Transcript of gr-l09
==========


_0:08_: Hello everybody.

_0:12_: This is later nine.

_0:13_: We are remarkably enough and I

_0:16_: cannot congratulate myself enough

_0:17_: for this on time, which is very good.

_0:21_: So we have a full 3 lectures to cover

_0:23_: Part 4 which is back to the physics of GR.

_0:27_: So the intense the intellectual payoff

_0:31_: of the. Of the weeks up to now.

_0:34_: Before we started on that, however.

_0:36_: And according to supervisions,

_0:40_: for historical reasons,

_0:41_: the way that the general course had

_0:43_: been covered by provisions has been

_0:46_: different from the other courses.

_0:47_: It's been a separate.

_0:50_: You have supervision sessions rather

_0:52_: than small groups of provisions, but.

_0:56_: There's some flux in the way

_0:58_: that these will be managed,

_0:60_: and it has occurred to me that

_1:02_: it might be useful to have it.

_1:04_: Haven't previously done office hours,

_1:07_: but either an office hour

_1:10_: I could announce or else.

_1:13_: As some others have done for the Honors

_1:16_: courses bookable slots at a certain time.

_1:19_: Does anyone have any feelings about this

_1:21_: or any suggestions on this sort of topic?

_1:23_: Is that a good idea that a bad idea?

_1:25_: Of those two possible models,

_1:27_: which one is is an obviously better one or?

_1:31_: And. Yeah. OK. Business.

_1:35_: And think otherwise.

_1:38_: OK,

_1:38_: the world is I'll I'll identify a

_1:40_: suitable time and I don't really know

_1:43_: your timetables are the afternoons out.

_1:47_: Or other mornings out? Or is there a?

_1:49_: Is there an obviously bad time to

_1:52_: organise to just an officer? Anyone.

_1:54_: I don't know your timetables.

_2:00_: He's normally bad 10 to 1225 bad.

_2:04_: OK, so if I said picked one o'clock one day,

_2:09_: would that be a suitable sort of?

_2:12_: Seems of notes OK I think.

_2:15_: I think with that and sort that out.

_2:20_: I think that is all.

_2:21_: Are there any other questions or thoughts

_2:25_: or anxieties to do with to do this that

_2:29_: you have anxieties not to do this are?

_2:33_: Excluded, OK and let's get going and I will.

_2:38_: Before I I said part 4,

_2:39_: but before I do that I will just

_2:42_: mention a couple of things about

_2:44_: the the the very end of part three.

_2:47_: At the end we're racing for the,

_2:49_: we're racing for the finish line as it were.

_2:52_: And I I I, I I skipped through the bit on

_2:55_: on Judy Deviation but there's a couple

_2:57_: of things I do want to just mention.

_2:59_: I won't work through the.

_3:02_: The derivation of this or the corroboration,

_3:05_: I'll ask you either to look at the

_3:07_: notes or just take it and trust

_3:09_: the point of duty deviation. Is.

_3:12_: You might remember this picture

_3:15_: from part one.

_3:16_: We talked about the idea of of two things

_3:20_: falling toward the center of the Earth.

_3:22_: And because they are falling

_3:23_: to the center of the earth.

_3:25_: The heading toward the same point, and so.

_3:31_: Although they are both in freefall.

_3:33_: Each of the two observers in

_3:35_: at either end of that well see,

_3:37_: spaceship or whatever,

_3:38_: we'll see the distance to the other

_3:41_: one decrease at an increasing rate.

_3:44_: They will be strictly in quote,

_3:46_: accelerating toward each other in

_3:48_: the specific sense that the second

_3:50_: derivative of their separation

_3:52_: will be nonzero.

_3:53_: But they will not be accelerating.

_3:55_: They will not be feeling any push

_3:57_: if they were holding a Plumb Bob,

_3:59_: but that wouldn't work.

_4:00_: If you're holding some sort of

_4:02_: accelerometer then they wouldn't

_4:03_: detect anything as they were

_4:05_: falling even though they were.

_4:06_: There's a second derivative

_4:08_: being non zero happening here,

_4:10_: and that is because these two observers

_4:12_: at either end of the spaceship or

_4:14_: train carriage or they want to call

_4:16_: it are both following geodesics.

_4:18_: But those eugenics,

_4:20_: the separation between those judaics,

_4:22_: is changing as a function,

_4:24_: as a quite complicated function of time.

_4:26_: Because of the curvature of the

_4:29_: space they're falling through.

_4:31_: And we're going to talk about the

_4:33_: relationship between curvature

_4:34_: and gravity in a moment.

_4:35_: But so just like jumping ahead of ourselves,

_4:38_: the gravity that the this earth,

_4:41_: this planet here. Has is for.

_4:47_: Has the consequence that the

_4:49_: space-time around it is curved in

_4:51_: a way we'll learn about shortly.

_4:53_: And so as those particles followed

_4:55_: udic through it,

_4:56_: those geodesics are straight lines in

_4:59_: the space, but nonetheless curved.

_5:01_: And so it's tidal effects like this,

_5:03_: so-called tidal effects that are where and.

_5:07_: The effects of gravity emerge.

_5:12_: That's the picture I used to to

_5:14_: set up the scamper through the

_5:17_: last couple of pages of the note.

_5:19_: The idea was, you recall,

_5:21_: that there were multiple geodesics

_5:23_: and were able to.

