Transcript of a2-l11
==========


_0:09_: Hello everybody this is lecture 11 as planned.

_0:14_: Little bit behind schedule, but not too much

_0:18_: was to finish off Chapter 7 today. There's just a little bit

_0:22_: left to do an important bit, but it we can finish that off and

_0:26_: then move on

_0:27_: to talk about General General Relativity. In the second-half

_0:32_: of this lecture and in the remaining 4 lectures of this sub

_0:36_: course

_0:38_: I'll first mention I see that there are questions appearing on

_0:42_: the padlet as good as the one that appeared this morning. I

_0:46_: answered that I take the point about handwriting but the goal

_0:50_: is to show the maths on the on the on the scribbles rather than

_0:54_: you know how beautiful notes. I have in the past

_0:59_: put up sky and diversion of those scribbles but the folk

_1:02_: last year the substitute later on committee said ohh there's

_1:05_: too much stuff in the middle. Oh my God, I'm sorry I said not to

_1:08_: do that, but if folk would like would have a different opinion

_1:11_: this year, I can scan those if that would be useful or if that

_1:14_: sort of not really necessary,

_1:18_: you wouldn't feel strongly about that. I mean me.

_1:24_: So where we are now is

_1:29_: getting on toward, well, finishing off chapter six and

_1:32_: seven when kinematic endemics and

_1:35_: moving on.

_1:36_: Any questions either organisational or where we got

_1:40_: to last time?

_1:45_: Yes,

_1:47_: point would be examinable,

_1:49_: right? I

_1:52_: I'm not. I haven't really thought about that yet. I've

_1:56_: only just finished marking assignment one, right, like at

_1:60_: four o'clock 4:00 last night. So I haven't got as far as that,

_2:04_: but I think in previous years I have.

_2:12_: I I think in previous years I've I've only gone as far As

_2:17_: for six and seven, six and seven. So there's suffering

_2:20_: today. So class is still that's at the end of November 2020.

_2:24_: Third right. Ohh deadline suit.

_2:27_: Thank you. I have to think about that

_2:31_: I I will make a ruling about that shortly. I know when when

_2:36_: as as some point, but I think it would basically be just 66 up to

_2:42_: up to the stuff stuff today but they're not

_2:47_: possibly I'll see what I sent the thing to see, but good

_2:50_: question. Thank you. Anything else there?

_2:54_: OK, let's get going.

_2:57_: OK.

_2:60_: That and

_3:03_: you.

_3:05_: OK Where we had got to

_3:08_: I think was talking about

_3:10_: this

_3:12_: and uh

_3:15_: the

_3:17_: this this collision where if you're if I recall we're looking

_3:21_: at a simplified collision here of two of relativistic party two

_3:26_: things which come in with former mentor

_3:31_: whatever they are and merge into one outgoing object with

_3:34_: appropriate formentor. And we simplify things by saying that

_3:37_: this is going to happen along the X axis. So we're going to

_3:40_: ignore the the, the, the, the Y&Z components will be 0

_3:44_: and these things will be colliding head on into some sort

_3:48_: of outgoing single particle. And what we reduced

_3:53_: last time from the

_3:57_: the conservation of momentum which I said with the other

_4:00_: physical statement that we're making in these lectures was

_4:04_: this table

_4:06_: where the

_4:10_: incoming

_4:13_: velocity

_4:16_: so so the the the the target particle is stationary. The

_4:19_: incoming particle is at nice convenient 1517, so the speed of

_4:23_: light and we watched all the all the other numbers

_4:27_: using the the expressions we have accumulated so far.

_4:32_: And the point is, as we're just seeing the very end last time,

_4:37_: the momenta

_4:38_: are conserved so that you add up the incoming momenta component

_4:43_: by component and you get the the outgoing energy momentum of

_4:47_: particle.

_4:50_: That allows you to deduce the velocity of the outgoing

_4:53_: particle, because it's just the ends up being just the

_4:59_: the X important divided by the you can put it. You'll see the

_5:03_: notes for details.

_5:05_: And

_5:07_: the

_5:10_: masses of the incoming particles are as expected, so that if we

_5:19_: obtained the mass respecting the equation that E squared equals P

_5:23_: ^2 + m ^2, which was one of the things we deduced last time. We

_5:28_: discovered that the mass,

_5:31_: the mass by that by that formula is 8, which is what we put into

_5:35_: this if you recall.

_5:38_: And the oddly enough,

_5:40_: the masses don't add up.

_5:43_: So the energy of a mentor

_5:46_: add up component by component, but the mass it doesn't add up.

_5:50_: And I introduced the terms that the

_5:55_: mass, the Andrew Mentum, is

_5:58_: conserved, meaning is the same before and after a collision.

_6:02_: We will discover that the that that some that things which are

_6:08_: the same indifferent frames such as the length of the energy

_6:13_: momentum vector are frame independent

_6:17_: or invariant is another way of putting that. So I'm introducing

_6:21_: terminology here. So invariant means the same in all frames,

_6:25_: conserved means both we means the same board before and after

_6:29_: a collision, and constant such as the speed of light means it's

_6:32_: the same everywhere. That's the folks of full House of of

_6:36_: invariant and conserve and everything.

