic

_20:28_: way of of saying about that,

_20:30_: but the the point,

_20:31_: the point of saying it is just to

_20:34_: mention this word affine and to trying

_20:37_: to link the that mathematical property,

_20:40_: that of those are a family of

_20:43_: solutions to the duties equation

_20:46_: to something more more physical.

_20:49_: Umm.

_20:54_: That is what it will have to see,

_20:56_: but for the moment about geodesics

_20:58_: is that if anything we can add.

_21:01_: Or. OK, then let's move on,

_21:03_: because now we come to the as were the

_21:06_: main event talking about coverture. Uhm.

_21:11_: And I think it's important

_21:12_: to have a clear idea of.

_21:17_: OK, another parameter is the time coordinate,

_21:19_: some inertial system.

_21:20_: And remember I said by national

_21:23_: system I meant as a system in which,

_21:26_: well for example you jumping up and down

_21:28_: a system in which Newton's laws work.

_21:30_: A system in freefall is international

_21:32_: system and then I think parameter,

_21:35_: the parameter of a geodesic is

_21:37_: a time parameter in in that and

_21:39_: we'll come back to that notion

_21:41_: of of that particular statement

_21:43_: of this implicitly later on.

_21:47_: Um given key points,

_21:50_: we can divide duties equation

_21:53_: as so the process of asking.

_21:59_: Where does if I if I throw

_22:00_: something in in a given space,

_22:02_: where does it go that,

_22:03_: that, that, that, that is?

_22:06_: You're solving the duties equation

_22:07_: to find an equation of motion.

_22:09_: So the answer to the duties equation,

_22:11_: the solution to the Nudestix equation,

_22:13_: is essentially an equation of motion.

_22:16_: Is the creation of a A line

_22:18_: in your space that's the.

_22:22_: Sort of vector version

_22:23_: of the of the equation.

_22:24_: That's simply the the component

_22:26_: version of of the same equation

_22:28_: with with with a a tangent vector

_22:30_: which is parallel transported along.

_22:36_: To remain being attention vector. Anyway.

_22:41_: Moving on talking about. Coverture.

_22:48_: I've drawn there.

_22:51_: The surface of a sphere,

_22:52_: for example the earth.

_22:55_: And if you imagine starting at the equator.

_22:59_: And pointing north.

_23:02_: Your point north.

_23:04_: What would have?

_23:08_: And you you, you, you,

_23:09_: you keep walking in a straight line,

_23:11_: in other words along a geodesic.

_23:13_: So a geodesic is,

_23:15_: I think I said, this area.

_23:17_: A geodesic is a straight

_23:18_: line in Euclidean space,

_23:20_: A geodesic is a straight line.

_23:22_: In the conventional fashion it's with the

_23:25_: with the signature of Euclidean space.

_23:27_: A straight line is the shortest

_23:29_: distance between two points.

_23:29_: In Minkowski space, or something with

_23:32_: the signature of special relativity,

_23:34_: a straight line is the longest distance.

_23:36_: June 2 points all of the alternative

_23:38_: versions of going from HB in

_23:40_: Minkowski space are shorter than the

_23:42_: in the straight than the straight

_23:44_: line and surface of a sphere.

_23:46_: A straight line is a great circle.

_23:48_: It's it's that that has the same

_23:50_: signature as you clean space and

_23:51_: so it's the shortest distance.

_23:53_: So I started the equator,

_23:54_: I point north and I walked NI.

_23:57_: Just keep walking in a straight line and

_23:59_: eventually I will get to the North Pole.

_24:01_: And at that point,

_24:03_: if I started at zero latitude

_24:05_: at that point I'm pointing

_24:07_: toward the 100 degree latitude.

_24:11_: I then start walking sideways.

_24:13_: And head down back down to the equator,

_24:16_: still pointing at the original

_24:18_: direction so I don't turn around.

