Transcript of a2-l06
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_0:10_: This is like the six, which is at which point we go on to

_0:14_: Chapter 5 talking about the Lorentz transformation.

_0:18_: And this, if you like, is the central mechanical algebraic

_0:23_: mathematical tool of special relativity. It's where we learn

_0:28_: how to go from

_0:31_: events which have coordinates one frame

_0:34_: to events which have coordinates in another frame. I'd rather

_0:38_: abstract thing to to worry about, but it's the thing that

_0:42_: ties together the

_0:44_: the things we're learning about if you know more or less

_0:47_: qualitative way in the lectures up to now.

_0:50_: So what we learned about length contraction time relation first

_0:55_: in a in A in a as a qualitative way

_0:59_: with the argument about the trains going past and people

_1:02_: thinking that person's watch is going faster that that that

_1:05_: carriage is shorter.

_1:07_: Then we added some numbers to that and and worked out what the

_1:10_: length contraction and time relation equations would be,

_1:13_: that factor of gamma.

_1:14_: So in a sense we're not adding anything more to that, we're

_1:17_: just doing the same sort of thing

_1:20_: in different form. But it's a crucial form which and and and

_1:25_: the the number of consequences to doing it that way.

_1:31_: And before going on, I want to just

_1:34_: recall a little bit from the the the the last chapter of the last

_1:38_: lecture. Well, I introduced space-time. I talked about the

_1:42_: Minkowski diagram and so on and

_1:45_: think about it. I think that is one of the harder lectures of

_1:48_: this course.

_1:50_: I mean, it seemed to me, partly because it seems quite bitty,

_1:53_: seem to remember different things happening there, and

_1:55_: suddenly, magically these Minkowski diagrams are appearing

_1:58_: in the terribly important and we're talking about space-time

_2:01_: and geometry and all that stuff. And so there is quite a lot

_2:04_: happening there.

_2:06_: There's not a lot of maths and there's not a lot of astronomy,

_2:10_: but there are a lot of physics happening there,

_2:13_: which it does take time to it, which is a challenge and it's

_2:16_: not. It's a challenge for second years and you know, unless years

_2:20_: it would be easy. I think it would be a challenge for anyone

_2:24_: in any year because it is

_2:26_: intellectually

_2:28_: difficult to get your head around those things because

_2:30_: you're being asked to think about things in a way that you

_2:33_: fact wouldn't have been in in previous years. The this, this

_2:36_: is, this is where I think in the first lecture I said this wasn't

_2:39_: wasn't a mathematical course.

_2:41_: It's not a mathematical court, in the sense there isn't any

_2:44_: complicated algebra, there's not any complicated mathematics.

_2:47_: It is quite a mathematical course, in the sense that the

_2:50_: way you've been great to think about things is quite abstract.

_2:53_: And that's and that is in a nontrivial challenge

_2:56_: project, that's if you're looking at the last thing going,

_2:59_: I have no idea what's going on, then that's fine,

_3:04_: work hard. It just take a bit of effort. But it does pay off in

_3:09_: wonderful insights into the nature of being. Anyway,

_3:15_: this

_3:18_: one of the things I one of the important things which may have

_3:21_: seemed a curiosity last time

_3:24_: is the idea of the very interval.

_3:28_: And I said, if you recall, the point of the interval was that

_3:32_: it was like Pythagoras theorem

_3:35_: in the sense that if you turn round

_3:38_: distances don't change. If you change your coordinates from

_3:41_: that coordinate system to that coordinate system, the length of

_3:44_: things that you measure you know in the squared distance between

_3:47_: two points

_3:48_: is invariant. And that is a deep fact about the geometry of

_3:53_: Acadian space. Euclidean space is just nothing more than the

_3:57_: the space we're used to the the the space where expert plus y ^2

_4:01_: is an invariant of translations and rotations.

_4:05_: There's this species that you're used to.

_4:09_: So putting that Euclidean space, I've, I've been nothing more

_4:12_: exotic than than, than than that which you never had to give a

_4:14_: name to before.