_5:24_: This is another version of what we

_5:26_: we what we stepped through fairly

_5:28_: painfully in the last lecture of taking

_5:31_: our vector for a walk around a closed path.

_5:34_: In this case, the end result of that

_5:36_: calculation was what would have been

_5:38_: if we'd gone through step by step,

_5:40_: would have been a calculation of.

_5:41_: So much of this connecting vector at the top,

_5:44_: which links points with equal

_5:46_: parameter on the two geodesics.

_5:48_: How much that changes and the answer.

_5:51_: Jumping to the end.

_5:54_: Was an expression like this,

_5:56_: which I, I, I, I, I, I,

_5:58_: I want to put up here so that I

_6:01_: can refer to it later on when we

_6:03_: when we discover when we can use

_6:06_: that expression in a simple case.

_6:09_: All it's saying is that the second

_6:11_: derivative of the separation

_6:13_: between the the the the the points

_6:16_: on 2 neighbouring diuretics.

_6:18_: Separation is a function of.

_6:21_: The tangents to the duties.

_6:24_: The separation itself,

_6:25_: you know we're pointing here to here,

_6:28_: and the Riemann tensor,

_6:29_: so the so that the reason why

_6:31_: this emerges naturally at the

_6:32_: end of Part 3 is this is another

_6:35_: case where the Riemann tensor is

_6:37_: telling us what we want to know.

_6:39_: About the shape of the space

_6:40_: that we're going through,

_6:41_: and this this time it's telling

_6:43_: us stuff about the change

_6:45_: that this connecting vector.

_6:49_: This.

_6:52_: This connecting vector here,

_6:53_: how much that changes because of the

_6:55_: curvature of the space that these

_6:57_: duties are limited are limiting,

_6:59_: so we'll see an example of that.

_7:02_: Do you do you, do you geodesic

_7:05_: deviation equation in not let

_7:07_: me lecture after this one when

_7:09_: you will be illuminated, I hope.

_7:15_: Um. Key points. Any questions about that?

_7:23_: So I I acknowledge I have

_7:25_: effectively missed a bit out there

_7:27_: because because the, the, the,

_7:29_: the details are less important

_7:31_: than the the the conclusion. So.

_7:38_: Onward.

_7:41_: So here we're back to physics. And.

_7:47_: The two things I want to to to,

_7:48_: to cover really in this part are first

_7:52_: of all the equivalence principle.

_7:54_: We touched on the equivalence principle in

_7:56_: part one where I said that all free falling

_7:58_: inertial frames are equivalent for the

_7:60_: performance of all physical experiments.

_8:01_: I think I actually said that in part one.

_8:03_: If I if I didn't then prior to

_8:05_: have done but you, you, you,

_8:07_: you probably have seen that express

_8:09_: somewhere and we find that the

_8:13_: we're able now to put some.

_8:17_: Mathematical flesh on the physical

_8:19_: bones of that of that statement,

_8:23_: that of the code principles.

_8:24_: So we see where the as it were to

_8:27_: use an overused image with with.

_8:29_: This is where the rubber hits the road,

_8:30_: as it were.

_8:32_: And.

_8:35_: My goal,

_8:35_: the overall goal of this of of G1

_8:38_: is to get to the point where we

_8:39_: can write down Einstein's equations

_8:41_: and say that's the question,

_8:43_: now go and answer it.

_8:44_: But it's unfair to do that and not

_8:47_: give our solution to IN equations

_8:50_: and which describes our space-time

_8:52_: of interest to us.

_8:53_: And the one we, we, we,

_8:54_: we,

_8:55_: we look at is how we can recover

_8:57_: nuisance theory of gravity from

_8:59_: Einstein's theory so as the

_9:01_: low energy and low speed.

_9:04_: A bit of of of a against alien

_9:08_: theory of gravity.

_9:10_: Um.

_9:14_: This is challenging, as I've said before,

_9:17_: the point of the the objectives and the

_9:19_: aims are closely linked to assessment,

_9:21_: and this is challenging to assess because all

_9:24_: the calculations are hideously long. So the.

_9:28_: The assessment tends to be explained X.

_9:32_: And, you know, show that you understand X.

_9:34_: Basically, you persuade me that

_9:36_: you have a clue type assessment

_9:38_: rather than other things.

_9:40_: But there are some simple dynamical

_9:43_: calculations that you'll be able to

_9:45_: do in certain contexts, which is.

_9:48_: So there's a there's some simple calculations

_9:51_: that are possible at the end of this.

_9:53_: Um. OK.

_10:02_: And you're in Newton's gravity.

_10:03_: The source of gravity is matter.

_10:06_: Is mass. The the the one of the

_10:09_: terms in in in in the theory of.

_10:13_: Law of universal gravitation or two

_10:15_: of the terms are EMS, they're masses.

_10:16_: So it's not surprise,

_10:18_: no surprise that it's a mass that

_10:19_: is that we have to to to think about

_10:21_: and deal with in this context.

_10:23_: However, let's be relativity.

_10:24_: We can't talk about mass.

_10:25_: The natural thing is talk about energy,

_10:27_: momentum, the so it's not

_10:29_: just the mass of something,

_10:30_: but the energy the intimate

_10:31_: has by virtue of his movement.

_10:33_: So when something is moving rapidly.

_10:36_: There's more energy momentum in

_10:38_: it and so it gravitates more is

_10:41_: what we're going to discover.