_6:41_: That's what we got last time.

_6:43_: No.

_6:45_: That is the so-called lab frame,

_6:49_: and this is a particle physics term. It's called the lab frame

_6:51_: because you know you're hitting something which is stationary,

_6:54_: and if you have a target

_6:57_: of whatever you want and you have an accelerator, then the

_6:59_: target is stationed in the lab frame, obviously. That's why

_7:02_: it's called frame. There's nothing profound to that,

_7:06_: but we can change frames.

_7:12_: Really confusing.

_7:14_: So

_7:15_: let's do that

_7:18_: and.

_7:29_: Repeat that committee that we can see that. Let's see if we

_7:32_: can get this to the right.

_7:34_: OK.

_7:37_: So

_7:40_: by conservation of momentum, we're going to see that the

_7:45_: the Andrew Mentum of this outgoing particle 3

_7:49_: through P3

_7:51_: is going to be gamma 1M1 plus.

_7:57_: Gamma 2M2

_8:00_: is the key component

_8:02_: and

_8:04_: gamma 1M1V1 plus gamma 2M2V2 is the X bullet. And remember the

_8:11_: Y&Z components are 0 here the this gamma one is just gamma

_8:20_: of of of the of the you know corresponding to the velocity of

_8:23_: the first particle and so on. So that all that I've written down

_8:26_: there is the result of momentum management and conservation.

_8:30_: OK, if you incoming memento

_8:34_: PNP two then Oakland 11 is that.

_8:37_: But let's change frame. Now

_8:40_: let's go to another frame. So look at the same

_8:44_: a vector 4 vector in a different frame A-frame which is moving

_8:48_: with speed V.

_8:51_: So what are the components of this?

_8:55_: Victor? There's four vector in that new frame. That's easy,

_8:59_: that's just mechanical. That's just looking at the

_9:02_: transformation equation the the matrix expression at the

_9:06_: beginning of Chapter 6,

_9:09_: and we discover that what we can we can find what the

_9:16_: the 0 component of

_9:20_: made them three in the prime frame is. We can discover that

_9:24_: whatever it is, and we can look at the

_9:27_: one component, the X component of

_9:30_: particle 3 in the primed frame. So this is quite a compact

_9:34_: notation here.

_9:36_: So So what, what? And that will be gamma

_9:42_: V with that's the the speed of the of the new frame

_9:45_: times P31 minus VP3

_9:53_: 0.

_9:55_: I just want to unpack that. I'm not doing. I'm not pulling a

_9:59_: fast one here. This is just the transformation equation that

_10:03_: goes from one frame to another.

_10:06_: So this is one of the lines of that matrix at the beginning of

_10:09_: chapter

_10:10_: 6,

_10:11_: and this looks like

_10:14_: the

_10:15_: and the. The range transmission equation that looks like T by

_10:18_: SVX

_10:19_: is basically what that is, and notice that's P31,

_10:23_: not P3 part, not P3 prime.

_10:27_: This is where my handwriting is fairly neat because maths has to

_10:31_: be fairly neat even if you you can't interpret it massively.

_10:34_: P31 primed is the one component of momentum 3 in the other

_10:38_: frame.

_10:39_: OK, what does that look like with this expression for the for

_10:44_: P3? So we have this is P3

_10:48_: zero and this is

_10:50_: P31.

_10:53_: OK, the one component of P3 that is gamma

_10:59_: V

_11:01_: P31 is gamma 1M1V1 plus gamma 2M2,

_11:08_: E 2 -, V

_11:11_: gamma 1M1 plus gamma 2M2

_11:17_: like that

_11:19_: job done. OK, so that that's that's the the components of

_11:23_: this

_11:25_: but the same incrementum to 4 vector in the other frame

_11:30_: of arbitrary V, speed of that speed along the X axis at speed

_11:34_: V

_11:35_: But we can pick a. We can make a sensible choice of what that V

_11:40_: is, because if we decide, let's pick the frame in which P3

_11:45_: primed,

_11:47_: yeah, P31 primed is equal to 0.

_11:52_: Let's pick the frame in which the outgoing particle has 0

_11:57_: spatial momentum.

_11:59_: They're going the the frame in which outgoing particle is at

_12:02_: risk.

_12:03_: OK, so so so, so, so that will be not the lab frame,

_12:08_: but it will be the frame in which the which is moving just.

_12:11_: So just right for the after the collision,

_12:14_: you're moving alongside the outgoing particle

_12:18_: in that frame.

_12:20_: This would be 0,

_12:21_: and in that frame that tells us what V is,

_12:25_: V will be.

_12:37_: And

_12:38_: what we have that expression over that expression and using

_12:42_: the the velocities that that we have chosen. In this particular

_12:46_: example,

_12:48_: that will be.

_12:56_: Right,

_12:57_: so in this particular example, remember the two incoming masses

_13:00_: M1 and M2 are both equal.

