Transcript of a2-l12
==========


_0:09_: Ohh, this is lecture 12, so we're heading in the home

_0:13_: straight.

_0:16_: A couple of things to mention. There's still a trickle of

_0:19_: things appearing on the paddle. That's good. I think I've

_0:22_: answered everything except the one that appeared yesterday on

_0:26_: the class test. I've been given this dude liberation. I think

_0:29_: I'm going to deem everything up to the end of chapter of Chapter

_0:33_: as being in school for the last Test. So that's up to the end of

_0:37_: the kinematics chapter, but not the dynamics chapter. That's not

_0:40_: because the dynamic chapter isn't important, it's extremely

_0:43_: important, but because I think the idea is in the end a little

_0:47_: more time to ferment in your head and the revision time is

_0:50_: excellent. Time for you to go back over that, start again and

_0:53_: let those things settle down. I think it would be. The

_0:56_: interesting question is because there are nice interesting

_0:60_: questions we can ask about the dynamics chapter.

_1:03_: But the interesting question is would I think, be a little bit

_1:07_: too forceful for just a couple of weeks after you've been

_1:11_: introduced the ideas. First of all, so I could only ask boring

_1:16_: questions in Chapter 7 if I deemed that in scope. So up

_1:19_: Chapter 6.

_1:22_: Other questions of that sort or technical can go in the padlet.

_1:25_: That's good. I like questions on the padlet partly because it

_1:28_: means everyone can see them and everything. It can be reassured

_1:31_: that that that that all someone has phrased my question better

_1:35_: than I could have or that's what I was going to ask for something

_1:38_: like that. So questions are all good.

_1:42_: Anything else?

_1:47_: Move that to say it

_1:49_: and do this.

_1:55_: We talked a bit about

_2:01_: the equivalence principle, last name

_2:05_: and how important it is. Remember that I said the way I

_2:09_: introduced this was talking about being in a box far away

_2:13_: from all gravitating sources, ocean space out of that,

_2:18_: and the idea that while you were in the box, just floating there,

_2:22_: Newton's laws work in the sense that things moving in a straight

_2:26_: line carry on moving the straight line until they hit

_2:29_: something, if it goes. May all those things work perfectly well

_2:33_: when you're in a box

_2:35_: far away from every.

_2:37_: I said then that if you put a rocket engine under that box and

_2:41_: accelerate the box gently at to pick a number at random 9.8

_2:45_: metres per second squared,

_2:48_: then what you would feel,

_2:50_: you you you the the box will accelerate toward the occupants

_2:53_: of the box. They would end up being pushed, pressed to to the

_2:56_: what suddenly has become the floor, and being accelerated at

_2:59_: 9.8 metres per second squared.

_3:02_: And they wouldn't. They would think I might. Perhaps on Earth

_3:05_: I might have. I fallen asleep and I'm suddenly back on Earth.

_3:08_: And the equivalence principle.

_3:11_: You know, version one

_3:13_: says. Not only that would be tricky, but it would be

_3:16_: impossible to tell the difference because the

_3:19_: equivalence principle says

_3:21_: uniform gravitational fields, such as the uniform

_3:25_: gravitational field we feel standing on the on the ground on

_3:29_: Earth,

_3:31_: are equivalent

_3:33_: to frames the accelerate uniformly relative to inertial

_3:36_: frames. Not that it's hard to tell the difference, but they

_3:39_: are indistinguishable.

_3:41_: OK,

_3:45_: that's good. That's interesting. And that also explains

_3:49_: that

_3:51_: equivalence, that statement, these things are equivalent.

_3:54_: Then explains

_3:55_: why all things fall at the same rate, why there isn't a

_3:59_: difference between the inertial mass and the gravitational mass

_4:02_: that I mentioned briefly last time, only to dismiss them as as

_4:06_: as prompted dismissed them as being unimportant, different

_4:09_: from each.

_4:11_: OK, that's what that equilibrium. What explains why

_4:13_: that happens?

_4:15_: Now we can make another version of that, a stronger version of

_4:19_: that,

_4:20_: and see all local free falling, non rotating laboratories, which

_4:24_: we're going to call local inertial frames are fully

_4:27_: equivalent for the performance of physical experiments.

_4:31_: Now all the words in that are important.

_4:35_: Local

_4:36_: means

_4:38_: that there's a a boundary to to your laboratory.

_4:42_: We're not talking about the whole earth here. We're going to

_4:45_: talk about our our frame, which is

_4:47_: whatever size. Now it might be that size, or it might be 10s of

_4:51_: kilometres in size, but there's a boundary, and there's a

_4:54_: boundary in time as well, so we're not going to observe

_4:57_: forever.

_4:58_: So local means a bounded box. OK,

_5:03_: we're not talking about global features here.

_5:07_: Free falling means specifically that it's moving only under the

_5:11_: influence of gravity.

_5:14_: Now that might include not moving. So being way out in

_5:17_: space, far away from all gravitating matter, it's also

_5:20_: moving under the influence of gravity. You know there's none

_5:23_: to affect it, but it's still that. That's that's the only

_5:26_: thing that's going to to to to matter. If there's nothing else,

_5:29_: there's no, there's no engine on.

_5:33_: It also refers to something which is moving purely under the

_5:36_: influence of gravity on Earth, for example such as that.