_10:42_: We shouldn't be surprised at that,

_10:43_: because if the energy momentum the, the,

_10:47_: the, the, the, the momentum 4 vector,

_10:50_: that is the key,

_10:52_: the thing that's important in

_10:53_: your study of relativity.

_10:55_: And the way we we edge up.

_10:58_: So we'll have to learn how to

_10:60_: discover how to talk about the

_11:02_: energy momentum in extended objects

_11:05_: in a rustic relativistically

_11:08_: satisfactory we we would do that,

_11:11_: we would talk about dust.

_11:13_: Which are beautiful you I want to just

_11:15_: check I'm not missing something out here.

_11:17_: I've I've I've notified of some remark

_11:20_: I might want to make sure I made early.

_11:23_: Ohh yeah.

_11:24_: So just a minor thing.

_11:25_: And I at this point up up up to

_11:28_: this point we've been talking

_11:30_: in about N dimensions.

_11:32_: Of course,

_11:33_: the reason we're talking

_11:34_: interventions is to turn into n = 4,

_11:36_: and so from now on we're talking about four

_11:39_: dimensions with signature of of space-time,

_11:41_: which in this case would be minus,

_11:42_: plus, plus, plus.

_11:44_: And I'm not going to use,

_11:46_: I'm going for indexes.

_11:47_: I'm going to switch to using

_11:49_: Greek indexes for space-time

_11:51_: Dimension 0123 and Latin ones

_11:53_: for it's space only ones.

_11:55_: That's just a slight notational

_11:57_: tweak to fit in with the conventions.

_12:01_: OK, so just. Matter is structured.

_12:06_: There are all sorts of things that.

_12:10_: That their whole disciplines

_12:12_: devoted the structure of matter.

_12:14_: That's complication.

_12:15_: You can simplify that and talk about

_12:18_: a perfect fluid which something with

_12:21_: no viscosity but which has mass,

_12:24_: mass, density and has pressure.

_12:27_: You can simplify that so further.

_12:30_: By talking of our of of of dust,

_12:32_: which in this context is something which has.

_12:37_: No internal structure,

_12:39_: so there's no viscosity and it's all,

_12:42_: there's a there's a frame

_12:44_: in which it is at rest,

_12:45_: so there's no pressure.

_12:46_: So and the idea is a collection

_12:49_: of dust particles.

_12:50_: So this time we push has mass density.

_12:54_: And nothing else.

_12:57_: So let's imagine our box of this.

_13:01_: There's a box of what?

_13:05_: Plenty dot X dot Y dot Z. And.

_13:08_: You can imagine it being moving at some

_13:12_: constant speed with respect to it,

_13:14_: or or we moving at some

_13:15_: constant with respect to it.

_13:20_: Now in that box will therefore

_13:23_: length contract as your

_13:25_: recollection or special activity

_13:28_: will will immediately remind you.

_13:31_: So I want to go through

_13:32_: this in careful order.

_13:37_: So the. And. The the volume.

_13:45_: Ohh of in this other in other frame

_13:49_: we just X prime delta Y prime delta

_13:53_: Z primed with an extra factor of.

_13:57_: The. Look at this update 10 no.

_14:04_: With an X Factor of gamma.

_14:05_: But that's the usual range factor.

_14:07_: So the thing will length contract to the

_14:09_: volume will go down by a factor of gamma,

_14:11_: which means the density goes up by

_14:13_: a factor that the number density

_14:15_: goes up by a factor of gamma. Um.

_14:21_: So if there is a, if we look at this

_14:25_: and ask what is the flux of particles

_14:28_: through the end wall of this with

_14:30_: this thing is moving with respect to.

_14:33_: I think the perspective is wrong in that.

_14:36_: I think it's actually I need to

_14:38_: redraw that I cannot tolerate

_14:40_: perspective being wrong and then

_14:43_: the the flux of particles through

_14:45_: the the the end wall will be the.

_14:49_: Volume of this X to the Y.

_14:55_: Doctor Z.

_14:58_: Divided by divided by.

_15:00_: The flux is the number per

_15:03_: unit area per unit time.

_15:06_: Which they're going to end up with a flux.

_15:10_: In the X direction. Being.

_15:20_: And.

_15:26_: Yeah, but I'm gonna let you

_15:28_: jump to the end being gamma.

_15:30_: In the X where N is the.

_15:42_: The the number density in the rest frame.

_15:49_: Turn off auto focus. OK.

_15:54_: And. One can go through that in

_15:58_: more step by step, I hope you you,

_16:01_: you you agree that's that,

_16:01_: that that that's reasonable.

_16:02_: The point is that that that the

_16:04_: number density is proportional to

_16:06_: the velocity but with an extra

_16:08_: factor of gamma because of the

_16:10_: of the increase in the number

_16:11_: of density because of the length

_16:13_: contraction of the side of the box.

_16:16_: And. That looks sort of familiar

_16:20_: that that that looks like.

_16:22_: N times.

_16:26_: Gamma X and so we can jump to the

_16:30_: conclusion that the there is a.

_16:33_: Our flux vector here.

_16:36_: Which is the number density times the.

_16:41_: For velocity of the.

_16:46_: Or the motion. So we're not just,

_16:48_: it's not just general,

_16:50_: not just specific.

_16:51_: This is to the X direction.

_16:54_: We'll jump to that conclusion.