_4:16_: So we're going to use the fact of the

_4:20_: invariance from invariance of the interval, the t ^2 -, X ^2

_4:26_: and see

_4:28_: if we start from that point of view

_4:31_: and and, and in a sense that's just a restatement of the second

_4:35_: axiom because it follows from the second, the the, the, the

_4:38_: 2nd axiom that I introduced in the first or second lecture. So

_4:41_: there's in a sense nothing new, but this is a new way of

_4:44_: starting from that. If we start from that, what do we get?

_4:52_: Any questions of that or worries about from last time or or the

_4:57_: one thing that I do remember someone pointed out that in the

_5:02_: notes from lecture you know for for for, for chapter 5. At the

_5:06_: very end of it, some of the figures have ended up on the

_5:09_: wrong side of the page. There's a slightly cut off that's just I

_5:13_: didn't run later enough times so that the I have replaced the

_5:17_: the, the

_5:18_: at the file with the same the same content, just an

_5:22_: externality, so that should be prettier. Just now.

_5:27_: I think that's the only thing I could remember to see.

_5:32_: Something else may occur to me. OK,

_5:37_: what we

_5:38_: why I believing up to there

_5:41_: is that?

_5:43_: Well first of all but before going on I want to just make

_5:46_: clear what the the problem is that we are trying to solve

_5:49_: here. What? What? What we're trying to achieve here

_5:57_: is offering

_5:59_: the next direction and in that time direction and the why and

_6:03_: the Z suppressed. And there's an event and that has coordinates

_6:08_: T&X. It happens at a time and at a position,

_6:13_: an expedition and attending the clock. And what those numbers

_6:17_: are depends on where you choose your origin. You're you're

_6:22_: exactly as is the case in anything else. But there's also

_6:26_: and as you know, familiar with the idea of Mankowski diagram.

_6:30_: There's also another set of of

_6:33_: axes we could choose

_6:37_: and in that frame. That same event

_6:41_: has coordinates T frame and X prime, so the observers in that

_6:44_: other frame would measure that event, which is something like

_6:47_: that. Anything which happens a place and a time,

_6:51_: and they would scrap it different place and say to the

_6:54_: different coordinate.

_6:55_: That's what we're trying to do is given one of these pairs of

_6:58_: coordinates, what's the other one?

_7:01_: OK, that's the the mathematical problem we've got. We're trying

_7:04_: to solve

_7:05_: that. What we can hold on to

_7:08_: the point where we start

_7:10_: is seeing that whatever happens. Ohh yeah, so so so that If you

_7:14_: also imagine

_7:15_: another event which happens at the origin, that's at

_7:20_: position zero

_7:22_: from the special origin and times zero.

_7:25_: And because these two frames are in standard configuration

_7:29_: that origins coincide, the special origins coincide at the

_7:33_: temporal origin. So at X = 0, T equals 0X, prime is equal to 0,

_7:36_: and T prime is equal to 0. That's the definition of static

_7:39_: configuration.

_7:41_: What that means is that there is a different a distance, a

_7:45_: separation, an an interval between those two events and we

_7:49_: know that the

_7:53_: go to t ^2 method delta X ^2.

_7:56_: In both cases one end of it is the origin, so we know that

_8:01_: delta that t ^2 -, X ^2

_8:03_: t prime squared

_8:06_: my ex. Prime

_8:07_: square.

_8:10_: That's our starting point. So the goal is to find an actual

_8:14_: expression for T&XT prime and X prime in terms of T&X,

_8:18_: which makes that true. Now that looks pretty similar, something

_8:22_: we already know.

_8:24_: It looked pretty similar

_8:26_: to

_8:27_: X ^2 + y ^2 = X prime squared plus

_8:32_: my prime squared, the Python green one,

_8:36_: and we knew and

_8:41_: transformation

_8:44_: which preserves that. Can we turn one of these things into

_8:47_: the other thing? Yes we can if we instead of talking about T

_8:51_: we talk about

_8:54_: L which is just T * I, the the imaginary, the unit imaginary.

_9:01_: And I've also into the lecture root folder. On the middle I've

_9:05_: uploaded 2 extra documents, one on maths, one on. We've got our

_9:08_: recipe, which I'll come to in a moment, but the maths one is

_9:12_: just a little bit of of revision. So if you already know

_9:15_: about the actual numbers and hyperbolic trigonometry, which

_9:19_: we'll come to in a moment,

_9:22_: I think just to to to just for religion.