_13:02_: They're both equal to 8, and so they they cancel out. We get

_13:08_: gamma 1V1 plus gamma 2V2 over gamma one plus gamma 2. And in

_13:13_: our in the using the numbers we've chosen that's equal to 3

_13:20_: 5th

_13:22_: which matches what we saw

_13:28_: in this table

_13:37_: that the outgoing particle is moving at speed 3/5.

_13:41_: So just to reiterate, what we've done is we've chosen

_13:46_: to go to A-frame in which which is moving at speed 3/5 in and in

_13:50_: that frame the outgoing particle is dictionary.

_13:54_: OK. And that and

_13:58_: so, so, OK, hold on to that though.

_14:04_: In that frame,

_14:07_: we can look at what we can find out when you calculate what this

_14:11_: table is in that frame

_14:13_: by, you know, transforming these expressions component by

_14:17_: component or or otherwise. And I won't go through that step by

_14:22_: step. There's an exercise which encourage you to do so

_14:26_: and

_14:31_: and we get an expression like this

_14:37_: and here

_14:39_: yeah so so the term of handle turning

_14:43_: to get this

_14:45_: but you can go through it. I heartily encourage you to go

_14:48_: through the steps of this because very instructive just to

_14:52_: get those mechanics in place. What we see is

_14:56_: some different numbers,

_14:58_: but there's also a pattern to these.

_14:60_: You notice that in this

_15:03_: in this frame, the speed of the outgoing particle is 0.

_15:07_: That's by definition. That's because that's how we choose

_15:09_: this frame.

_15:12_: It means that the the speeds of the incoming particles are equal

_15:16_: and opposite because they have equal mass. So in this frame,

_15:19_: the particles are coming in from opposite directions, merging and

_15:23_: just staying there. Because momentum,

_15:25_: Sir,

_15:27_: the energies of the two particles are different, The

_15:30_: spatial momenta the two particles are equal and opposite

_15:33_: and the but they are still conserved in the collision,

_15:37_: so that the energy of momentum of the product particle

_15:43_: is still the sum of the energy momentum of the two incoming

_15:46_: particles. The components are different

_15:50_: because this is the because although this is the same

_15:54_: vector,

_15:56_: the management of it's it's the same P3

_15:60_: because you're looking at it in a different frame. The

_16:03_: components are different but the same rules applied. Andrew

_16:05_: Mentum still conserved

_16:08_: and

_16:10_: and notice the masses are the same,

_16:13_: so massive although the components E&P are

_16:16_: different. The masses of the incoming particles and the mass

_16:21_: of the product particle are as they were before,

_16:26_: of course, because that mass is just the squared length of the

_16:34_: engagement particle.

_16:36_: So it doesn't matter what components what what framework

_16:39_: you're picking. The components will will will, will change, but

_16:42_: they'll change in such a way that this length stays the same.

_16:46_: OK, so that illustrates quite a number of things about the

_16:49_: changing of frames and about and about four vectors.

_16:55_: And that seems a bit like slate of hand because because I

_16:58_: haven't gone through each of those calculations and that's

_17:01_: why I I, I encourage you to to go through those afterwards just

_17:04_: to reassure yourself there's nothing tricky happening here.

_17:09_: Any questions about that?

_17:13_: Thank

_17:14_: and part of this is good about this because there's a lot more.

_17:24_: For example the collisions at CERN I, the collision vertices

_17:28_: at the AT, the four experiments rob around around around the

_17:32_: LHC. They all they are designed so that you have two

_17:36_: conversating beams which hit each other head on. So the lab

_17:40_: frame in that context is the same as the as the centre of the

_17:45_: centre of mass

_17:47_: because precisely in order to maximise the energy available

_17:51_: for for collisions in a linear accelerator like slack, which is

_17:55_: just a long straight accelerator which is a target, the last

_17:60_: frame and the central map centre mental frame are not the same.

_18:05_: But this is astronomy, not particle physics, so we're not

_18:09_: going to go into that too much,

_18:12_: no?

_18:14_: Did anyone think there's anything odd about this

_18:17_: apart from the obvious things? But there's there's a a key

_18:20_: thing that seems very strangely odd about this

_18:24_: in the final column.

_18:30_: That's right. That's it.

_18:32_: So the mass is not conserved.

_18:36_: There appears to be more mass afterwards

_18:39_: in the world before,

_18:41_: so there's more gravity there than there was before.

_18:45_: Where's this mask come from?

_18:47_: And that is strange.

_18:50_: And you can push away at that

_18:53_: short circuited by seeing that mass is not the source of

_18:56_: gravity.

_18:58_: Gravity doesn't come from mass.

_19:01_: Gravity comes from energy momentum.

_19:05_: So it's the amount of energy momentum

_19:09_: that is the source of mass, not the motor stuff.

_19:15_: OK. And if you imagine this,

_19:20_: and

_19:24_: so how do how do we explain that? I I've I've ordered this

_19:28_: fairly carefully here.

_19:34_: Yes. So you you you you you you have this this this box. The two

_19:37_: human particles, this collision two incoming particles and

_19:40_: they're going particle and you put it in a box.

_19:43_: OK,

_19:45_: the

_19:47_: afterwards you've got just got this see in the centre of

_19:50_: momentum free.