_5:41_: So if I'm falling, if I'm jumping, if I'm leaping from

_5:44_: point to point in the gazelle like fashion, then I am moving

_5:49_: under under influence of gravity. And in that while I'm

_5:52_: doing that

_5:54_: I am in free fall

_5:56_: and this version and and I'm not twisting. I'm not doing

_5:59_: gymnastics and twisting at the same time. OK,

_6:04_: because rotation is is detectable you you you can tell

_6:07_: if you're rotating.

_6:10_: Ohh, your your ears can tell if you're rotating.

_6:13_: So

_6:15_: all we're gonna call things which are bounded in that sense

_6:18_: and moving under gravity local and national frames. Local

_6:22_: because the local inertial frames because the inertia that

_6:25_: Newton describes in Newton, Newton Newton's laws still works

_6:29_: OK and they match the inertial frames that we have been talking

_6:33_: about in the context of special relativity.

_6:36_: So just to

_6:39_: reiterate, the thing that's the different about the last five

_6:42_: lectures is that we're no longer talking to the special case of

_6:46_: no gravity, but we are explicitly bringing gravity into

_6:49_: the conversation. And the thing about the the equivalence

_6:52_: principle is that how we bring gravity in

_6:55_: the equivalent. The equivalence principle is the bit where we

_6:58_: make the connection between the physics we understand using

_7:01_: physics and the physics we don't gravitational physics.

_7:05_: Now, Newton had an answer for that, but we're not going to

_7:08_: follow Newton and and his law of universal gravitation. We're

_7:10_: going to use this, this, this principle, this statement about

_7:13_: the universe. To make that jump

_7:17_: through all such LAF are fully equivalent, but the performance

_7:21_: of all physical experiments.

_7:24_: That means you can't tell the difference between an experiment

_7:27_: done in one LF and experiment in another lab.

_7:30_: OK, not it's hard, But you can't. And that all physical

_7:32_: experiments. We're not talking about mechanics here. We're not

_7:35_: talking about juggling here. We're not talking about

_7:37_: electromagnetism. We're not talking about biology here.

_7:39_: We're talking about all things that you could do. There is no

_7:42_: experiment you could do could tell the difference.

_7:45_: OK, in other words this is saying something about the

_7:47_: structure of our universe.

_7:50_: There's a very important and and and the difference between this

_7:54_: and the previous version is that the previous version was the the

_7:57_: stepping stone to this. We're talking about uniform

_7:60_: gravitational fields and that uniform gravitational fields is

_8:03_: a statement about non locality is saying this gravitational

_8:06_: fields are the same everywhere. And the word local is saying

_8:10_: we're not going to talk everywhere. We're just going to

_8:12_: talk about a box,

_8:14_: OK, because global effects are are complicated in Geo.

_8:20_: There's another version of this that we'll come to later on. But

_8:23_: this is a a a key version

_8:25_: and and it's worth thinking about that and and where it lies

_8:29_: in the argument

_8:31_: possible times.

_8:34_: I've got two versions of that.

_8:37_: Strange. Anyway,

_8:40_: next thought experiment.

_8:42_: So you're back out in this

_8:44_: box floating in space,

_8:48_: and you have a late. You're just floating there

_8:52_: because there's no gravity. You're just moving purely and

_8:54_: then some influence of the gravity that isn't there.

_8:57_: You train the laser pointer.

_8:59_: You hold up up up to the up to the wall.

_9:03_: You showing it across the room.

_9:04_: It's horizontal because it's perpendicular wise, yes. But you

_9:07_: you just say it's horizontal,

_9:09_: it's going to hit the opposite wall

_9:12_: at the same height as it started off. If if if it's a right angle

_9:15_: to the wall and all that,

_9:18_: you can set that up so it's that that happens.

_9:22_: So the light crosses the the box and hits the other side at the

_9:26_: same height. Thank you. Nothing complicated with that at all.

_9:34_: And that's exactly as you'd expect.

_9:37_: No,

_9:38_: Let's say that you're doing this in a falling lift shaft.

_9:43_: Now, of course the objection is if you're in a falling lift off,

_9:45_: you have other things to worry about. But you might want to

_9:48_: distract yourself thinking about general activity.

_9:50_: If you do so, you will notice that if you did that in the lift

_9:54_: shaft, then

_9:56_: the question is where would the

_9:59_: laser pointer point on the opposite wall?

_10:02_: And the equivalence principle tells you the answer. It points

_10:05_: to the same place

_10:07_: because if you're falling,

_10:09_: if you're in this box, you're following.

_10:11_: That is a local national freedom if indistinguishable from our

_10:15_: frame, which is otherwise moving entirely under gravity. We have

_10:19_: in space and so the same logic would work.

_10:23_: So the equivalence which tells us this would be true in a

_10:26_: falling frame as well.

_10:28_: It would hit the other side

_10:31_: at the same point, because if it didn't you could tell you were

_10:35_: falling. This is another variant of of of the

_10:38_: the the the the first axiom of special relativity that you

_10:41_: can't tell you move.

_10:43_: I can't tell you falling anyway,

_10:46_: but how would that look from the point of view of someone who's

_10:51_: standing outside this left on in safety on the on the floor as

_10:55_: you see the, the, the, the, this lift accelerate away from them.

_10:59_: How would that look to them?