_24:20_: I end up back at the equator,

_24:21_: this time pointing along the equator,

_24:24_: and then walk backwards through 90

_24:26_: degrees and end up back where I started,

_24:28_: but this time still pointing

_24:30_: along the equator.

_24:31_: So I haven't changed the direction

_24:33_: I'm pointing at any point.

_24:35_: And I've been walking in

_24:37_: straight lines in each case,

_24:38_: but of course when I get

_24:39_: back to where I started,

_24:41_: I'm pointing in a different direction.

_24:44_: If the angular went through

_24:45_: the the North Pole was smaller,

_24:47_: then this deflection would be smaller,

_24:50_: but it would still be still be reflection.

_24:52_: In other words, going for a walk.

_24:56_: And you're saying pointing the same

_24:58_: direction allows me to pick up some

_25:01_: information about the the curved surface.

_25:03_: If it's the fact that the earth is

_25:05_: curved that I can tell the story.

_25:06_: I go up North Pole going to

_25:07_: come down and back and clean,

_25:09_: I have to go all the way to the North Pole.

_25:10_: I could I could clearly do this in any

_25:12_: but any any any circular circular route,

_25:15_: but so,

_25:16_: so the the deflection of this.

_25:21_: This this vector.

_25:22_: Through a circuit is telling us

_25:25_: something about the curvature of.

_25:28_: It's telling us something obviously

_25:29_: telling something about the the

_25:31_: curvature of that surface in a way

_25:33_: which is completely independent of.

_25:35_: Components. So we haven't talked

_25:38_: about components or coordinates or

_25:40_: anything like that at this point.

_25:41_: So there's a, there's a,

_25:43_: a geometrical.

_25:45_: This is a geometrical thing we're

_25:47_: picking up by by this process.

_25:49_: So what we need to do now is find our way of.

_25:53_: Capturing that intuition in a

_25:55_: mathematical form to get some expression,

_25:57_: some something we can calculate with that,

_26:00_: we'll talk about the curvature.

_26:02_: So that's what we're talking

_26:04_: about of a space.

_26:08_: So the way we do that is a rather intricate,

_26:13_: but not fundamentally deep process.

_26:17_: With this poor like you.

_26:21_: So I'm going to set up a path

_26:24_: to go around. Let's have two.

_26:29_: Coordinates here.

_26:31_: We'll call this first one.

_26:34_: X Sigma and that's the,

_26:37_: for example X versus Y or R versus

_26:39_: Theta would it whatever you like

_26:41_: and that line there is the line

_26:43_: where X Sigma is equal to a.

_26:45_: I would draw another line through

_26:47_: the SpaceX Sigma equals a. Plus.

_26:51_: Delta E So that's just just moving

_26:54_: along a bit. And we'll have.

_26:57_: Another period of lines of

_26:60_: constant coordinate lines where X.

_27:03_: Lambda equals B.

_27:05_: Next line equals B plus.

_27:08_: Delta B well and and as you can

_27:10_: guess these this delta and Delta

_27:11_: B I'm going to make small later.

_27:16_: And so at this. Is the.

_27:20_: Tangent vector corresponding to the.

_27:24_: Sigma coordinate.

_27:25_: So that's the the direction in which

_27:28_: the Sigma coordinate changes and this.

_27:32_: Is the. Direction in which the.

_27:38_: X Lambda coordinate changes.

_27:39_: And we'll start off with a with a vector.

_27:43_: V at this point. 8.

_27:47_: And we'll do what we do,

_27:48_: what we described for.

_27:50_: In this case, we'll take this

_27:52_: vector for a walk around this path.

_27:54_: We'll take it to. To be.

_27:57_: We do get to see what you did

_27:60_: and we'll bring it back to.

_28:02_: A and discover that.

_28:07_: Quite unfortunate location of these.

_28:13_: There's a change in the vector which is

_28:15_: picked up as it goes around that circuit.

_28:22_: And we want to, and it's,

_28:24_: it's picked that up by virtue of

_28:26_: going round that curved space.