_19:52_: Afterwards you've got this big lump of of stuff sitting there,

_19:56_: which probably very hot and and so on. There's a lot of a lot of

_20:00_: gravitating stuff before you seem to have less mass.

_20:04_: But the particles were moving very rapidly.

_20:07_: In other words, there was lots of oomph in there but before the

_20:11_: collision. So you have two things colliding and then just

_20:13_: stopping you. You end up with one heavy particle afterwards,

_20:17_: before you're too lighter particles, but they were moving

_20:19_: very rapidly

_20:21_: and so it's that energy Momentum

_20:24_: that is contributing to the grip, the the the gravitation.

_20:27_: So it's not mass that gravitates the energy momentum and the fact

_20:30_: that you have a lot of energy in the box,

_20:33_: not all of it in the form of mass is what gravitates

_20:38_: and we'll we'll come back to that in in, in in the other

_20:41_: lectures. But that's an important thing to to stress and

_20:44_: and and to to think about and you walk home

_20:49_: great. I'm I'm keen to to press on

_20:53_: and I'm going to.

_20:60_: Is not mass at the source of gravitation but the formentor?

_21:05_: OK

_21:07_: no I'm also I'm I'm going to quick fairly quickly mention

_21:12_: another change of units but this is much less confusing than

_21:15_: natural units

_21:17_: you will have

_21:19_: yeah I'm sure you'll recall I have you come across the Janski

_21:23_: has that been you know the Janski and the Janski just it

_21:26_: sounds exotic unit but all it is is a a convenient name for our

_21:30_: our convenient small number and a convenient small small amount

_21:34_: of of course of course you you observational astronomy and and

_21:38_: it's the it's units of flux per square metre

_21:42_: mumble mumble. I I think so, but it's it's given a name because

_21:46_: it's a convenient small quantity

_21:50_: in the context of particle physics.

_21:53_: You have particles moving around all over the place and there are

_21:57_: convenient unit of energy is the electron Volt which is

_22:00_: that number of joules is not very many joules but it's that

_22:04_: number of joules. And one electron Volt is the amount of

_22:07_: energy and electron has one that's been accelerated through

_22:11_: 111 Volt.

_22:12_: So it's, you know, it's a nice sensible unit and I'm not gonna

_22:16_: say very much more about it other than that it's a

_22:20_: convenient unit to to use.

_22:23_: And uh,

_22:32_: yeah, there's there's much more that there's excitement about

_22:35_: that.

_22:37_: But if I do refer to it later on, then that's the section to

_22:41_: go back and remind you what I mean.

_22:44_: We'll finish off with a worked example.

_22:49_: A very important work. Example.

_22:52_: But Compton scattering, so-called

_22:56_: through this you are familiar with Thompson scattering. I

_22:60_: presume that's the that's why the sky is blue.

_23:03_: Our electromagnetic wave comes along,

_23:07_: accelerates our charged particle and which radiates and the

_23:12_: colour change and and the and the electronic wave is scattered

_23:17_: by different modes depending on on the frequency. That's not

_23:20_: we're talking about here.

_23:22_: We're talking here, is an electron

_23:27_: an incoming photon?

_23:31_: Which bounces off

_23:35_: the electron,

_23:36_: scattering it.

_23:39_: And this is a a quantum mechanical collision.

_23:42_: We didn't go into the details, but it is a quantum mechanical

_23:45_: collision in which the photon as a particle collides with

_23:48_: electron as a particle and the two recoil in the same way that

_23:51_: you are familiar with from Newtonian physics. But in this

_23:53_: case, we're one of the one of the the particles of the

_23:56_: collision is a photon,

_23:58_: No? We want to analyse this collision,

_24:03_: so we'll set up with the incoming

_24:07_: photon

_24:08_: having energy

_24:10_: Q1

_24:12_: because it has Planck's constant times

_24:15_: incoming frequency.

_24:17_: Outgoing fortune will call energy Q2HF2.

_24:23_: The outgoing electron will have mass M and outgoing energy E and

_24:30_: momentum P

_24:32_: at at certain

_24:35_: scattering angles, and this is just the.

_24:40_: Figure 7/3.

_24:44_: So we have all the the the dynamical information there. We

_24:49_: can write this down component by component and conserve the

_24:53_: incoming

_24:55_: momentum, energy momentum

_24:58_: through the collision

_24:59_: and find out and and and balance that balance. That

_25:04_: right.

_25:06_: What we discover is that the incoming

_25:09_: right?

_25:12_: And

_25:14_: what we write down is the

_25:16_: that before

_25:18_: the electron

_25:20_: is stationary, so the

_25:23_: zero the the the 0 component of the energy. Momentum is just its

_25:26_: mass,

_25:29_: there is

_25:31_: and it's not moving. So the XY&Z components of momentum

_25:34_: are 0

_25:36_: the.