_11:02_: Well it wouldn't take long for the light to cross the the the

_11:06_: the following lift cabin, but it would take a a non zero amount

_11:11_: of time

_11:12_: and in that non zero entertain the lift was fallen slightly.

_11:17_: So that means that the point at which the light arrives at the

_11:21_: other side of the lift cabin,

_11:24_: it's going to be slightly lower

_11:26_: and the point we started off. So from the point of view of the

_11:31_: person standing inside here watching all this and thinking

_11:34_: should I call 999 is if the other things think about that

_11:38_: the light appears to what does not appears to, it does land

_11:42_: slightly lower than it started off. In other words, the

_11:46_: trajectory that they see for the light is curved. In other words,

_11:50_: they are seeing light bend in the gravitational field

_11:55_: at a direct and immediate consequence of the equivalence

_11:58_: principle.

_12:00_: OK.

_12:01_: And that's surprising.

_12:04_: It's a very direct consequence of the principle, but it's

_12:07_: surprising.

_12:10_: Any questions?

_12:13_: I see a lot of This is good. Smiles. That's good. Alex

_12:16_: smiles.

_12:23_: So

_12:24_: let's not talk about light

_12:26_: Apprentice

_12:29_: gonna take some sort of spring guys some some spring loaded

_12:32_: thing with which fired the ball bearing or something across the

_12:35_: lift shaft.

_12:38_: If it's set up like that

_12:41_: with three light bulbs AB&C

_12:45_: and I point this thing this spring loaded thing across that

_12:48_: in non relativistic spring load of thing across the lift shaft

_12:51_: and

_12:52_: fire the ball beating across, is it gonna touch hit AB or C Who's

_12:56_: A

_12:58_: which say B

_13:00_: who says C

_13:03_: put the hand up yet.

_13:05_: All right, we'll pick one there. I pick pick one at random and

_13:09_: who's The

_13:11_: Who say be

_13:13_: which they see.

_13:15_: Talk to your neighbours.

_14:01_: OK,

_14:03_: I hate to break into animated conversations, but that's right.

_14:07_: Again, should aim

_14:10_: me directly. So aim at B

_14:14_: should aim at sea,

_14:16_: right?

_14:17_: I should aim at B because if I were doing this entirely, you

_14:21_: know we out in space

_14:23_: with nothing around me, then Newton's laws say that something

_14:28_: we just fired goes in a straight line until it hits something

_14:33_: so we're out in space.

_14:35_: I should fire I aimed directly at B because the the the the the

_14:39_: ball bearings are going to go non relativistic speed in a

_14:42_: straight line

_14:44_: and if this is happening in a falling lift in contrast

_14:48_: then

_14:50_: they could responsible tells us the same logic. It's the case,

_14:52_: it's not. It's not special to light

_14:55_: it. It works for for everything.

_14:58_: OK, so I should aim directly opposite again. So saying that

_15:02_: if you are in space without the feeling of gravity, yeah be the

_15:06_: same as three fold, yes, there would not be the same as

_15:10_: standing on the ground, not on the ground, no. Yes. So if if

_15:15_: this were not falling but we're stationary on the on the ground

_15:19_: and you fired the thing across then yes from that in that frame

_15:25_: the the ball bearing would would would drop

_15:29_: and So what you would expect is is is

_15:34_: and the the the the the ball would drop. You have to aim

_15:37_: slightly above it in order to get further for it to to to to

_15:40_: to to hit the the the middle bulb.

_15:44_: But the point is that the that the the the government principle

_15:47_: doesn't just apply to light. It applies to to to to to

_15:49_: everything.

_15:57_: OK, next there's another good one.

_16:02_: See, I have a mask.

_16:06_: And so, as we learned a couple of weeks ago, that has a certain

_16:10_: amount of energy equals MC squared.

_16:14_: OK,

_16:15_: just by virtue of its mass,

_16:17_: I let that fall

_16:20_: and it falls through a distance Z

_16:23_: right?

_16:26_: So the energy, the total energy of this mass M is now

_16:32_: the

_16:34_: we must aim plus the

_16:37_: kinetic energy. It's it's picked up by virtue of having fallen

_16:41_: through a potential of of of ZZ. So the energy of this mass at

_16:44_: the bottom of this drop is bigger than it was before, as

_16:48_: you learned a couple of weeks ago, that that's a special

_16:51_: artistic remark that the energy of the mass increases. OK,

_16:57_: but but this this term is is is Newtonian. There's nothing

_16:60_: complicated about that.

_17:03_: So then

_17:05_: turn this mass entirely into energy, into into our an ultra

_17:08_: realistic photon and fired it straight upwards

_17:13_: and he gets to the top.

_17:15_: I turned it back into our particle. No, this is

_17:18_: kinematically impossible. There's no means of doing this.

_17:22_: But say you could. We're in the energy balance here.

_17:27_: You turn it back into

_17:30_: from from from a photon of energy primed to turn it back

_17:33_: into a particle of mass M

_17:37_: which will therefore have an energy of MC squared or M. Now

_17:41_: either we have

_17:43_: invented a perpetual motion machine, which we haven't,

_17:46_: or else

_17:47_: the energy of the photon at the top of its ascent E primed is

_17:52_: just M again.

_17:54_: In other words, for this to all the whole thing to hand

_17:57_: together, it must be that the the photon loses energy

_18:01_: as it comes up, as it climbs up through a gravitational field.