_28:27_: And what we want to do is work out.

_28:30_: The the the size of that

_28:33_: vector there and how it?

_28:36_: How how it picks up information

_28:38_: about the space as we go around it.

_28:44_: Now, sorry, by transporting the vector

_28:46_: I mean parallel transporting it.

_28:48_: Remember I said we've got in this

_28:49_: case I'm parallel transporting

_28:50_: that back up to there, transport,

_28:52_: parallel transport down here and

_28:53_: parallel transporting it back.

_28:54_: So parallel transport reporting in each case.

_28:58_: OK, in other words,

_28:60_: in such a way that the derivative is 0.

_29:02_: So we know how to do that.

_29:08_: So by parallel transporting.

_29:12_: The vector V. Um.

_29:15_: Along this this fresh leg.

_29:19_: That is as seen.

_29:25_: That the that it will be parallel

_29:27_: transport in such a way that the.

_29:32_: The vector of V.

_29:35_: As parallel transport

_29:36_: it along EE Sigma this.

_29:38_: Right here will be 0.

_29:41_: And what that means is that um again, V.

_29:44_: I comma Sigma.

_29:46_: Will be equal to minus gamma.

_29:51_: IK Sigma V. Key, and that's just the

_29:56_: the the expression for the equate,

_29:59_: derivative and component form

_30:00_: rearranged for the case where it's 0.

_30:05_: Right. Now what? What? What was

_30:07_: the result of that going to be?

_30:09_: What is the? The. The vector.

_30:14_: Let's call that vector at a.

_30:17_: What's the vector? The value

_30:19_: of the vector V would be at B.

_30:24_: What that is going to be the vector.

_30:28_: Well, it's listing with, so we're

_30:30_: going to go with individual components,

_30:32_: include the, the, the, the,

_30:33_: the component I component at a. Plus.

_30:38_: The changes in the the that that

_30:43_: that component as we move along.

_30:46_: So it would be the integral from A to

_30:51_: B of DV. Yep, I by DX Sigma, DX Sigma.

_30:57_: So I'm just I'm just integrating the

_31:02_: derivative along along the curve to get the.

_31:06_: The results would be IA minus.

_31:13_: I Sigma. The. KBTX Sigma.

_31:25_: Um.

_31:28_: And. This A to B is moving from X = A

_31:34_: to X segment equals A+ Delta A. VI.

_31:38_: At a. My integral of a A plus. Dot E.

_31:48_: DX Sigma. Evaluated. At.

_31:53_: Along this line X. Lambda equals B.

_32:02_: And.

_32:05_: That that's the the the neat

_32:07_: version of what I'm growing here.

_32:10_: That's what I've just written down

_32:11_: and it's clear that we can buy

_32:14_: this beam means get the value of

_32:17_: the component of V at B. From E.

_32:20_: We can get to the value of the item point

_32:24_: of C from B in the same way and so on.

_32:29_: And end up. With.

_32:33_: I I ask you to think what that might be.

_32:39_: That is one of the exercises just

_32:43_: I encourage you to think through.

_32:46_: One of the quick exercises at the

_32:47_: at the at the end and I can't

_32:49_: remember off the top of my head,

_32:51_: but this is an exercise in in

_32:53_: keeping track of of signs really.

_32:60_: But the. End result.

_33:03_: When you when you get

_33:05_: all the signs right.

_33:06_: And go through step by

_33:07_: step is this pattern of.

_33:11_: Of of plus minuses and this set

_33:13_: of of of of specific intervals.

_33:16_: So this integrating from A to B,

_33:18_: that's integrating from point B

_33:19_: to Point C, Point C to point D,

_33:21_: and point D back to point A.

_33:25_: Messy, fiddly, but not, but nothing more

_33:29_: exotic is happening than than here.

_33:35_: Now we can take advantage.

_33:36_: We had at this point taking

_33:37_: advantage of the fact that

_33:38_: just A and delta B are small.