_9:24_: So

_9:26_: L is

_9:27_: T * I and L prime is equal to I times TT prime. Then

_9:33_: L ^2 equal minus t ^2

_9:36_: L prime squared is minus T prime squared and this

_9:42_: turns into

_9:44_: L ^2 + X ^2 = L prime squared plus

_9:49_: explain square

_9:51_: to have turned what we want. The problem we want to solve is the

_9:55_: problem we already know.

_9:57_: Because we knew how to, we have a a transformation

_10:02_: from XL to experiment and L prime which preserves that which

_10:08_: is just X prime is equal to

_10:11_: X Cos Theta

_10:14_: plus L sine Theta. L prime is equal to X sine minus X sine

_10:21_: Theta

_10:22_: plus L Cos beta for some angle Theta. If any angle Theta any

_10:27_: any value of Theta we turn, we take X&L, obtain X prime and

_10:32_: L prime, and the sum of the squares will be the same.

_10:39_: OK, that's fine. So that that's progress.

_10:43_: Now we're also then going to write.

_10:46_: Peter is equal to

_10:48_: I Phi.

_10:50_: So rather than an angle, we're going to see that this this is

_10:54_: the an an angle times the unit imaginary. And if you remember

_10:58_: well, if you you I I trust you, I have heard of hyperbolic

_11:03_: cosines, and hyperbolic you have here.

_11:08_: But what you may or may not remember is that the hyperbolic

_11:13_: trigonometric functions and the circular trig functions are

_11:18_: related by

_11:20_: sign.

_11:22_: I Theta is I

_11:24_: Saints.

_11:27_: I'm going to write that as say I Phi Phi and Cos

_11:35_: I Phi is equal to

_11:38_: koshi

_11:41_: and

_11:43_: we then

_11:45_: plug that into here.

_11:47_: I'm not gonna go through the the the the successive steps, but

_11:51_: the end result when we plug that replace Theta there with. If I

_11:56_: do this, swap this and then turn the handle a bit,

_12:01_: turn L&L primes back into T&T frames is we get

_12:06_: G prime is equal to

_12:09_: he cash Phi minus X,

_12:13_: sanctify

_12:15_: X prime is equal to minus T

_12:19_: Saints Phi plus X

_12:23_: and we're sort of done.

_12:26_: So this means that given a an X&AT coordinate for an event

_12:32_: in one frame,

_12:34_: we can work out what the

_12:37_: quadrant of the same event

_12:40_: and obtained by different observers in two frames of time

_12:43_: configuration are in such a way that preserves the invariance of

_12:46_: the interval at the top.

_12:49_: Job done.

_12:51_: And that basically is job done

_12:53_: in the sense that is the Makovsky, the the Romance

_12:56_: transformation derived

_12:59_: to the rest of this chapter.

_13:01_: You're looking at a couple of variants of that expression

_13:04_: that's that's not the most common expression for that and

_13:06_: it's not the expression of it you'll see most often.

_13:10_: And your used most often in doing all the exercises is ohh

_13:13_: that's. I think that's the other thing I meant to remember. I

_13:16_: meant to remember to say to you, we're gonna talk about the

_13:19_: exercises.

_13:20_: I mean, the exercise is at the back of my notes.

_13:23_: There's also a tutorial handbook I think would be part of the of

_13:27_: the you know up in the middle

_13:30_: and that has had exercises for the four sub part of of of a two

_13:34_: of operational astronomy starting the Spectra theoretical

_13:38_: astrophysics and relativity

_13:42_: and the exercise in there the relatively exercise and there

_13:44_: are good

_13:47_: I'm not going to refer to them because you know while they are

_13:50_: good that they're not keyed to my notes in the way that that

_13:53_: that that that that my now that they're they're not a hangover

_13:56_: but that they are being built up over the years for a two. So

_13:59_: just to be clear, when I talk about the exercise, I mean the

_14:02_: exercise and notes

_14:04_: as opposed to the exercises in the shoreline.

_14:07_: OK.

_14:10_: Who was I?