_16:55_: I mean, it will turn out to

_16:56_: that as a sensible thing to do,

_16:59_: as what I'm doing here is

_17:01_: motivating this as an idea.

_17:04_: So that's the.

_17:09_: Flux factor.

_17:13_: And the components of this.

_17:15_: Are going, they're going to be MN.

_17:19_: Yep, gamma, NVX, gamma,

_17:23_: NVY, gamma, NV. Is it?

_17:29_: How do we extract those components?

_17:34_: Of the of the of the of the flux

_17:37_: vector in exactly the the usual

_17:39_: way by plugging a what one of

_17:42_: the basis one forms into the.

_17:47_: Into the. It's ****** factor.

_17:50_: That's ****** vector. And.

_17:55_: With the alpha basis one form

_17:58_: plugged into it. And here as I say,

_17:60_: I'm switching to using the the

_18:03_: traditional thing of using Greek

_18:05_: letters for the the 00 to 3 indexes,

_18:08_: and I'll later use Latin letters for

_18:11_: the spatial components, so that's.

_18:14_: I'm just recapitulating stuff that

_18:16_: you've you've seen before about

_18:18_: the way we extract components or

_18:20_: vector from the vector itself and

_18:22_: and one of the basis one forms.

_18:28_: Umm.

_18:35_: So.

_18:39_: We've here talked about the change in the.

_18:43_: Numbered entity of the particles in the

_18:45_: box and that number density has gone up

_18:48_: because the size of the box has gone down.

_18:50_: Usually retract. However,

_18:53_: there's also a because these

_18:55_: these these these dust particles

_18:57_: in the box are moving at speed.

_18:60_: The energy that they have.

_19:02_: Is acquires another factor of gamma.

_19:06_: So the energy of the particles

_19:09_: in the box. Uh.

_19:13_: Let's write down the, the, the, the.

_19:17_: Energy ends up being gamma

_19:21_: N the increase to the. Yeah.

_19:32_: The increase in the number density.

_19:36_: And each of these particles has a mass M.

_19:39_: We inquired another gamma the energy in the.

_19:44_: Of the particles of the box has acquired

_19:46_: a gamma squared and that is a hint

_19:48_: that we are not going to be able to

_19:50_: describe this the energy of the particles

_19:52_: in this box by a vector because.

_19:58_: The. But by a rank one object,

_20:01_: because the transformation of a rank

_20:03_: one object from frame to frame would

_20:05_: only pick up one factor of gamma,

_20:07_: whereas it's clear we need two,

_20:09_: so we're going to need a higher rank object.

_20:12_: What do we have to to build

_20:14_: that higher rank object?

_20:15_: The things we have available are the

_20:18_: momentum 4 vectors of the dust of the dust

_20:21_: particles and the overall flux vector.

_20:24_: So the the the metal dust particles is.

_20:28_: It is telling us that in the in

_20:31_: another frame the momentum of those

_20:33_: dust particles would would change and

_20:36_: we also are told that there is a an

_20:38_: overall number density corresponding

_20:40_: to the flux vector which will

_20:42_: also change and so those are the

_20:44_: two vectors we have to play with.

_20:48_: And. So what we can do is.

_20:53_: Guess and here are motivating

_20:56_: again rather than deducing

_20:58_: that there is our tensor.

_21:00_: Which is the. The.

_21:06_: Momentum for vector of the dust particles.

_21:11_: Outer product with the. Sorry.

_21:16_: The. Flux 4 vector of the particles

_21:21_: as a whole. Which is going to be.

_21:26_: M. You. Cross. And. You from above.

_21:34_: Which is just. Role you.

_21:40_: Because you were rules. You could.

_21:42_: Amen. And I guess it's obvious M here

_21:46_: the rest mass of the particles and N

_21:49_: is the rest density of the particles.

_21:52_: In, in, in the dust.

_21:57_: And we can extract components of this.

_22:03_: Kendra in the usual way by applying.

_22:09_: Thesis.

_22:12_: One forms to it. Which extracted which

_22:16_: ended with being P. The. Xmu. Times.

_22:24_: In.

_22:27_: The. X. You.

_22:32_: In again illustrating the way that the.

_22:37_: The. We evaluate a natural product.

_22:40_: By plugging the. In this case,

_22:45_: one forms in turn into the two

_22:48_: components of the outer product,

_22:49_: and simply real number

_22:51_: multiplying the results together.

_22:53_: So that's that's that's an equivalent

_22:55_: saying rather than an equal sign really.

_22:60_: Did that. I don't see any looks

_23:02_: of shock and horror. That's good.

_23:04_: So the so the thing don't need to do again.

_23:08_: Reassuring you that the sensible

_23:09_: thing to do is to start looking at

_23:11_: individual components of this object,

_23:13_: which is the energy momentum tensor or

_23:16_: the stress energy tensor of the dust.

_23:19_: So this is the tensor which I assert

_23:22_: is characterizing the energy and

_23:24_: momentum of the dust as an assembly,

_23:27_: which we are going to discover is the

_23:30_: source of the gravitational field,

_23:32_: gravitational field that the the

_23:34_: the change to space-time around it.

_23:36_: So let's look, for example, T00.

_23:41_: There's usually a component. That um.

_23:48_: P.

_23:51_: DT times N. DT. And the.

_23:58_: The same component of the Momentum

_24:02_: 4 vector is gamma M If you

_24:05_: recall your special relativity.