_18:05_: So a photon of energy E here

_18:08_: ends up being found energy primed at the top, which is less

_18:14_: by MGZ.

_18:17_: So photons are red shifted by ascent through a gravitational

_18:22_: field.

_18:25_: Then we can turn we can we can write that down a little more

_18:27_: mathematical detail.

_18:30_: What that means is

_18:39_: have

_18:42_: M

_18:43_: drop M + M GZ.

_18:49_: The photon

_18:52_: E primed equals

_18:55_: and

_18:57_: that is on on screen. Good.

_19:02_: No. That means

_19:05_: that

_19:07_: and

_19:10_: the E = M equals

_19:15_: and

_19:19_: is equal to E

_19:22_: minus MG Z.

_19:29_: E prime is equal to E minus

_19:32_: MGZ

_19:36_: and

_19:39_: uh

_19:40_: which is M plus.

_19:44_: I get this,

_19:47_: but I'm about to write a wrong thing here.

_19:58_: And

_19:60_: and.

_20:04_: And it has a ohh yeah,

_20:12_: I'm about to write something wrong here.

_20:20_: A second

_20:22_: and close

_20:29_: would write him trivial.

_20:39_: That's right. OK,

_20:43_: right.

_20:44_: This

_20:46_: that that the E primed

_20:51_: up. Here

_20:53_: it was just really trivially rearranging that E / 1 plus

_20:59_: GZ. So we can write we we We can write down what the

_21:06_: change in the

_21:09_: in in the energy is at the photon rises from the bottom of

_21:14_: that well to the top.

_21:17_: We can go out on a little further

_21:20_: again, get this right.

_21:24_: You will recall, or what you're about to be told, that photons.

_21:28_: The quantum mechanics tells us that photons have a certain

_21:31_: energy.

_21:33_: The energy of a photon is Planck's constant frequency.

_21:36_: Have you encountered that by this stage?

_21:40_: Something, something, some slightly slightly nervous nodes?

_21:43_: We're not gonna depend on anything, anything deep there.

_21:46_: So

_21:49_: the photo on when it starts off at the bottom here

_21:52_: will have frequency F

_21:54_: which is the energy divided by Planck's constant

_21:58_: and the fortune at the top

_22:00_: will have energy.

_22:03_: You prime equals H,

_22:05_: F prime

_22:09_: and if

_22:16_: and

_22:22_: right if E then here

_22:29_: is equal to M which is E primed

_22:33_: one plus

_22:35_: G and I'm going to write that as the I'm going to change change

_22:39_: rotation here rather than talking about height there. And

_22:42_: we're talking about radial distance R away from the centre

_22:45_: of the Earth, for example

_22:49_: GTR.

_22:50_: Where that's there for that.

_22:54_: Hey there Delta R

_22:58_: and switching from from

_23:01_: E to F that's going to be

_23:04_: each F is equal to HF primed 1 + U Delta R

_23:13_: And I'm then going to say OK

_23:16_: if that's true for our height delta R that is true for an

_23:20_: infinitesimal height Dr

_23:28_: and

_23:29_: F prime is equal to F plus

_23:33_: some infinitesimal mode

_23:36_: PDF. So the FF will be equal to

_23:39_: F plus

_23:41_: TF

_23:43_: 1 + g

_23:45_: Dr.

_23:48_: I can expand that

_23:50_: F plus

_23:51_: PDF

_23:53_: plus

_23:56_: FGDR plus DF

_24:01_: yeah,

_24:03_: but because these are infinite and quantities I'm going I can

_24:05_: ignore

_24:07_: 2nd order in them

_24:15_: and get a DCF

_24:18_: over

_24:19_: R

_24:21_: radio DF

_24:23_: over F

_24:25_: if you could minus G

_24:28_: Dr

_24:30_: which it looks good nice. It's. It has a a simple form

_24:36_: then I'm going to see

_24:40_: that

_24:43_: and again make sure that it's step.

_24:47_: And

_24:51_: if we

_24:53_: rate the gravitational force in terms of the

_24:58_: gravitational potential

_25:00_: then the

_25:03_: I'm missing out some some minor steps that are in the in the

_25:08_: notes will be the Phi by Dr.

_25:11_: So the the the gravitational force we feel is the radical

_25:16_: change of the gravitational potential as we we ascend

_25:21_: through a A

_25:24_: height

_25:25_: and take down trust for the moment

_25:30_: which means this turns into DF by the DF by over F is equal to

_25:35_: minus.

_25:37_: Define

_25:38_: so the change in the frequency of this photon as it rises from

_25:45_: from below to above is proportional to the change in

_25:48_: the gravitational potential. And that's just nothing exotic there

_25:51_: about the gravitational potential. That's just basically

_25:55_: nuisance gravitational potential,

_25:57_: OK,

_25:59_: And we could integrate that

_26:01_: again. I'm gonna miss it the the intermediate steps and get that

_26:04_: the

_26:06_: change in frequency

_26:09_: he's

_26:13_: make sure I've got this right. We up

_26:22_: that if I go from somewhere with the gravitational potential of

_26:26_: F0

_26:27_: movement some of the gravitational potential of F1 of

_26:30_: Phi 1,

_26:31_: then the frequency of the of the light when it goes from the

_26:35_: bottom to the top will be. We'll go from F naughty to F1

_26:38_: Question.