_33:40_: So we can do that.

_33:41_: The way we do that is by.

_33:44_: And. Saying that.

_33:49_: The expression such as this.

_33:53_: For. Thank you.

_33:56_: Have a question I'm picking here.

_34:03_: Alright.

_34:06_: And. Yes, I think I'm, I'm,

_34:09_: I'm, I'm picking specifically.

_34:14_: This one here to illustrate.

_34:19_: The.

_34:23_: Showing OK.

_34:26_: The. IG Lambda

_34:31_: VGAT evaluated at X Sigma equals

_34:35_: E Plus delta east and we can just

_34:39_: use Hello theorem to discover

_34:41_: that is going to be IG Lambda V.

_34:47_: G. At X, Sigma equals A+ Delta A.

_34:54_: The body X Sigma gamma I.

_34:59_: G Lambda VG at X,

_35:03_: Sigma equals A+ order. And.

_35:10_: All I'm doing is Taylor theorem to

_35:13_: work out what this. This one is.

_35:18_: In terms of of this. And you can

_35:23_: see that now work out that so the.

_35:27_: The that that particular expression

_35:30_: evaluated A+ Delta E is going to

_35:33_: be that expression plus a bit.

_35:35_: So that when I subtract these two things.

_35:38_: What I'm left with? Is this? OK.

_35:45_: Um.

_35:48_: I end up with a simpler expression

_35:50_: for which I'm not going to put

_35:52_: which is is a numbered in the notes,

_35:54_: but between three 47348,

_35:55_: which I I encourage you to set through,

_35:58_: but we end up.

_35:59_: I I'm not going to go through the

_36:02_: the the index manipulations here

_36:04_: because they're not terribly edifying

_36:06_: that just you to watch me try to

_36:09_: copy indexes from from my nose,

_36:11_: but you but you have the notes and

_36:13_: you go through them very carefully.

_36:14_: But the point is that we end up.

_36:21_: With more things cancelling.

_36:23_: And an expression. For.

_36:28_: This the change in the ith

_36:31_: component of this vector,

_36:33_: in other words the.

_36:39_: The ith component of this change vector here.

_36:44_: In terms of the size?

_36:49_: Of this. The size of this?

_36:54_: The vector we started off with and this.

_36:58_: Rather fiddly expression involving the

_37:00_: christophel symbols, which as you recall,

_37:02_: you tell us information about the way that

_37:06_: the coordinate changes as we move around.

_37:08_: But now we'll look at this and

_37:10_: stare at it a bit and we realize.

_37:14_: This. Component here. Is a number.

_37:19_: I mean is, is, is a component of a vector.

_37:21_: But is a is a number. What number is it?

_37:23_: Is the number we get by applying our.

_37:26_: But by taking a vector and

_37:29_: applying A1 form to it.

_37:32_: So so this this one of the

_37:35_: basis one forms which is.

_37:37_: And one of the this is the ith.

_37:42_: Component of that vector.

_37:44_: We obtained it by applying that vector.

_37:48_: So V. Is equal to. Delta V.

_37:55_: Applied to the.

_37:59_: XI.

_38:01_: So this depends on A1 form.

_38:04_: It also clearly linearly

_38:07_: depends on this number DXL.

_38:10_: But this?

_38:14_: This. To write that. This displacement here.

_38:23_: Is going to be delta A. He Sigma.

_38:28_: So this displacement vector.

_38:32_: Is something which.

_38:33_: Is a vector which has as

_38:36_: its as its size delta E,

_38:38_: so that vector displacement.

_38:40_: There is also something that's

_38:42_: gone in to a thing to get this.

_38:45_: So this is also linearly dependent

_38:47_: on the size that size delta E.

_38:49_: The size there's a B which was written.

_38:53_: I put it here and it's

_38:55_: literally dependent on this,

_38:56_: the size of this vector we started off with.

_38:58_: In other words, this number,

_39:01_: this real number Delta VI is a

_39:05_: number which depends on one,

_39:06_: one form and three vectors.