_24:07_: And the time component of

_24:10_: the flux vector is gamma N.

_24:16_: Which gives the.

_24:20_: The. Term that, that,

_24:21_: that that we sort of in a slightly

_24:25_: hand WAVY way deduce earlier on.

_24:27_: So that's reassuring.

_24:30_: Um, the.

_24:35_: T0Y and again this is me using

_24:39_: Latin letters to indicate the

_24:41_: space components of the tensor.

_24:43_: That's so is 1 two or three?

_24:47_: It's going to be P.

_24:51_: ETN.

_24:54_: The XI. Which will be gamma M.

_24:60_: And gamma. And VI.

_25:06_: And if we, you know that's different from.

_25:11_: Here. And we look at that

_25:15_: that has the dimensions of.

_25:22_: Mass. Per unit area per unit time.

_25:30_: So that is the. And match being in

_25:34_: this context the same as energy.

_25:36_: That's the energy flow per

_25:38_: unit area per unit time.

_25:39_: So that's the energy flow across.

_25:43_: A surface of constant X per unit time.

_25:49_: So that's the essentially the mass flow.

_25:54_: That's been. The represented by the.

_25:59_: The mass flow of the dust. And.

_26:03_: That we can have a similar sort

_26:06_: of thing argument for what TI0 is.

_26:09_: That's the the spatial component of the

_26:11_: momentum and the time component of the dust.

_26:13_: And you can. It's supposed to reassure

_26:16_: yourself that is the same as.

_26:18_: At T0I you know there was this

_26:20_: tensor is symmetric, turns out.

_26:24_: And we can look at T IJ so the space

_26:30_: space components of the tensor.

_26:32_: Which will be equal to P. The XI.

_26:39_: And. The X. G. Uh, which will be um?

_26:49_: M. Gamma M Phi I times.

_26:56_: Gamma

_26:58_: NVJ which basically from the

_27:03_: dimensions ends up being the.

_27:07_: The. Flux the flux of momentum

_27:10_: per unit area per unit time,

_27:12_: which ends up being the force per unit area.

_27:15_: And. There's typically argument argument

_27:19_: here but but that that is picking up the,

_27:22_: the, the, the, the,

_27:24_: the the force that there would be on a

_27:27_: wall of if if this body of flux but the

_27:30_: this body of dust particles hit a wall.

_27:33_: And that's that's. Force.

_27:38_: Per unit. Area. Umm.

_27:46_: So more more generally,

_27:48_: what we're doing here is we're

_27:51_: interpreting the various components of

_27:54_: this tensor as the flux the way I put the.

_28:02_: And the flux of the the the T Alpha beta is

_28:06_: the flux of the alpha component of momentum,

_28:10_: so energy or momentum across the beta.

_28:15_: Our surface of constant beta the constant

_28:17_: constant time I into the future or surface

_28:21_: of constant of constant position, IE.

_28:25_: The flow you know through the space.

_28:32_: OK. I'm aware this is all

_28:36_: slightly hand waving,

_28:37_: but it's I as I say again,

_28:40_: it's intended to motivate

_28:41_: the the identification these

_28:43_: things rather than reduce them.

_28:49_: Now I said this was a perfect,

_28:52_: this was just it had no pressure.

_28:56_: If we relax that slightly

_28:57_: and talk of a perfect fluid,

_28:59_: a perfect fluid has no preferred

_29:01_: direction so that the spatial part.

_29:03_: So for the for the incrementum

_29:05_: tensor describing that perfect fluid

_29:07_: therefore cannot have any asymmetry,

_29:09_: must be symmetrical.

_29:12_: All directions in the all spatial directions.

_29:15_: So that for a perfect fluid.

_29:18_: And the.

_29:21_: IG components the spatial.

_29:25_: Spatial sector of the energy momentum

_29:28_: tensor has to be proportional.

_29:31_: To the to the identity

_29:34_: matrix and by comparing what?

_29:42_: What we've just seen is that

_29:45_: it's plausible to identify the.

_29:48_: The the spatial part of the

_29:51_: energy momentum tensor as the.

_29:53_: And at having the dimensions

_29:56_: of pressure per unit. Time.

_29:60_: Of of force per unit time,

_30:02_: which is pressure.

_30:04_: Then the spatial sector.

_30:07_: Is it good to pee?

_30:11_: Delta IG. And for a for dust put,

_30:15_: the pressure is 0, therefore the spatial

_30:18_: sector is going to be 0, so it's

_30:20_: detrimental density is is very simple.

_30:24_: Umm.

_30:28_: Key now.

_30:33_: In the.

_30:36_: And.

_30:40_: I. OK.

_30:46_: Through for further. So yeah,

_30:50_: like hand waving, pushback.

_30:51_: I don't like this hand waving,

_30:53_: but it's, it's, it's, it's sort of

_30:55_: necessary to get to the the, the,

_30:57_: the end point we can end up concluding.

_31:04_: That.

_31:09_: And.

_31:13_: For a perfect fluid.

_31:18_: That the form for the energy momentum

_31:20_: tensor of our perfect fluid.

_31:22_: So that's something which has no

_31:25_: viscosity but does have pressure.

_31:27_: Is ends up being a very simple form,

_31:30_: which is just that you cross you.

_31:32_: With terms involving the mass

_31:35_: density in the restream. The.