_26:50_: So the the, the, the the the expanding the brackets. OK and

_26:55_: there I'm just expanding the brackets so in the usual

_26:60_: fashion. So F * 1 plus D, F * 1 plus

_27:06_: F * 3,

_27:09_: but F times GDR plus DF times GDR.

_27:14_: We're just expanding that

_27:20_: just

_27:23_: or and

_27:25_: so I'm I'm discarding that because because the

_27:27_: infrastructure quantities in two two small things multiplied

_27:31_: together are small are ignorable small.

_27:34_: So that that that that's the importance of going to of of

_27:38_: talking about of going from Delta R as of macroscopic

_27:41_: difference in our that would be

_27:52_: yes. So the FA council. So zero is equal to 2DF plus FDR.

_28:00_: So

_28:03_: address

_28:04_: cancel.

_28:07_: Ohh yeah so. So yeah, do manoeuvring there at once.

_28:12_: So I'm not gonna do anything more with that, but the I think

_28:16_: the point of

_28:18_: mentioning that is to negative concretise that this difference

_28:21_: in frequency is

_28:23_: directly linked to the difference in the gravitational

_28:26_: potential at the two heights. And you can measure that in a

_28:29_: laboratory,

_28:32_: one of the early collaborations of this in 1950s using the the

_28:37_: most power effect, which is an effect where our photon is

_28:42_: absorbed by a crystal with insane precision, Insane

_28:47_: resolution in terms of the crystal will absorb photons of 1

_28:52_: frequency only with incredibly narrowly.

_28:57_: And what folk did

_28:58_: was the fired X-rays donor tower which was 22.8 metres high, so

_29:03_: not high, I mean a couple of 10s of metres. And found that in

_29:07_: order for the photons to be absorbed in the crystal of the

_29:11_: bottom, they had to be tuned extremely precisely at the top

_29:16_: so that the change in frequency was exactly matched. This so

_29:21_: that is directly experimentally measurable

_29:26_: in a beautiful experiment called based on the most part effect

_29:31_: and it was pounded into Ribka. We did this in 1959

_29:37_: rather beautifully.

_29:39_: OK.

_29:40_: Any questions about that?

_29:43_: So

_29:44_: a couple of mathematical steps,

_29:48_: I think. Quite instructive mathematical steps, but you

_29:51_: know, not something to test you on. OK, I I mentioned that

_29:54_: because I wanted to be very clear that there's no slate of

_29:57_: hand here. You can get to an experimental measure measurable

_30:01_: outcome here really quite promptly.

_30:06_: OK,

_30:08_: good.

_30:13_: Thank you.

_30:21_: And

_30:25_: the next thing I'm going to mention because I think it's

_30:28_: important, but I'm not going to go through in detail because the

_30:32_: out, the, the, the end result of this argument is more important

_30:35_: than the than the details. The details are interesting and I

_30:39_: encourage you to go and look at the relevant section of the

_30:42_: notes. But this the the the point here, and they're referred

_30:46_: to Shields. Photons

_30:48_: is what? If you do this and you do it twice. So you fire a

_30:51_: photon from one place

_30:54_: up, and then a little while later you do the exactly the

_30:57_: same thing again, you fire a another photon from that same

_31:00_: place up.

_31:02_: You can draw that Anand Koski diagram.

_31:04_: So the starting point of this zaxis is pointing upwards. The

_31:09_: time axis is pointing upwards on the graph. So this is going from

_31:13_: a low point to a high point, a photo on going from,

_31:19_: you know, below to above, moving at 45 degrees in the usual

_31:21_: fashion, and then a short while later the same thing happening

_31:25_: again.

_31:26_: And you can see the

_31:29_: time difference. Let's say the time difference between these

_31:32_: two

_31:34_: emissions at the bottom

_31:36_: is some number of

_31:39_: Ohh of of periods.

_31:42_: So at no over FN over that frequency.

_31:46_: And

_31:48_: as we've just established, when the Photon gets to the top, the

_31:54_: the freaking ball have changed.

_31:56_: But everything will still be in feet,

_31:60_: so that the difference in time between B&B primed will also

_32:04_: be a whole number of periods of frequency.

_32:08_: But because the frequency has changed,

_32:12_: a whole number of periods of the light of the light at

_32:17_: height A and a whole number of periods of the light at FB

_32:21_: will be different.

_32:23_: In other words, the time difference between A and A

_32:25_: primed

_32:26_: will be different the time difference between B&B

_32:29_: frame.

_32:30_: So that looks like a parallelogram,

_32:33_: but it's not.

_32:35_: It looks like a prime and B prime should be the same length

_32:39_: and the other same length and periods because the frequency of

_32:42_: different they're not the same length in time.

_32:45_: And that is the first hint

_32:48_: that the geometry of Minkowski diagram in the present of

_32:51_: gravity is a bit a bit a bit different from what you expect.

_32:55_: It's not just flat

_32:57_: because what you you saw in the midfield diagram the the

_33:00_: geometry of the of Minkowski space in special relativity. OK,

_33:03_: it was a bit strange with this, with this odd idea of the

_33:07_: metric, but it was flat in the sense that a parallel

_33:10_: parallelogram would a parallelogram,

_33:13_: and this construction is telling you that's not actually true. So

_33:16_: this is where Cover Show has come in here already.