_14:11_: Ohh, yes, so so the the, the form of the of the Lorentz

_14:14_: transformation that you

_14:16_: and

_14:18_: will use is not not that, not that. We'll come on to that in a

_14:22_: moment,

_14:23_: but the rest of this chapter

_14:25_: is all about using this to do these things and getting

_14:29_: thoroughly familiar with what it means about the the geometry of

_14:34_: the of of the space and time that we're looking at.

_14:39_: And I'll also mention that the, you know, OK, I'll, I'll, I'll

_14:42_: go on, go on a bit.

_14:45_: Um,

_14:47_: no this. See this event here

_14:52_: was happening

_14:54_: that there's a.

_14:57_: We would like to change the scenario and now the

_15:02_: the event we're talking about.

_15:04_: We'll choose always to have X print equal to 0.

_15:07_: In other words, it's at the origin of the moving frame,

_15:11_: which means that the position of that event in the platform

_15:15_: frame, the stationary whatever will always be at

_15:23_: X, equals

_15:25_: Vt. This P times times something which is happening at the origin

_15:29_: of the moving frame is always going to be at that position in

_15:32_: the stationary frame.

_15:35_: OK? Or in other words, OK tricky bit of algebra. Here V is equal

_15:40_: to X /

_15:41_: t If we look at this expression, here the second

_15:46_: the second one

_15:48_: for this event, then X prime we're saying is equal to 0.

_15:52_: So T sange Phi equals X cosh Phi.

_15:57_: Turn the handle a little bit, which over the V

_16:00_: is equal to

_16:04_: times Phi. In other words, we've filled in the remaining bit of

_16:08_: the remaining bit of the transformation, which says that

_16:13_: if these two frames are instant configuration and the second one

_16:16_: is moving at speed V,

_16:18_: then the value of Phi which is called the rapidity. The Phi

_16:22_: which corresponds to that is arctangent of of V and then you

_16:26_: plug that into the into this and get your

_16:31_: and get the answer.

_16:34_: It's we will not we we will in fact not end up using a lot of a

_16:40_: lot of

_16:41_: I've worked trigonometry you may or may not be relieved to hear.

_16:45_: Because

_16:48_: we're gonna get all that all our hyperbolic trig trig done up

_16:52_: ahead of time

_16:54_: and uh

_16:56_: use this to to rewrite

_16:59_: this transformation in a stage different form which I'm not

_17:02_: going to go through the steps. So I think one of the I think

_17:05_: exercise cycle one of 5.2 is encouraging you to go through

_17:07_: these these steps just to show you there isn't anything being

_17:10_: there's no slate of hand here.

_17:13_: But that turns into

_17:16_: the prime is equal to Gamma

_17:20_: t -, v ** prime is equal to Gamma

_17:24_: X -, v T where gamma is as usual. Now it's actually cosh

_17:29_: Phi as it turns out,

_17:33_: 1 -, v ^2, the minus

_17:36_: and and and get it? One way of tangent squared the minus half,

_17:41_: blah blah blah. And correspondingly, T is equal to

_17:45_: Gamma T prime plus VX prime.

_17:48_: X is equal to gamma X prime plus Vt prime.

_17:57_: OK, so the all I want to stress that going from there to there

_18:01_: is just a matter of algebra. Go through set yourself nothing,

_18:05_: not the real fast one being pulled. But this is the form of

_18:09_: the transformation equations that you will use again and

_18:13_: again and again. And that predated Einstein.

_18:18_: A bit of bit of history, a bit of a bit of history of physics

_18:21_: here that predated Einstein. Einstein didn't invent that from

_18:23_: from scratch.

_18:26_: And remember I said that the the macro equations, the description

_18:32_: of electromagnetism and light? Where

_18:37_: it was clear at the end of the century,

_18:39_: Maxwell, I've done the right thing. This this worked as a

_18:42_: description of light, and the puzzle

_18:45_: that prompted

_18:48_: relatively true marriage

_18:50_: was that maximal equations didn't behave the right way.

_18:54_: They didn't behave in the way that a Galilean transformation

_18:58_: said they should.

_19:01_: And that was a puzzle and thought were aware that was, you

_19:03_: know, a terrible problem.