_31:40_: The pressure, so P here is pressure

_31:43_: rather than momentum slightly

_31:45_: unfortunately with an extra term

_31:47_: which is the just the metric.

_31:50_: And in the case of dust

_31:53_: where the pressure is 0.

_31:56_: That ends up being simply rule. And.

_32:11_: 000 for justice.

_32:13_: In specifically the momentarily comoving.

_32:17_: Reference frame MRF.

_32:20_: So that's a long way of of saying

_32:23_: that this energy maintains a

_32:25_: clearly can be quite complicated.

_32:28_: There's also things that could be in there.

_32:32_: But the key things are the.

_32:35_: Masters there.

_32:37_: The actual objects there,

_32:39_: which are expressed in things like

_32:41_: the density, the mass density.

_32:43_: And other motion,

_32:44_: other sources of energy such as pressure.

_32:48_: Which appear when in the case of a

_32:50_: fluid but not in the case of dust and

_32:53_: in the case of of the of the of dust,

_32:56_: that energy metric tensor becomes,

_32:58_: you know, reassuringly simple.

_32:60_: It's it ends up with only one known 0

_33:05_: component in the the framework which

_33:07_: is not moving and Lawrence transform

_33:11_: Lorentz transform into other forms.

_33:14_: Similarly that's.

_33:16_: So this is a diagonal matrix there.

_33:20_: OK,

_33:21_: going on to slightly more substantive things.

_33:24_: No.

_33:25_: Section 412 is another way of arriving

_33:28_: at that sort of sort of inclusion,

_33:32_: which is is more satisfying in some

_33:35_: respects and less satisfying others.

_33:38_: This way of of getting

_33:40_: to that conclusion does.

_33:42_: It celebrates our return to physics

_33:45_: here by by returning to slightly

_33:47_: heuristic physical arguments.

_33:49_: The other way of approaching this is to.

_33:53_: Talk about the.

_33:57_: The way in which you can describe

_33:59_: volumes and describe volumes in

_34:02_: a geometrical satisfactory way

_34:04_: within relativity using one forms

_34:06_: and two forms and similar things,

_34:09_: because one forms as the the clues in

_34:11_: the name at the bottom of a tree of

_34:14_: more complicated geometrical objects.

_34:15_: There's nothing else I want to say about

_34:17_: that other alternative approach to this,

_34:19_: but I but it doesn't have a

_34:21_: dangerous because you might be

_34:22_: interested and it's sort of relevant,

_34:24_: but I'm not gonna say

_34:26_: anything about it just now.

_34:28_: And.

_34:32_: OK.

_34:34_: Would you were doing well?

_34:37_: And now what we're going to do is

_34:39_: to actually talk about the laws

_34:40_: of physics in curved space-time.

_34:41_: So this is really where

_34:45_: we start to discover how.

_34:48_: How all the maths we've done

_34:50_: so far links to to physics,

_34:53_: what we want to understand.

_34:55_: Before we do that,

_34:56_: we ought to make a detour back into

_34:59_: into math by defining defining

_35:01_: a couple of extra objects. Umm.

_35:06_: But the they're fairly routine things,

_35:08_: so they are just contractions

_35:10_: of Riemann tensor,

_35:11_: which turned to be of more ready

_35:14_: significance. So #1 is. And.

_35:20_: What? Is it the remote sensor?

_35:25_: Alpha Beta you knew and contract

_35:28_: it over the 1st. And 3rd indexes.

_35:36_: Then we get an object with a.

_35:43_: We've got a rank two object.

_35:46_: With with the two unmatched indexes,

_35:48_: which is called the Richie Tensor.

_35:50_: So just a contraction of the of

_35:54_: the Riemann tensor you plug in,

_35:57_: plug plug two of the

_35:59_: components into each other.

_36:01_: If we further contract that

_36:05_: and take the the Richie tensor.

_36:09_: And contract over those.

_36:12_: Remaining 2 indexes, then we get a

_36:15_: scaler called the Richie Scaler.

_36:21_: Um.

_36:24_: And this from the from the

_36:26_: symmetries of the Riemann tensor

_36:28_: that we saw briefly last time,

_36:30_: we can deduce that that.

_36:35_: Richie Tensor is symmetric,

_36:37_: so R Beta nu is equal to RU beta.

_36:45_: By we also last time discovered that

_36:48_: in the local and national frame.

_36:51_: That is an explicit expression for the

_36:54_: Richie tensor, which is in terms of

_36:58_: derivatives of the metric. So if we.

_37:03_: And this is not obvious if we ask what about?

_37:10_: Peter. You knew? Comma Umbra and

_37:15_: and differentiate the Richie tensor

_37:18_: with respect to one of the other.

_37:24_: Yeah, according to functions.

_37:27_: And. Do so. Alpha beta.

_37:31_: New Lambda or new? Ohh. Of a beta.

_37:39_: Under you, you we discover.

_37:43_: Or rather, Yankee discovered.

_37:46_: That's zero, that there is a.

_37:51_: Asymmetry of the Riemann tensor,

_37:53_: of of the of the of the of

_37:55_: the differential ruin tensor.

_37:56_: So so these this is the each

_37:59_: of these expressions is a mess

_38:01_: of derivatives of the metric,

_38:03_: so I think it's not too hard to

_38:05_: work that out for yourself to to

_38:08_: to to do the calculation yourself

_38:10_: and discover that it is all cancel,

_38:11_: but it's not terribly exciting.