_25:41_: Before

_25:43_: the mention of incoming photon

_25:46_: is going to be Q1, but it's energy and it's moving entirely

_25:50_: in the

_25:52_: X direction and so in order for the length, the length squared

_25:58_: of this formant to be 0, we can deduce that the

_26:03_: X component of its

_26:06_: energy momentum

_26:08_: is going to be the same

_26:11_: because it's moving and and it's all in the important because

_26:15_: it's moving along the X axis. So they are Q 1 ^2 -. Q one squared

_26:19_: is equal to 0 as a photon 4 momentum must be

_26:23_: afterwards

_26:25_: the.

_26:26_: Similarly the the the electron's outgoing electron has energy. E

_26:32_: mean we've just decided to label that component east and the

_26:40_: components of their spatial momentum

_26:44_: are going to be peak Cos Theta & Cos Theta.

_26:48_: We're just doing the same thing that you've you've done, you've

_26:51_: rehearsed in previous years with

_26:56_: conservation of momentum. The only different thing, the only

_26:58_: difference here is we've got a fourth, a fourth component in

_27:01_: here,

_27:02_: Peter. Gamma is going to be

_27:05_: Q2 the energy of the outgoing photon and similarly the the the

_27:11_: the 2X and Y components of the spatial momentum Q2

_27:16_: course Phi Q2.

_27:20_: Fine, fine.

_27:22_: And again you can see that Q 1 ^2 -, Q two Cos Cos Phi squared

_27:26_: minus Q2 sine Phi squared is going to be equal to 0

_27:32_: as it has been the case for a photon.

_27:37_: And we can balance this component by component

_27:40_: M

_27:42_: M plus Q1 is equal to east plus Q2,

_27:52_: Q1

_27:55_: If you could do a P

_27:58_: Cos Theta

_28:00_: plus Q2.

_28:03_: Of course Phi

_28:07_: is equal to

_28:09_: app sine Theta

_28:13_: plus Q2

_28:16_: saying Phi and so on

_28:19_: and I won't go through the IT would a little more time. I

_28:22_: would go through the the the the step by step. I won't. I won't

_28:25_: do that. It's in the notes. The the point is that what we're

_28:29_: doing here is exactly the same as what you've done in previous

_28:32_: years in Newtonian physics. And we

_28:35_: add everything up, Do a bit of algebra, Slightly fiddly

_28:38_: algebra, but not not hard,

_28:40_: and discover that

_28:44_: went over

_28:47_: Q 2 -, 1 over

_28:49_: Q1 is equal to 1 minus

_28:53_: cost 5 over

_28:56_: M

_28:58_: Or in terms of. You know, if you remember that the

_29:03_: Q is just the

_29:06_: in terms of frequency and thus in terms of the wavelength. So

_29:10_: that Lambda

_29:12_: 2 minus Lambda one is equal to

_29:16_: thanks constant over M

_29:18_: 1 minus

_29:20_: course 5

_29:23_: Compton Compton formula.

_29:27_: And this is not

_29:30_: a a Doppler shift. So the the the outgoing photon has changed.

_29:34_: Its

_29:36_: a wavelength, it changes its energy.

_29:39_: But it's not a Doppler shift because we we're not, we're not

_29:41_: talking about changes of frames here. There's no Lorentz

_29:43_: transformation here.

_29:45_: What we're doing here is conserving momentum, the rest of

_29:50_: us momentum. But we we can discover the prediction for the

_29:55_: change in the photon energy in this case, which is amply

_29:59_: verified by experiment. And this is you can do this in the lab.

_30:06_: This process is also important Astro physically because the

_30:10_: process of of so-called inverse competence gathering where you

_30:14_: go the other way around and

_30:16_: thermal in Blackpool accretion discs are thermal photon can be

_30:21_: scattered off a high energy electron. So this is the

_30:26_: opposite process where our photon collides with our a

_30:30_: realistic electron and increases its its energy. So you get X-ray

_30:35_: emission from Blackpool accretion discs because of the

_30:39_: inverse of this process.

_30:42_: Yeah,

_30:44_: in a way that a particle physicist would would relate at

_30:46_: an astrophysicist would like delight at.

_30:50_: So I I skimmed over a couple of of of of algebraic details, but

_30:54_: you can sort those out.

_30:59_: So that brings us to basically the end of the special activity

_31:03_: part of this course.

_31:05_: We're going to go into Gianni in just a moment, but

_31:11_: it brings, it brings it then rather neatly with an

_31:14_: application of all this stuff that you've been learning about

_31:18_: relativistic trains and and and so on in a way which is

_31:22_: important for particle particle physics and for astrophysics.

_31:27_: There are other sorts of other applications of this to.

_31:37_: To to the realistic version of quantum mechanics is Rossford.

_31:41_: Quantum mechanics is what allowed people to discover the

_31:46_: idea of the of the neutrino and so on. Quantum field theory,

_31:50_: which modern particle physics is based heavily on, is founded on

_31:55_: special activity.

_31:57_: So in the sense that

_31:60_: review theory is just particle, it just quantum mechanics redone

_32:04_: in with the assumption that the universe is based on special is

_32:08_: structured and special relativity.