_39:09_: In other words, this is the.

_39:14_: Would have written like this.

_39:16_: This number is therefore the

_39:19_: the what you get when you.

_39:23_: Take a a tensor which we'll call R

_39:26_: in different rieman, and plug into.

_39:28_: It's a 1/3.

_39:29_: To answer,

_39:30_: we plug in our basis one form vector,

_39:34_: started off with and there's two sizes

_39:37_: of the two vectors which describe

_39:40_: the shape and size of this detour.

_39:44_: And this vector here is

_39:46_: called the Riemann tensor.

_39:48_: It has components this.

_39:52_: And it picks up it.

_39:54_: It encodes information about the,

_39:57_: the, the, the, the, the,

_39:59_: the way that the vector changes as

_40:01_: we move it around around the circuit

_40:03_: in a way which picks up from the.

_40:06_: Christophel symbols.

_40:09_: And we put pictures value from the

_40:12_: console symbols in other words,

_40:13_: which we know which you already

_40:15_: know contain information,

_40:17_: encode information about the way

_40:19_: that coordinates change as you

_40:21_: move around this space question.

_40:25_: So you can create a full path, yes.

_40:27_: So and there will be because what we've

_40:31_: done is these. We'll set this up.

_40:34_: We've constructed this so that these

_40:36_: curves here that we're we're moving along

_40:39_: are the curves of constant coordinates,

_40:42_: so given a coordinate system.

_40:45_: Given a coordinate system and then

_40:47_: you will always be able to to

_40:49_: describe curves of constant cost.

_40:51_: Curves cost X and covers cost

_40:54_: Y or covers of constant R.

_40:56_: And covered, of course,

_40:57_: theatre or or or or whatever or latitude,

_40:59_: longitude.

_40:59_: So you'll be able to set up that grid.

_41:03_: And so, but but of course.

_41:06_: What coordinates you pick?

_41:07_: What coordinate functions you pick?

_41:09_: Will affect the number you get here,

_41:11_: but that's fair enough.

_41:12_: But this is clearly the.

_41:13_: This is clearly the the components

_41:15_: of the Riemann tensor in a

_41:18_: particular coordinate system.

_41:20_: You know this is the basis one

_41:21_: formed in that coordinate system.

_41:23_: These are the basis vectors

_41:24_: in that coordinate system.

_41:25_: So the this number is clearly

_41:27_: a coordinate dependent thing.

_41:29_: But what we've argued here is

_41:31_: that it has nonetheless what we've

_41:33_: indicated that we nonetheless

_41:35_: it's our a geometrical object.

_41:38_: Which just has coordinate

_41:40_: dependent components.

_41:44_: And.

_41:49_: Uh.

_41:53_: And I think that's. Yeah.

_41:59_: And and and and that's a very important,

_42:01_: very important important tensor which is

_42:04_: the includes the information about the,

_42:06_: the, the, the, the curvature.

_42:07_: Now there's another way we can

_42:10_: define that Mark 43 annoying me.

_42:12_: There's another way we can define the.

_42:15_: The Event Center,

_42:16_: which I'm going to just look.

_42:24_: And which I'm going to mention because we

_42:26_: come back to it briefly in a in a moment.

_42:32_: Which is that the human tensor can

_42:34_: also be defined in such a way.

_42:37_: I'm going to not go through

_42:40_: this in detail. It's not it.

_42:42_: It is. It's somewhat peripheral.

_42:47_: We'll write this down and then

_42:49_: explain what it is. And uh.

_43:01_: This is station here. Is a commutator.

_43:05_: Defined so that a B is equal to a B -, B.

_43:10_: E. And I'm not going to you know go

_43:12_: through and calculate it with this,

_43:14_: but the I I mentioned that just in order

_43:17_: to show that there is another way of of of

_43:20_: getting to the same point which in a sense.