_33:19_: Non Euclidean geometry has come in here already.

_33:23_: We're only you. I know We're into instruction to to general

_33:26_: activity.

_33:30_: One can go, one can step through that more slowly and I encourage

_33:33_: you to to to look at the relevant bits of the notes, but

_33:36_: that that that punch line is the important point there

_33:41_: and so we quick question similarly on that.

_33:46_: Umm,

_33:47_: a couple of quick questions to help you. You think it through.

_33:51_: The next thing, and this is getting us right to the the the

_33:53_: heart of the problem,

_33:55_: is what if you have two things high above the earth to objects

_34:00_: high above the earth,

_34:04_: and you let them fall?

_34:07_: They're going to fall directly toward the centre of the Earth.

_34:11_: Straightforward.

_34:13_: Not really. I'm not taking anything that exotic

_34:16_: through the start off

_34:18_: a distance XI apart from each other,

_34:21_: OK?

_34:23_: And as you get closer to the centre of the earth,

_34:26_: because they are converging another point, that side of T is

_34:30_: going to get smaller.

_34:34_: So the separation between the two of these two objects will

_34:37_: decrease. It will decrease

_34:40_: faster and faster. Are these things accelerate towards the

_34:43_: earth?

_34:45_: OK,

_34:49_: but so so so the second derivative of PSI

_34:54_: will be known 0

_34:55_: and and the the notes you walk through the step by step. It's

_34:57_: not complicated, but you know I'm not gonna do it.

_35:00_: So the second derivative of X is going to be non zero D2 XI by DT

_35:05_: squared equal not equal to 0.

_35:08_: Does that mean we're accelerating? This is the second

_35:11_: derivative of of of of a separation is we're not zero?

_35:13_: OK, they're accelerating. They're not accelerating

_35:16_: because these two things are freefall.

_35:19_: So if if these two things were in free fall, then they're not

_35:23_: going to feel

_35:25_: push it. You know if you were inside a box

_35:29_: falling like that, you wouldn't feel any acceleration

_35:32_: you've been free for.

_35:34_: So this is a case where the 2nd derivative of a of a distance is

_35:39_: non 0 but there is no acceleration.

_35:42_: What is going on here?

_35:46_: The difference is that we have to be careful what we mean by

_35:48_: the word acceleration.

_35:51_: And by acceleration I do not mean

_35:54_: merely second derivative, opposition being 0 being non 0.

_35:58_: What I mean by acceleration is a push.

_36:01_: So if I if I push you, you are accelerated. If a car pushes me

_36:05_: then the the seat pushes pushes my back and and we move forward.

_36:09_: If I hit the ground, I'm accelerated and there's I I can

_36:13_: feel it. If I turn round, if I rotate,

_36:17_: I can. I can feel it.

_36:19_: So acceleration is force applied

_36:23_: that you can feel and there's and there's nothing relative

_36:26_: about acceleration you can you can definitely feel

_36:29_: acceleration. It's not frame dependent thing. If there's

_36:32_: acceleration is enough to to break something then it breaks.

_36:35_: There's no, there's no frame in which that thing won't break.

_36:39_: OK. So when I say due to what acceleration, I mean

_36:43_: that

_36:45_: but this the two objects A and B are not accelerating towards

_36:49_: each other.

_36:50_: In that sense, because they're both in freefall,

_36:53_: they don't feel any push.

_36:57_: All that there is is a difference in the second

_36:59_: derivative of the separation of separation.

_37:04_: So that is also. That's another thing that's that's that's

_37:09_: pushing us toward an interesting thing here that that that the

_37:12_: coordinates here are not straightforward or linked to the

_37:16_: the, the, the physics. We expect

_37:18_: that that this coordinate of of of physical separation is not

_37:22_: quite what we expect. So we have to be careful what we mean by

_37:25_: the difference in two coordinates

_37:28_: and that have gone. Very important.

_37:31_: So this is all rather

_37:34_: abstract and it's rather a bit of a of a toy setup.

_37:39_: But what if the thing was happening here was not

_37:43_: two things falling towards Earth, but two particles in the

_37:46_: accretion discount of black hole somewhere you know the other

_37:49_: side of the universe.

_37:52_: What we want to do is we want to understand what's happening

_37:54_: there,

_37:55_: understand the physics of what's happening there

_37:59_: and we know that what's happening there is in A-frame,

_38:03_: which is highly governed by the gravity of the the black hole

_38:07_: the seclusion disc is is moving around. So what we want to do is

_38:12_: understand how the how the physics works.

_38:16_: And one way in which that gets complicated is say

_38:21_: I'm in box A

_38:24_: and someone else is, I know it's in box B.

_38:27_: They have some coordinate attached to the frame, they're

_38:30_: framed, some coordinate attached to the box. You know the they've

_38:34_: drawn lines

_38:36_: on the on the side of the box and they've got, they've got a

_38:40_: watch. I want to understand the physics of what they're

_38:44_: measuring

_38:45_: in their books

_38:47_: based on what I'm measuring in my box and my knowledge of the

_38:51_: relationship between these two things.

_38:54_: Because see they were on the other side of the earth, then we

_38:58_: both be quote accelerating towards each other in the sense

_39:01_: that secretive below 0. We've both been 3-4 and each

_39:04_: understand that.