_19:06_: The ultimate resolution was special activity. Einstein,

_19:09_: clever chap. But there were other folk, including the

_19:12_: president, Lawrence and Pointer,

_19:15_: and Fitzgerald,

_19:18_: an Irish physicist and their third name, who was not coming

_19:23_: to me,

_19:25_: who said, well, migrations do sort of work.

_19:29_: If you use not the Galilean transformation to go from one

_19:32_: frame to the other, but you use this and this sort of plucked

_19:35_: out of the air as well, this works.

_19:39_: And you and you and you can. There's another route by which

_19:41_: you can get to that that answer, but it was sort of plucked out

_19:44_: of the air that this makes maximal equations work in the

_19:46_: sense that Maxwell equations work in the moving frame the

_19:49_: same way they work in the stationary frame.

_19:52_: But there was number reasons for that to be true. But so far this

_19:56_: works. But you know what was supposed to do here? And there's

_19:59_: some suggestion that.

_20:02_: It might be that if things were moving through the ether this

_20:07_: mysterious medium that that light was the waves in

_20:12_: that if things were moving through either they get squashed

_20:14_: a bit

_20:15_: you know sort of

_20:16_: hand waving way. But in this precise we would just right to

_20:20_: make length contraction work and and I'm actually equations all

_20:24_: happened but there was no

_20:27_: they'll get you just start again start again. From that point,

_20:30_: there was no real

_20:32_: motivation for that.

_20:34_: What was special about Einstein's approach was he

_20:36_: started from a different place.

_20:38_: So you started from these two axioms from, you know, a

_20:42_: fundamental understanding of what was a fundamental statement

_20:46_: of what was going on here, and derive these in a very natural

_20:51_: way.

_20:53_: And that's what's important about special activity is much

_20:56_: more natural than the ad hoc way in which the the the the

_20:59_: peculiar peculiar behaviour of material equations were

_21:02_: different.

_21:08_: And another question there and the bottom line there. The one

_21:12_: take squared is that one as in the speed of light. Yes, that's

_21:16_: right. Yeah. Well or rather in units which which aren't we're

_21:19_: light metres are not our units of time. That would be v ^2 over

_21:23_: square.

_21:24_: So, so, so, so so it's it's one as a unit 1, but but here this

_21:28_: is,

_21:31_: there's sort of 1 / C ^2 that's invisible there because C ^2 in

_21:36_: the right units is one. Yeah, but yes, so. So yes, this is in

_21:40_: units where C is equal to 1 as as will always be the case.

_21:47_: And

_21:49_: there's a lot more one could see. But the history of special

_21:52_: relativity I I'm

_21:54_: with exceeding great restraint in not spending on the whole are

_21:57_: talking about that because quite exciting,

_22:00_: right.

_22:02_: That's important. I'll move on. I'll keep going. Keep going.

_22:05_: Keep

_22:10_: right now we're. As I said, we weren't going to use that

_22:15_: trigonometric version too much, but we can do one more useful

_22:20_: thing with it which is if we

_22:24_: adding substract are these together and subtract them from

_22:27_: each other

_22:29_: we then we get.

_22:32_: You pray my ex prime is equal to

_22:36_: E to the five

_22:38_: t -, X

_22:40_: T

_22:45_: is equal to each of the Mace by

_22:49_: another another. That's another version of the same

_22:51_: transformation

_22:53_: which

_22:55_: any particular useful except that

_22:58_: it lets us see.

_23:01_: But if we add,

_23:04_: if you ask OK,

_23:06_: the trains go through the station platform. On the train

_23:09_: someone is running,

_23:11_: so that means the three frames here, station platform, the

_23:14_: train and the person running.

_23:17_: And so we we we now know how to get from the vision platform to

_23:21_: the train frame and from the train frame to the the the the

_23:24_: the athletes frame.

_23:27_: Do we know how to get from the train frame direct to the

_23:29_: athletes frame?

_23:31_: Yes we do,

_23:33_: because if we have 3 frames

_23:36_: then the same thing will be true for getting from the.

_23:40_: There's a second one to the third, which will be T double

_23:45_: prime minus X double prime equals E

_23:49_: Fly two. Let's call it

_23:51_: T prime minus X prime which will be E by 2 * e Phi one t -, X

_23:56_: where? So here if if I wanted the rapidity with the speed of

_24:01_: the train and the platform frame and by two of the rapidity of

_24:07_: the athlete in the train frame. And that's very simple.