_43:24_: Manifestly, it doesn't depend on

_43:26_: coordinates that this this process here,

_43:29_: you know, uses court,

_43:30_: the truffle symbols it was done in terms of

_43:33_: coordinate functions and so on. It's yeah,

_43:36_: it feels like it's somehow codependent,

_43:39_: but I I will assert that there's

_43:41_: another way of getting to the the

_43:44_: the the renter in that way.

_43:48_: And more or less parenthetically.

_43:51_: Um, no, if we stick with this version here.

_43:57_: Then you will recall, I hope,

_43:60_: that in a particular.

_44:04_: And.

_44:07_: In particular coordinate system.

_44:10_: You can calculate the values of the grateful

_44:15_: symbols using the driving from from the

_44:18_: components of the metric. So we can.

_44:23_: If this would have on the next slide.

_44:26_: It's not annoying.

_44:29_: Which I wish I had a slight seeing this,

_44:31_: but I don't really want to write

_44:33_: this whole thing down but. Uh.

_44:39_: OK, I'll do it. Is this is

_44:47_: 355 RIJKL is half G. IL comma,

_44:52_: JK minus and and and other stuff.

_44:55_: No write the whole thing the whole day.

_44:56_: I should have. I always had a

_44:58_: slide of it and now that involves.

_45:05_: Is that a question there?

_45:07_: And I think I'll can we

_45:09_: should auto focus back on to.

_45:12_: So the improve that I'm not sure.

_45:16_: So this is the. Um.

_45:22_: The quick confusing this.

_45:23_: So if we recall the the the expression

_45:26_: we found for the Christoffel symbol in

_45:29_: terms of the metric and its derivatives

_45:32_: and do this fairly calculation,

_45:34_: we end up with an expression like

_45:38_: this and if we do this in a.

_45:40_: A local national frame,

_45:42_: which makes the whole thing simpler.

_45:44_: We end up with an expression for

_45:47_: the components of the of the

_45:50_: Riemann tensor in in that frame.

_45:53_: Now we can't do very much

_45:55_: with that in that frame,

_45:56_: and we can't use the comical semi

_45:58_: colon rule that we found last time,

_46:01_: which said that in our local natural

_46:03_: frame we could turn simple derivatives

_46:06_: into covariant derivatives that

_46:08_: works in with single derivatives.

_46:11_: Not, but not with second derivatives.

_46:14_: But what we can do is look at this state

_46:17_: different while and discover a number of.

_46:22_: Symmetries of the Riemann. Tensor.

_46:27_: If you swap the first the the the

_46:30_: the the first pair of of indexes,

_46:33_: you change the sign of the of

_46:36_: this component if you swap the.

_46:39_: Last pier.

_46:41_: That you get you the saying if

_46:43_: you swap 22 like that you get.

_46:46_: An expression which is that that that

_46:49_: that component is is the same so that

_46:51_: although there are N by N by N by

_46:53_: N components in the Riemann tensor,

_46:56_: there are lots of that a lot

_46:57_: of them are equal.

_46:59_: And you can also discover that

_47:02_: if you permute the GKLL GKG if

_47:06_: you promote the the 3rd 3.

_47:09_: Components.

_47:09_: You end up with something which is 0 again.

_47:13_: By staring at this and the point is,

_47:16_: although this is we calculate calculate

_47:18_: this in the local national frame where that

_47:20_: calculation relatively relatively easy.

_47:22_: It's merely tedious and error prone.

_47:26_: That's not a tensor equation

_47:27_: because of the second derivatives,

_47:29_: but we can workout.

_47:32_: On the basis of that,

_47:33_: we can work out these symmetries because

_47:36_: they don't involve any derivatives,

_47:38_: particularly don't involve

_47:39_: any second derivatives.

_47:40_: These are tensor equations.

_47:42_: So these,

_47:43_: although we calculate them using

_47:44_: this in the local national frame,

_47:46_: these are not specific to that frame

_47:49_: and are and correspond to to tensor.