_39:07_: And that brings us to the key thing here,

_39:11_: which is

_39:15_: how do we go from geometry to general activity.

_39:18_: And this is the plan.

_39:20_: First, what do we mean by the geometry of space? I think you

_39:23_: you right, you know roughly what I mean,

_39:26_: but we need to be a little more precise about that. We're going

_39:28_: to, we're going to be in a moment.

_39:31_: How do we describe geometry mathematically?

_39:34_: You know, how do we do calculations with it? And that's

_39:36_: that's not easy.

_39:38_: But that's a separate, separate step.

_39:42_: And how do we make the link from geometry to to gravity?

_39:46_: So what? What we can perhaps describe the the strange shape

_39:50_: of, for example, the Minkowski space in that shows photon

_39:54_: setup. But how do we

_39:57_: talk about that? Step 2 and step three make the link between that

_40:02_: and gravity. And that's the plan for GR.

_40:06_: For general activity

_40:09_: and

_40:11_: step one involves stepping back a bit and think. Having to think

_40:15_: Step 2 is mathematically hard.

_40:18_: Step three is Einstein being clever.

_40:22_: OK,

_40:24_: so we can do step one. We can do steps two. We can do step one

_40:28_: and Step 3

_40:30_: here.

_40:31_: We can't do Step 2 because it's just too hard.

_40:34_: If you carry on your study of of massive astronomy or whatever,

_40:38_: then the generativity course

_40:40_: which does 10 weeks just on Step 2 there

_40:44_: into one and then another 10 weeks in semester 2. Talking

_40:47_: about the consequences of that,

_40:50_: because it's quite challenge,

_40:51_: excellent fund, but a bit, but a bit, a bit, a bit much for a

_40:54_: second year.

_40:56_: So we're going to gesture toward Step 2 and concentrate on step

_40:59_: one and Step 3

_41:04_: because and and and and and and and in a sense, this is the is

_41:07_: the is the payoff for the way I've I've been talking about

_41:11_: special relativity because I have been stressing general

_41:14_: geometry. And where is the physics? All the way through the

_41:17_: discussion of or special activity. So that this section

_41:21_: of GR is we can we can talk about the important ideas in GR,

_41:24_: the important technical ideas in GR, even if we can't go into any

_41:28_: of the details. So this is the payoff for the route I've taken

_41:32_: into special activity.

_41:35_: OK.

_41:37_: Umm,

_41:42_: that is the important bit.

_41:45_: OK,

_41:47_: geometry. I'll just start on this just now.

_41:52_: And

_41:57_: so geometry is the mathematical description of shapes.

_42:01_: And you've you've learned a bit about geometry. You you learn

_42:05_: about angles and sines and cosines and Pythagoras theorem

_42:08_: and parallel lines and all that stuff. And in the 19 century, if

_42:12_: you were an undergraduate, you'd be given Euclid vaccines and be

_42:15_: chased up and down the steps of the Academy,

_42:20_: this, that, and the other, using Euclid's axioms over how to

_42:23_: construct this construction, this using rules and straight

_42:26_: edge, and so on. So there's a lot of intricate details you can

_42:29_: learn about geometry

_42:32_: and because there's a lot of it, and I started off in classical

_42:37_: Greece and it was developed and made more intricate through over

_42:41_: the course of

_42:43_: of my of mathematics from classical race through multiple

_42:47_: different cultures to 13th century and it was thought to be

_42:52_: essentially done.

_42:54_: What can you say? What you say geometry, that hasn't been said

_42:57_: for 1000 years

_43:02_: and it was in the 19th century with Riemann who

_43:07_: discovered that.

_43:10_: Well, one of the of the of the standing puddles of that branch

_43:14_: of of geometry, mathematical geometry was how to prove

_43:18_: Euclid's parallel postulate.

_43:21_: Nucleus Panel postulate said that if you have two lines which

_43:25_: are parallel

_43:27_: and then if I meet.

_43:30_: So that's sort of the definition of parallel means you know you

_43:32_: have two lanes to start off the point of them direction you

_43:35_: could go along those lines and they've just never meet.

_43:38_: And that's perfectly true as far as Euclid is concerned and it

_43:41_: was thought this was hopefully feeling a bit trivial. So there

_43:44_: was a there was like 1000 years of folk trying to say, well

_43:47_: obviously this must just be a consequence of the other axioms.

_43:51_: And so people trying extremely hard you dying poppers because

_43:55_: they were cranks and they said the eye finally proved Euclid

_43:58_: parallel postulate

_44:00_: and but eventually Riemann.

_44:04_: I think we're well,

_44:06_: I think that's slightly unfair, but it wasn't really. We remain

_44:09_: but

_44:10_: the market associated with them. So there's more history of

_44:13_: mathematics there which we'll talk about. But he said,

_44:16_: actually you can't, because it's an independent postulate it it.

_44:19_: It is something you can decide to be true or not to say to be

_44:22_: true.

_44:24_: And if you just say it's true, then what you get is you could

_44:28_: geometry, Euclidean space, the flat space we're used to on

_44:32_: paper and in three-dimensional space.

_44:35_: If you don't say that's true. If you say that you can have two

_44:38_: lanes which startup in parallel and which you could produce in a

_44:42_: straight line and the end up meeting

_44:45_: or or not meeting or something, then you can get a consistent

_44:48_: geometry that way.