_24:12_: So now we have that

_24:16_: that's just E the Phi one plus Phi 2IN

_24:22_: other words,

_24:23_: And there's just a different range transformer.

_24:26_: So it's still, but it's still the right transformation as it

_24:29_: really has to be. Otherwise you could tell you were moving if

_24:32_: the if the transformations went all wrong. But typically simple

_24:35_: version of it

_24:37_: I would also means is that and I'm again I'm not going to

_24:43_: I I guess from hints for how to to to step through that but that

_24:47_: also tells us

_24:48_: but but but by telling us that the rapidity of this of the

_24:52_: athlete in the station frame is just the sum of the realities of

_24:56_: the train in the station frame and the athlete in the train

_24:60_: frame. We therefore have a relationship between the speeds

_25:04_: of the station of the of the of the train in the station frame

_25:07_: and the speed of the athlete in the train frame,

_25:10_: which is

_25:14_: that the speed

_25:17_: of the athlete in the train in the station stream is V1 plus V,

_25:24_: 2 /

_25:27_: 1 plus V1.

_25:29_: You too. And no, I'm not going to go through the steps of that.

_25:33_: You can do that in in in the exercise. But the important

_25:37_: thing to note there is it's not just V1 plus V2.

_25:41_: So we are used to

_25:45_: speech adding in the street in an uncomplicated way. If you are

_25:48_: going through a A station at 60 miles an hour and you are so

_25:52_: thuggish as to throw them out of the window at 20 miles an hour,

_25:56_: it'll hit someone on the station platform and you'll be arrested.

_25:59_: But it'll do at 60 + 80 at 60 + 20 miles an hour.

_26:05_: OK? And this tells us that they don't, that that isn't true when

_26:09_: things are moving at relativistic speeds. As things

_26:11_: are moving relativistic speeds,

_26:15_: you end up well. If you're still in that for a bit, you you you

_26:19_: convince yourself that

_26:22_: when thinking of moving at near the speed of light when when V1

_26:25_: and V2 are nearly one,

_26:29_: the combined speed is still less than the speed of light, still

_26:32_: less than one.

_26:33_: So no matter how fast you go relative to things, some of

_26:36_: which is going fast,

_26:38_: you still aren't going fast this prelate,

_26:41_: OK?

_26:43_: And that will keep me relevant. I'm not gonna make a big thing

_26:46_: about it, just that that does all hang together,

_26:51_: right?

_26:53_: Any questions about that

_26:56_: question there?

_27:01_: What does buy represent right? And I don't think it's anything

_27:05_: that's particularly visualizable really. Here

_27:09_: it's introduced here really just as the the the bridge

_27:16_: from the. Look this expression for the range transformation in

_27:21_: terms of the hyperbolic functions

_27:25_: to this version in terms of the things we are actually more

_27:29_: interested in. So

_27:31_: and I think you could think of it as just architecture the

_27:34_: speed

_27:35_: so it's a function of the speed which happens to to to make

_27:38_: things work out in that form. So I I don't I don't think that

_27:41_: anything any any physical intuition that really available

_27:45_: by by. I think with that but but.

_27:51_: Except as maybe the the you know strictly in skateboards, the

_27:55_: rotation angle for for that. Because that is a a good point

_27:59_: that what we're looking at here isn't anything more than what

_28:03_: we're looking at when we see when we rotate our axes and get

_28:07_: different coordinate.

_28:10_: If I hold that up in front of you, I think I did mention this

_28:13_: last name, hold up that up in front of you and I rotated the

_28:16_: coordinates in your framework of the of the ends are different

_28:19_: but the the length is the same. You're not surprised of that

_28:22_: because all that's happened is has been a rotation.

_28:27_: What this is describing is a rotation and that's not strictly

_28:30_: a rotation but it but it it predicts it's close enough to

_28:33_: Puritan. That's a good thing to think about. And what that's

_28:37_: telling you is the and I'm repeating myself here that the

_28:40_: geometry of the space we're looking at is different from the

_28:43_: geometry the familiar with. But the thing I want to really drive

_28:47_: home is, because I'm saying it several times, is it is just a

_28:50_: matter of geometry.