_47:55_: Identities, right?

_47:59_: We are almost. Out of time.

_48:05_: In order to not run over I could I I

_48:09_: do really want to finish this chapter.

_48:13_: Today I'll go through the next part

_48:16_: fairly quickly and I don't think

_48:18_: there are any objectives which depend

_48:20_: on the details of this section,

_48:23_: but do you want to just talk

_48:25_: through them quickly? Um.

_48:29_: If you think back to chapter one,

_48:31_: we have I'm going to this in one minute.

_48:35_: Actually, I'm not. I'm.

_48:36_: I I will pick this up again

_48:38_: again next time I want to try.

_48:39_: At least I have get to go to

_48:41_: the end of the of the chapter.

_48:42_: We remember that diagram from chapter one,

_48:44_: but we were things which were in freefall.

_48:49_: A significant difference apart

_48:50_: from each other if freefall

_48:51_: toward the center of the Earth.

_48:53_: They are distance separation between

_48:56_: them decreased as time moved on,

_48:59_: and the 2nd derivative of that

_49:01_: separation was nonzero even though

_49:04_: the things weren't accelerating.

_49:05_: These as we'll come to discover,

_49:07_: we haven't.

_49:08_: We don't know this yet because we haven't

_49:10_: talked about about general relativity yet.

_49:11_: These paths are geodesics.

_49:13_: The the path that's something

_49:16_: follows as it falls it geodesic and

_49:19_: so the fact that the separation

_49:21_: between the UCS changes as the as

_49:26_: the objects move through space-time.

_49:29_: Is is is telling us something about the

_49:32_: coverage of the space they're moving through?

_49:35_: And So what we discover is through another

_49:40_: similar construct we can talk about.

_49:44_: Appears of geodesics and talk about

_49:48_: the vector. Of course you know joining.

_49:52_: A family of duties.

_49:55_: I'll talk about the vector joining

_49:57_: points on neighboring eugenics

_49:59_: which have the same parameter,

_50:01_: and ask how does that vector change.

_50:05_: We can say that what we have here is also.

_50:09_: A circuit in the space so the remain

_50:13_: coverage tensor is going to tell us how

_50:16_: this connecting vector here changes as

_50:18_: we move through that I'm I'm making.

_50:21_: That's a rather handwaving

_50:23_: remark I at this point.

_50:25_: But the end point is that.

_50:30_: Jumping to the end,

_50:32_: is that the the second derivative of?

_50:37_: That connecting vector joining

_50:40_: points in two neighbouring geodesics.

_50:43_: Ends up. Depending on the.

_50:46_: The change of vector to the geodesic the X it

_50:49_: depends on the on how far apart they start,

_50:51_: off the side of the of the connecting

_50:54_: vector at the beginning and on the

_50:56_: Riemann curvature tensor.

_50:58_: So the recovery tensor is not just

_50:60_: telling you about the shape of the well,

_51:02_: it is telling you the the curvature,

_51:04_: the shape of the.

_51:06_: Space you're moving through.

_51:08_: And one of the ways it does

_51:10_: so is by saying how?

_51:12_: How quickly.

_51:14_: These two.

_51:16_: Two objects.

_51:20_: Two things moving along along geodesics,

_51:22_: how quickly that connecting vector changes.

_51:27_: That's a tidal force.

_51:28_: That's things like these two things falling

_51:31_: towards the center of the Earth and getting

_51:33_: closer together without accelerating.

_51:35_: I think there's basically no objective

_51:37_: depend on the details of that.

_51:39_: I, I, I encourage you to look through

_51:40_: the corresponding part of the notes,

_51:42_: but I I think that is all I'll see with that.

_51:46_: So I think we can therefore declare part

_51:49_: three finished or go into Part 4 next time.

_51:53_: I think the overview video

_51:56_: is up on the stream site.

_51:59_: If it's not,

_51:60_: I'll check and I'll make sure that

_52:02_: the notes are up promptly and I'll