_44:50_: And thus was non euclidean geometry born.

_44:54_: And I shouldn't spend much time talking about but it is

_44:57_: interesting.

_45:03_: But what is important is I talk about what geometry is, and what

_45:08_: that means is I talk about what spaces

_45:13_: this is. This is good stuff.

_45:15_: So a space is

_45:19_: space to speak. I mean it's a thing you can move around

_45:22_: and

_45:24_: you can always attach coordinates to that space. So

_45:27_: you can see OK in this space here I I can draw a line on on

_45:30_: on on the ground mark it off

_45:33_: like the walls and and I could identify a coordinate for

_45:37_: everything in that space.

_45:40_: And I can ask questions like well, how far apart is that

_45:43_: point there? From that point there,

_45:46_: and I know how to do that in in Euclidean space, I take the

_45:49_: difference in the X coordinate,

_45:51_: the difference in the Y coordinate, difference in the

_45:54_: coordinate

_45:55_: XL plus y ^2 + ^2 = r ^2, the 3D version of Pythagoras theorem.

_46:02_: And that worked. That's that's a definition of space

_46:06_: in including space in terms of Euclidian or Cartesian

_46:09_: coordinates, the XYZ.

_46:13_: And we've discovered that you could do the same in Mickey's

_46:16_: face, but the only difference is there's a there's a - in there

_46:19_: as well. So it all gets a bit, you know, a little bit

_46:22_: confusing, but the same general idea.

_46:27_: So what the way I can write that down

_46:30_: is I can see

_46:32_: that have not been used too much space here. Alright, that was

_46:36_: 12.2.

_46:38_: Is you can see

_46:40_: that in Euclidean space. Whoops,

_46:46_: the

_46:47_: unit of distance EA squared is equal to DX squared plus

_46:54_: y ^2.

_46:56_: And that's not surprising. That's a differential form of

_47:00_: Pythagoras theorem

_47:04_: and then you can ask questions like say we have

_47:08_: and

_47:11_: this is X.

_47:13_: Why?

_47:15_: See, we have two points. P Naughty

_47:18_: And

_47:22_: P1 and that's that

_47:24_: point. X Naughty.

_47:26_: Next one.

_47:29_: What's the difference between those two? What's the distance

_47:31_: between those two points?

_47:33_: Let's do it the hard way. OK.

_47:37_: The distance between those two points,

_47:41_: imitation rate call

_47:44_: is the integral

_47:45_: from

_47:48_: be naughty to P1

_47:52_: of the

_47:56_: differential

_47:57_: different

_47:60_: step from one point to the next.

_48:04_: What is that? That's going to be the integral from X naughty to

_48:09_: X1 of

_48:11_: DX squared plus the Y ^2 just just using that definition of

_48:16_: our our our differential version of Pythagoras theorem.

_48:21_: But in this part here do you dy is 0,

_48:29_: so that becomes just integral from X naughty to X1 of

_48:35_: square root of DX squared, which is the X which is equal to X

_48:41_: makes not to X1, which is of course X1 minus

_48:45_: X0, which you which you could just said right from the right

_48:48_: from the outset.

_48:49_: So that seemed to be part of a long way of getting to it to an

_48:51_: obvious answer,

_48:53_: but the point is, it shows that the length between between two

_48:58_: points is the integral of the of the differential lengths as you

_49:02_: step along that path. So as you step along the path from one

_49:07_: place to another, the total distance distance you go is the

_49:11_: sum of all the steps you take

_49:16_: and and and this is just a. You know that. That's all I'm seeing

_49:19_: there.

_49:20_: It looks quite exotic. We have written it, but it's an exotic

_49:23_: we're writing something that makes that should, I hope makes

_49:25_: sense to you.

_49:27_: And that's how the metric gets

_49:29_: is used. So the metric is is is defined in terms of of of the

_49:33_: separation between 2 points and infinitesimal difference

_49:37_: distance apart in terms of the coordinates.

_49:42_: Now I can.

_49:46_: That's not the only way I can talk about, I could talk about

_49:48_: distance. I and I I will say this quickly. I know you're all

_49:51_: rushing off to get back, and I could see that get the end of

_49:53_: the next bit quite, very quickly.

_49:55_: In polar coordinates I can see that that distance is equal to

_49:60_: the r ^2 plus

_50:02_: and outward.

_50:05_: Do you think the squared for two things which are separated but

_50:09_: separated in polar coordinates by different distances and

_50:13_: different angles? So

_50:16_: E, Theta,

_50:19_: Dr

_50:20_: and I can do, I can ask what's the the, the, the, the length,

_50:24_: the distance between these two things in the in the same way

_50:27_: you know P1. Yes.

_50:29_: And I end up seeing it X 1 -, X. Naughty

_50:33_: by a long by security route which comes to the same answer.

_50:37_: The point being and I will shut up right after I finish. Saying

_50:40_: this, that although these are two different ways of writing

_50:43_: the same

_50:45_: distance, two different ways of writing what the difference is

_50:49_: between 2222 coordinates, they end up with the same answer for

_50:52_: the total distance.

_50:55_: Met you in that I'll, I'll just touch on that very briefly next

_50:58_: time to use that as the launch party. We're going to the next