_28:53_: The, the, the, the, the, the, the, the geometry, the, the, the

_28:57_: spatial rules of space-time are just like those of ordinary

_29:00_: cleaning space, but different.

_29:04_: And the way that things distort contract elite. When you're

_29:07_: looking at, I think a movie, different speech, it's sort of a

_29:11_: perspective effect in the same way that this seems to get

_29:14_: shorter when you look at it from a different angle.

_29:19_: And that requires going for a long walk and thinking about it

_29:22_: and and and and and, and and talk about just to let that

_29:24_: settle down in your head

_29:29_: because this is a big change. Because before,

_29:34_: before the 20th century,

_29:35_: the picture of the universe that we were left with promotion was

_29:39_: that there was a three-dimensional space which

_29:42_: was sort of marching through time.

_29:45_: We all shared that same 3 dimensional space and time went

_29:48_: tick tick, tick, tick tick tick tick for all of us at the same

_29:51_: time

_29:53_: and so. So space of three-dimensional time was

_29:55_: one-dimensional. End of story.

_29:58_: And what this is saying is that's just not true.

_30:01_: That is genuinely the whole thing is a four dimensional

_30:04_: thing, and one picture I find is quite useful, though I'm I'm not

_30:08_: sure I necessarily explained. I may not explain it very well,

_30:12_: but I invite you to think about it is. If you imagine two cars

_30:16_: heading across a desert,

_30:22_: see one of them's going.

_30:26_: No one's going in that direction

_30:29_: and when I was going

_30:32_: in that direction, so it seems to be different. Velocities

_30:34_: seems to be different direction

_30:36_: and that car

_30:41_: E

_30:43_: is heading in that NNN

_30:47_: NW it's going north northwest faster than carb is it's it's

_30:51_: it's heading in that direction as it more towards the north

_30:55_: northwest than carbo

_30:57_: so it's going faster.

_30:60_: Bicarb is heading to the north northeast

_31:03_: faster than car is,

_31:06_: So they're both going faster than the other one

_31:09_: really.

_31:11_: And that's not a contradiction because they're they're their

_31:14_: notion of straight ahead is different

_31:17_: at first. Car is concerned

_31:19_: N NW is IS is straight ahead and other ones going off at a

_31:23_: different direction. As far as club is concerned, N NE is

_31:26_: straight ahead, London different direction. That's why, it's

_31:30_: because of the change of frame

_31:33_: that they can be both, both of them be moving both faster and

_31:36_: slower than the other.

_31:38_: And that is what's happening

_31:41_: when you change frames in Russian context, you've changed

_31:44_: frames. So your your notion of what is straight ahead in time.

_31:48_: You know, I'm standing here looking at my watch. That's

_31:51_: straight ahead in time.

_31:53_: Your notion of of that is different from that of someone

_31:55_: who is looking at the same event in a different way

_31:59_: and there's nothing more happening that's happening

_32:01_: there. I think. I think I I think that's quite I I had a

_32:04_: little haha.

_32:07_: That picture occurred to me.

_32:12_: Any question about that? I like, I like questions.

_32:17_: I think there's a, there's a a quick question slide somewhere

_32:21_: in here, but the well, but I'll just stick with the, the, the,

_32:25_: the, the

_32:27_: at the moment. The other thing we're going to mention before we

_32:31_: move on to applications around transformation is the idea of

_32:34_: proper time.

_32:41_: If there's two of you, there's two events, one

_32:44_: here and one

_32:47_: here, right?

_32:49_: So the separate, the separate in space and in time

_32:53_: and any 2 frames I'm repeating myself a year will have

_32:55_: different ideas of what those the time, space and time

_32:58_: quadrants of those are. And we can now turn one turn to the

_33:01_: other. But there's one special frame that we can pick out. One

_33:05_: we can pick out special

_33:07_: who's the frame

_33:09_: which was present at both these where that event took place at

_33:12_: the origin in both cases. So it's if. If there's a frame

_33:15_: that's moving in the right direction at the right speed,

_33:18_: then this event will happen at it's already at its origin and

_33:22_: at its origin over over here. OK, so that that that frame is

_33:25_: sort of special.