_32:11_: So you won't. So in a sense you might not use special whatever

_32:15_: again in a specific in this is where my special effects come

_32:20_: in. You won't have to to calculate the speed of rustic

_32:23_: trains,

_32:25_: but in order to understand where Rusty 1 mechanic comes from, in

_32:28_: order to understand where quantum field theory comes from,

_32:31_: you will have to be thinking in a in a special artistic world.

_32:36_: And the interesting thing and interesting thing is that what

_32:39_: we've covered here

_32:42_: is basically all the special activity there is

_32:45_: that isn't sort of advanced special relativity,

_32:48_: right overly in the special case of new acceleration is

_32:52_: sort of done

_32:55_: OK.

_32:57_: The the more general case of relativity with acceleration or

_33:00_: relativity with gravitation is what general relativity is, that

_33:04_: that's what the general is in general relativity. And that's

_33:07_: what we're going to go and talk about that in a moment.

_33:11_: But there's not more of it as such. And I hope that you have

_33:17_: at the beginning of an actual one. I said that that there

_33:22_: wasn't a lot of

_33:25_: hard maths in this course in the sense that what the what would

_33:29_: addition to fraction, multiplication, division and

_33:32_: square root. And OK, we got 4 vectors as a sort of extension

_33:36_: to three vectors and that's a bit of maths but that's

_33:39_: basically all there is.

_33:43_: I hope you realise that's that's true, I said. But also I think

_33:47_: you're also congratulate yourself having got here.

_33:50_: Because although the the the mathematical bricks and mortar

_33:53_: that you're using are quite simple, the way that you've had

_33:57_: to put those together and think in our from

_34:00_: very apparently straightforward principles, you know the two

_34:04_: axioms

_34:05_: to some really quite strange ideas is quite hard work

_34:10_: and in some ways equip mathematical way of approach.

_34:14_: And I think it is strange to look back and think those two

_34:18_: axioms are both plausible.

_34:21_: But you get those. You put those into your head,

_34:24_: step forward and dropped in a rabbit hole

_34:27_: and you think what we is up.

_34:30_: It's very strange. So it is strange and discomforting. But I

_34:34_: hope you have the the haven't been any missing steps along the

_34:38_: way. We're sort of tiptoed through that whole landscape

_34:42_: and got somewhere really quite exotic

_34:45_: step by step. So well done.

_34:50_: Umm,

_34:53_: any questions before we move on?

_34:57_: OK

_34:59_: then we shall. Let's go back to here,

_35:06_: did the picture of conference gathering, right.

_35:16_: Thank you. No,

_35:23_: at this point we somewhat change gears because we we have to

_35:28_: chapters one to seven about special activity. As I've just

_35:32_: said chapters 8 and 9 out about general activity. Now I said

_35:35_: I've just finished seeing that the maths of special relativity

_35:39_: is nice and simple. It's school maths.

_35:42_: The same is not true for GR.

_35:45_: The master GR is advanced undergraduate or graduate level

_35:49_: maths, and if you carry on with doing astronomy, masters or the

_35:52_: theoretical physics course or a couple of other things of map, I

_35:56_: think maths and astronomy, I'm not sure the variety of courses.

_35:60_: Then you will have the opportunity to do the general

_36:03_: activity course in either your 4th or 5th year for you. It'll

_36:06_: be in your fifth year if you if you do that. And I actually

_36:10_: teach that course as well, but it kept it to there because it's

_36:13_: it's the maths is challenging enough that you need a lot of

_36:17_: practise to go up to that point. So we're not going to touch that

_36:20_: maths here.

_36:23_: But because of the way I've done special activity

_36:27_: focusing on the geometry, it's not the only way you can

_36:30_: introduce special activity, but we're focused on the geometry

_36:33_: because of that I think. Well, the point of that, the reason

_36:37_: why I've done it that way is because I think it makes a

_36:40_: Natural Bridge into talking about GR.

_36:43_: So we're not going to do many of the details of GR here, but

_36:47_: because of the last 10 lectures in the last few weeks of of of

_36:51_: of relativity, we can go into a lot of the ideas

_36:55_: with a lot more sophistication than any sort of popular

_36:58_: account.

_36:59_: OK, so, so pop accounts, you know, we'll we'll leave the

_37:02_: hands about, you know, curved rubber sheets and all that

_37:05_: stuff. And I'm sure you've seen those sort of things on

_37:08_: television or or or whatever. We can do better than that.

_37:12_: So in a sense the the payoff OF11 payoff of the the last 10

_37:17_: lectures is that we can cover quite sophisticated if not very

_37:22_: technical or if not high not although not hyper technical

_37:26_: account of GR.

_37:28_: So enough rubbing.

_37:31_: One other thing. And so there are a very important aims to

_37:35_: this

_37:38_: appreciate, understand, understand. And those are the

_37:42_: point of all this.

_37:44_: The objectives, however, are thin

_37:49_: because it's not terribly easy

_37:52_: to

_37:53_: right

_37:55_: exercises or homework or class tests or exams which cover this.

_37:59_: So there's a limited number of things that I'm going to be able

_38:03_: to say that are accessible,

_38:06_: but I'm not gonna let that stop me.