_38:15_: But this is being done in the.

_38:18_: Local, natural, free.

_38:22_: In the local inertial frame

_38:23_: because the expression for the

_38:25_: Riemann tensor in terms of the

_38:27_: metric only applied in the in

_38:29_: the in the local metal frame.

_38:30_: In the local natural frame,

_38:31_: however, these are.

_38:35_: In the local natural frame

_38:38_: covariant differentiation.

_38:39_: Is the same as ordinary differentiation.

_38:41_: So in that frame it turns out that we can

_38:46_: write turn these commas in semi colon.

_38:49_: This is a trick we did also last time.

_38:52_: Because the in the local inertial

_38:54_: frame the the the gammas that appear

_38:56_: when we go the other way are all zero.

_38:59_: But that is a tensor equation.

_39:04_: So that although we worked

_39:05_: out in special frame,

_39:06_: we've worked to attentional

_39:07_: equation in that special frame.

_39:09_: Which means that it's not just true,

_39:11_: it's it's it's true not just

_39:12_: in that special frame.

_39:18_: And and this. Expiration here is

_39:22_: called the the Bianchi identity,

_39:24_: and it is used for at various times

_39:26_: in the algebra of doing the the

_39:28_: the calculations here to to to get

_39:30_: rid of terms, rearrange things,

_39:32_: and make things disappear.

_39:35_: And if we do this contraction

_39:39_: here on the UM bank identities,

_39:42_: we get similar one which

_39:44_: I'll just write down South.

_39:45_: It's on the sheet due to the new.

_39:49_: Semi colon Lambda +3. Minus R.

_39:54_: Beta Lambda semi colon nu plus R.

_40:01_: Which new lambdas? And accordingly

_40:04_: you contracted Bianchi identity.

_40:07_: And if we can track that again.

_40:10_: We get an expression G alpha beta

_40:15_: semi colon beta. Equals zero.

_40:18_: And you know I I'm missing out a

_40:20_: lot of very boring algebra here

_40:21_: where this could this tensor G.

_40:26_: Is. Formed from the Richie tensor.

_40:32_: And the metric and the Richie scaler.

_40:36_: And this tensor here.

_40:39_: Has this property that is that that

_40:41_: that as a deduction of from this

_40:44_: that is convenient derivative is 0.

_40:47_: And that change is called

_40:49_: the Einstein tensor.

_40:50_: And as you might not be astonished

_40:52_: to discover, that tends to plays

_40:54_: quite significant part part in our

_40:57_: in what we are about to discover.

_40:60_: But the what?

_41:01_: The what the the Riemann tensor? Uh.

_41:07_: There are a lot of tensions happening here.

_41:09_: The Riemann tensor.

_41:11_: Is the one that we discovered contained

_41:14_: all the information about how a vector

_41:16_: changes as it goes around a path.

_41:18_: So you you you plug your your question

_41:20_: in this path that this vector go

_41:22_: around this path you plug that that

_41:24_: that data into the rementer and

_41:26_: what comes out is an is information

_41:28_: about how much that that that

_41:31_: vector vector changes and or else

_41:33_: you talk about the tangent to two

_41:36_: geodesics and the link joining them.

_41:38_: You plug those in and what comes

_41:41_: out is information from the Riemann

_41:43_: tensor about how much that.

_41:45_: That, that, that,

_41:46_: that connecting vector between these

_41:48_: two things, how much that changes.

_41:50_: So there's a lot of information in

_41:53_: the Riemann tensor, the Ricci tensor.

_41:56_: Has fewer degrees of freedom.

_41:58_: A lot of the of stuff had been contracted

_42:01_: away so that that there are animatrix.

_42:03_: There are 4 by 4 components

_42:04_: to the Richie tensor,

_42:05_: where there are 4 by 4 by 4 by 4

_42:07_: components to the Riemann tensor.

_42:11_: But so, but the the virtue change is,

_42:14_: in a sense, all the bits of of information

_42:16_: in the human density that you care about.

_42:19_: It turns out that that that's the bit

_42:21_: that matches most closely the stuff that

_42:23_: is informed of to us about the curvature

_42:26_: of the space-time going through it.

_42:29_: And this Einstein tensor is just a adaptation

_42:32_: of tweaked version of that Richie Tensor,

_42:35_: which is the one with this

_42:38_: interesting property. Which?

_42:42_: Is of significance to us.

_42:48_: And and having. Carefully got to the.

_42:54_: One reason why it's important to get through

_42:57_: this step through that section 1441.

_43:01_: To an incremental tensor.

_43:03_: The previous thing is that the the sort of.

_43:08_: The the punchline of the.

_43:13_: Of the discussion of the elementum tensor,

_43:15_: is that the argumentum tensor?

_43:19_: It's such that. For.

_43:24_: So on physical grounds,

_43:25_: based on the idea that if you have a box,

_43:28_: the total amount of going

_43:30_: into it must be this.

_43:32_: Arrangement going into it must be

_43:34_: the same as the total amount in the

_43:37_: box minus A in the coming out of it.

_43:39_: And a mathematical statement of

_43:41_: that is that the? Convenient

_43:44_: derivative of energy momentum tensor.

_43:47_: Is 0. And the fact that the. The.

_43:55_: This property of the.

_43:56_: There's energy intensive

_43:57_: describing the amount of energy,

_43:59_: the amount of stuff in an area

_44:02_: of space that it has this.