_38:08_: So in these, in this part of the course, I will be seeing things,

_38:12_: a lot of things that aren't basically accessible because

_38:15_: they're not on that list.

_38:18_: So don't panic, right. I I'm going to go fairly rapidly

_38:21_: through this. This part I'm going, I'm going to be, we're

_38:25_: going to be jogging in in these last five lectures, but don't

_38:28_: get stressed because a lot of it isn't examinable, right. And I'm

_38:32_: telling you this because it's wonderful and beautiful and good

_38:36_: for your intellectual and moral development, right. It makes you

_38:40_: better people for having struggled with this, right? But

_38:43_: don't get anxious.

_38:46_: I I feel it's important to see that right, because people do

_38:49_: get anxious, right.

_38:51_: But pay attention to the objectives. Those are the things

_38:54_: that I think are fair, that I think will be fair. Again,

_39:00_: generativity. As I said, the general in general relativity

_39:04_: is not the special case of no acceleration and not the special

_39:08_: case of no gravity.

_39:10_: Because although when we've been talking about special activity,

_39:13_: we talk about trains move through stations,

_39:15_: You know, the trains are on the Earth and are held down by

_39:18_: gravity and and so on. But we've ignored the gravity bit, the

_39:21_: gravity but hasn't been important to the to to to to to

_39:24_: what's happening. Things are going all, all all the trains

_39:27_: are being moving along level train tracks. There's been no

_39:30_: gravity. We're just ignored it.

_39:34_: Newton has a theory of gravity which works very well. You can

_39:37_: get to the moon and back

_39:39_: with Newton theory of gravity.

_39:41_: OK, so it's not wrong. It's just as we discover about to discover

_39:47_: doesn't quite get to tell the full story.

_39:52_: But Newton

_39:55_: it's something false if I have an apple to pick an example at

_39:59_: random and I drop it.

_40:01_: Then, as you know,

_40:03_: the force of gravity

_40:06_: acting on that apple is proportional to the mass of the

_40:08_: Earth and proportional to the mass of the apple.

_40:11_: Much GM1 M 2 / R ^2. OK. So the bigger the apple they have or

_40:15_: the apple, the more the forces in proportion to the mass.

_40:21_: So there's a force,

_40:22_: so the apple then accelerates toward the ground.

_40:25_: How much is accelerate

_40:27_: if you go there? Me

_40:28_: the apple accelerates in

_40:31_: proportion to or inverse proportion to its mass.

_40:35_: So if I double the mass of the apple,

_40:39_: the force acting on it of gravity doubles

_40:42_: I think, but the acceleration halves,

_40:46_: you know, so they they they they they just balance out. In other

_40:50_: words, if I have a an apple and dirty great gold bar

_40:54_: or an apple and a feather in a vacuum, they will fall at the

_40:57_: same rate because those things cancel out.

_41:02_: That's not surprising. You may well have seen videos of of

_41:05_: Apollo 16 or whatever it was and and whichever astronaut was

_41:08_: dropping a hammer and a feather on the on the moon. I believe

_41:11_: they got terrible, terrible trouble from Mission Control for

_41:14_: doing that. They weren't supposed to do that. They

_41:16_: smuggled on board just because it was so much, so much fun to

_41:19_: do it on the moon.

_41:22_: But this is strange.

_41:24_: You think that's that's fairly obvious, right? But this is

_41:27_: strange because the two masses there are different things.

_41:32_: The mass that is acted upon by gravity is a sort of

_41:35_: gravitational charge. That's a that's a you can think of as the

_41:39_: gravitational mass of an object. It's how how much is it affected

_41:44_: by gravity.

_41:46_: The mass in the second equation

_41:48_: is inertial mass. It's a different thing really. That's

_41:51_: that's the the, the, the the the the the property of the Apple

_41:54_: hires that that lets it resist being accelerated and there's no

_41:57_: reason why they should have to do with each.

_42:00_: So you could imagine them being quite having quite different

_42:05_: relationships to the dynamics.

_42:08_: But it turns out in Nuisance Britannica nuisance and

_42:11_: dynamics, they are exactly proportion,

_42:14_: not just proportional ish, they are exactly proportional. So it

_42:18_: seems to be no difference between these two on the face of

_42:21_: a completely different things, a national mass and gravitational

_42:25_: mass. And that is strange. And Newton thought it was strange.

_42:29_: He noticed it and said I had. I have no idea why that's true

_42:33_: in the in the general scholium at the end of of of interview he

_42:36_: basically says. I don't know why that's true.

_42:38_: I don't know how gravity works, she said.

_42:41_: As Newton did not claim to understand gravity, he said I've

_42:44_: got an equation which describes it, and if I start with the

_42:47_: right place, all the maths works out and and and all works, but I

_42:51_: don't know what gravity is, he said.

_42:57_: And that puddle remained.

_42:60_: But let's step aside from that for a moment and go out into

_43:03_: space.

_43:05_: And imagine you're in a box

_43:07_: out in space,

_43:10_: breathable, see, So you're you're you're well away from

_43:13_: everything. You're in orbit or you're in between the stars or

_43:15_: or something. So there's no gravity,

_43:18_: right? And everyone just floats around

_43:21_: and if you, there's you, there's a clock, there's photons, the

_43:24_: electromagnetism, there's biology happening. There's

_43:27_: anything like happening in that box

_43:29_: and everyone's just floating around

_43:32_: and the thing that's the thing that is still true in that box

_43:36_: is that

_43:37_: Newton's three laws work. So if you

_43:41_: if if you sort of push the a clock across the across the

_43:44_: cabin, it'll move at a constant speed until it hits until it

_43:48_: hits the end.

_43:49_: If you pull something, it'll accelerate according to if it

_43:52_: was me, and so on. So nuisance laws work in that box in exactly

_43:56_: the way you'd expect,

_43:59_: of course,

_44:00_: because why wouldn't?

_44:05_: Then let's start a little rocket motor under the box

_44:11_: and you turn it on and then the box would move

_44:15_: but we are floating around inside it and nothing happens

_44:19_: until the the box reaches us

_44:22_: and and start pushing it. So before the box touches us,

_44:26_: we're not going to

_44:27_: be affected by the book obviously.

_44:30_: But when we get home at the box does reach us, then we'll be

_44:34_: pushed by the floor of the box. And if the rocket motor is is

_44:37_: set so that it produces an acceleration of 9.81 metres per

_44:41_: second squared, then we are going to feel as if we're

_44:44_: standing on Earth. We're going to be accelerated in the exact

_44:48_: same way

_44:50_: and that's exactly the same we

_44:52_: it's not ish,

_44:54_: it is exactly the same. We, we will not be able to tell the

_44:57_: difference

_44:59_: between

_45:00_: being accelerated in that way

_45:03_: and

_45:05_: being on being on Earth,

_45:07_: OK, And that is a physical statement. That's not a

_45:10_: mathematical statement. That's a physical statement. That's true.

_45:13_: But our universe?

_45:14_: In our universe, we cannot tell the difference between a uniform

_45:18_: gravitational field and an acceleration like that, and

_45:21_: that's called the Equivalence Prince.

_45:24_: And that explains the whole gravitational mass versus versus

_45:28_: inertial mass thing.

_45:30_: Because

_45:32_: in the case where everyone's floating around,

_45:37_: the methods aren't involved.

_45:39_: In the case where the rocket

_45:42_: is pushing the cabinet that that the cabinet up towards us,

_45:46_: everyone's going to inside the cabin is going to be

_45:48_: accelerating towards the base of the cabin in exactly the same

_45:51_: way, independent of their mass.

_45:54_: So the statement that these two things are equivalent

_45:60_: explains why the the the different gravitational national

_46:03_: mass doesn't matter.

_46:05_: And this statement, they equivalence principle, was

_46:08_: enunciated by Einstein and it's what is in it. It is what the

_46:11_: general theory of relativity is based on

_46:14_: that that that variant of that principle is what the whole

_46:17_: thing that that's the the starting point. If you're like

_46:20_: like the two accidental activity, there's equivalence

_46:23_: principle is the starting point for the whole rest of the

_46:27_: elaborate apparatus,

_46:32_: right? OK.

_46:36_: And as I say, I want it just I said I said at the end of it

_46:39_: again it's a physical principle. It could be otherwise, it's not

_46:42_: a mathematical necessity. It could be otherwise, certainly

_46:45_: was it would otherwise, for Newton could be otherwise. But

_46:48_: it is the case in our universe.

_46:52_: Then letters in the last moment, and I will. I will have to

_46:55_: return to this afterwards.

_47:01_: So. So uniform gravitational fields are

_47:04_: are equivalent to frames accelerate uniformly routed

_47:07_: inertial frames, That's the frame that's accelerating

_47:10_: uniformly. Is this rocket frame

_47:14_: another version of that?

_47:16_: And this is the one that

_47:19_: Einstein wrote down. All local free falling, non rotating

_47:22_: observatories are fully equivalent to the performance of

_47:25_: all physical experiments. And all those words are important.

_47:32_: The free falling

_47:33_: and the laboratory is

_47:37_: the laboratory in in in the in the first picture where the the

_47:43_: the the books that are in space.

_47:47_: I I I say clean this with the happiest I've I've I've got

_47:50_: order wrong here is actually it was a happy thought of his idea

_47:54_: was what happens if you are in freefall so just falling purely

_47:57_: under gravity

_47:60_: if you if you jump out of a window

_48:02_: in a in a box or or or you know lift shaft and the lift cable

_48:05_: breaks bad situation you're gonna have a bad day. But on the

_48:09_: way down you can think you can meditate on the on the on the on

_48:12_: the delights of of young relativity as you go and that

_48:16_: situation where you are falling freely under under gravity you

_48:19_: and everyone inside the lift cabin and and and your dog is

_48:22_: there whenever and able opportunities are all going to

_48:28_: or being Newton's laws perfectly in the sense of of of of

_48:30_: accelerating here and there So I'm gardening this I'll come

_48:34_: back I will come back to that point

_48:38_: I will come back to the point and see why all all the words of

_48:40_: that are are important and what and what they mean.