_44:05_: This property here,

_44:06_: and the fact that this potentially

_44:08_: there's the ancient tensor which

_44:10_: is based on the curvature of

_44:13_: the space-time has a similar

_44:15_: property is heavily hinting

_44:16_: that these two things are very

_44:18_: closely related to each other,

_44:20_: and we'll discover that that's true.

_44:26_: Um. OK.

_44:31_: We'll hold on to that thought because

_44:34_: we want to come back to that.

_44:36_: And. I keep forgetting I've got

_44:40_: these quick questions and then

_44:42_: so this is scrolling past them,

_44:44_: it's you have these slides.

_44:45_: It's very useful to have a

_44:47_: have a think about those quick

_44:49_: questions in the way I suggested.

_44:53_: In the past and doing the mental.

_44:57_: What's my first reaction to this?

_44:59_: Then thinking about it and going up?

_45:03_: I I. These questions are useful not just

_45:07_: because they remind you what's going on,

_45:09_: because they give you a bit of

_45:11_: a break and stop listening to

_45:13_: me talking for an entire hour.

_45:15_: Anyway. Um.

_45:23_: Um.

_45:32_: In the last couple of minutes.

_45:34_: I'm going to just edge into.

_45:37_: I'll start again from 2nd 4/4 to two but the.

_45:45_: What we have now.

_45:47_: Now we have all the materials that

_45:49_: we can start doing physics again.

_45:51_: Because we have some mathematical objects

_45:54_: which have properties that are that are.

_45:58_: And that we can attach physical meaning to.

_46:01_: And that have a continuity conditions

_46:04_: such as literal amount of energy,

_46:06_: mental which conserve.

_46:08_: That's conservation law.

_46:09_: Which is implausible.

_46:10_: We have something which describes curvature

_46:13_: in a way which we think is plausible.

_46:15_: Now we have to work out what's the

_46:17_: relationship between those and the

_46:18_: way we do that is by going back to

_46:21_: the ideas that we talked about in

_46:23_: the in part one that they brief,

_46:25_: part one we talked about the.

_46:28_: Things falling in lift shafts

_46:30_: and things being freefall,

_46:30_: and the idea of the local national

_46:32_: frame being the one you're in

_46:34_: when you're either away from or

_46:36_: gravity or in free fall or lift

_46:37_: shaft or or or something similar.

_46:39_: And what we what we've discovered

_46:41_: there is that you can make sense of

_46:44_: all these things happening by talking

_46:46_: about the equivalence principle.

_46:47_: And I think we saw a couple one or

_46:50_: two versions of the equivalence

_46:52_: principle in the in part one.

_46:54_: Which would be to do with seeing that

_46:57_: the experience of being free fall.

_46:60_: And the experience of being.

_47:04_: Of not being in a gravitational field at all.

_47:07_: Are equivalent. Not just hard to separate.

_47:10_: Not just impossible to separate,

_47:11_: but they are actually the same thing.

_47:14_: And So what we're going to do next

_47:15_: is see two further versions of the

_47:16_: code response 2 for the statements

_47:18_: of the equivalence principle, which.

_47:21_: Restate that in terms that we've in

_47:25_: slightly more sophisticated terms

_47:27_: that we have available to us now.

_47:29_: And also we tell us what to do next,

_47:31_: and spoiler what it tells us is that

_47:35_: the physics that we understand.

_47:38_: In special relativity.

_47:39_: Which is basically Newton's Newton physics,

_47:42_: except with with some tweaks.

_47:44_: So the physics we all of our our experience,

_47:46_: the physics we understand when

_47:48_: we jump up and down.

_47:49_: It turns out that because of this,

_47:52_: we can import that directly into the local

_47:55_: inertial frame in a in a curved space-time.

_47:58_: And because we can transform from that

_48:00_: local natural frame to anywhere else,

_48:02_: we at that point can use the fakes

_48:05_: we understand in free fall and

_48:07_: apply it in a curved space-time.

_48:10_: And what that leased with is

_48:11_: the problem of how do we,

_48:13_: how precisely do we discover what

_48:16_: the constraints are that matter

_48:19_: applies to that space-time.

_48:21_: In other words,

_48:22_: how does matter change the

_48:23_: curvature of that space-time?

_48:24_: At which point,

_48:25_: if we understand what the space-time is,

_48:27_: what the what the shape of the space-time is,

_48:30_: and kind of work out,

_48:31_: and we haven't done this yet,

_48:32_: you can work out how things will

_48:34_: move through that space-time.

_48:35_: Spoiler is logistics.

_48:37_: Then we have walked out how things

_48:39_: move in a curved space-time and

_48:41_: we have reached our goal.

_48:43_: So there's only a couple more steps to do,

_48:45_: and each of those steps is in a

_48:46_: way quite short, quite, quite,

_48:48_: quite, quite small step at this

_48:50_: point because of all the work we've

_48:52_: done about understanding duties,

_48:54_: understanding curvature and

_48:55_: understanding the metric.

_48:57_: So this is where it all comes

_48:59_: together to get Einstein's equations,

_49:00_: which are the instance,

_49:01_: the end point of who we are.

_49:03_: So the next time I aim to get to the

_49:06_: point where at least we're at two

_49:08_: and beyond the point where we are

_49:09_: talking about some equations and we

_49:11_: got onto one simple solution and all of G2.

_49:13_: Solutions to integrate.