_33:28_: It's special because the two events took place zero distance

_33:32_: apart

_33:34_: in that frame because they put you place the order,

_33:37_: so X prime

_33:38_: procedural there and and and the zero there.

_33:43_: And

_33:45_: if you had an observer

_33:49_: at those two those two events, then they could look at their

_33:52_: watch and write down what the time was.

_33:55_: But because we have no have the transformation

_33:59_: we can work out what that observers

_34:05_: coordinates, what got readings would be

_34:10_: and that observer, that special observer

_34:12_: who is Co located with both events

_34:15_: the the time on their watch is

_34:18_: sort of special. But also crucially, anyone in any frame

_34:21_: could work out what it was because of the transformer

_34:26_: and that time, that time interval between those two

_34:28_: things we refer to as the proper time,

_34:31_: then there's nothing fundamentally special about it.

_34:34_: It's just that there's a, there's a coordinate choice

_34:37_: which says that that that is a a nice, neat thing to refer to. So

_34:41_: that's when I talk about proper time. I don't mean it's

_34:45_: fundamentally distinguished time, but it is a I think

_34:48_: everyone can agree on

_34:50_: because it's happening allocated with two events

_34:58_: and.

_35:02_: Ohh, yeah, and. And that means that if we have two events which

_35:07_: are

_35:09_: separated by

_35:12_: there are two squared

_35:14_: and delta X of prime squared,

_35:18_: then the

_35:21_: interval between them is.

_35:23_: Is that exactly as before?

_35:26_: But if we're talking about about proper time, then separate the

_35:30_: physical separation between those two events will be

_35:34_: such that X prime does X prime is equal to 0

_35:39_: that you could delta D prime squared

_35:45_: zero and that time would refer to as

_35:51_: the proper time tour

_35:53_: To to I I will give you refer to the proper time and write down

_35:56_: as tall or taller or how we want to pronounce it. And it's just

_35:60_: another way of writing it. It's fundamentally just another way

_36:03_: of writing the invariant interval S,

_36:06_: but but it makes it clear that that that,

_36:09_: yeah, it's just another way of writing that. I'm not going to

_36:12_: bang on about that,

_36:14_: yeah,

_36:15_: but I want to lodge that idea with you.

_36:19_: Any questions?

_36:20_: Thank you.

_36:22_: Right

_36:26_: at

_36:29_: that that that that's as I said the that's the main bit of of of

_36:34_: mathematical machinery

_36:36_: for this for this chapter and for the chapter chapter to come.

_36:41_: The rest is applications, so think over that. Make sure you

_36:45_: it's all it's all the right way up in your head

_36:48_: and we're not going to applications and and way in

_36:50_: which you can use this to do things, to do other things.

_36:56_: One is to imagine.

_37:04_: Thanks

_37:06_: fine.

_37:10_: A rod

_37:11_: lead out along the X frame axis, so this is the axis of the

_37:15_: quartz moving free.

_37:18_: And the distance in that frame is going to be L0. And I'm going

_37:21_: to refer to that as the proper length

_37:24_: because in the that also is our to some extent a free moving

_37:28_: quantity in the sense that everyone could work out what it

_37:32_: is. It's the length of the rod in the frame in which it's not

_37:35_: moving.

_37:37_: OK, so if you're standing by the the, the, the, the, the, the

_37:40_: redness, not move, you have to do anything complicated about,

_37:43_: you know, multiple observers watching the that the ends at

_37:46_: the right time. It's just nice and simple

_37:48_: to an effect of the proper length. That's what I'm

_37:50_: referring to,

_37:53_: I think. I've I think I have used the term proper length

_37:56_: before part of this for sure,

_38:02_: right?

_38:04_: What we want to do is work out what is its length in the other

_38:08_: frame in the T frame in frame in frame rather than frame S prime.

_38:12_: And we want to rederive the length contraction formula.

_38:17_: What we can do

_38:19_: is

_38:21_: we'll have two observers in

_38:25_: South if you if you remember how this works.

_38:29_: I hope I measure the length works. I said that you have

_38:33_: something moving past your station platform and at a

_38:37_: certain time in your frame,

_38:40_: the observers who are at the

_38:43_: ends of that moving moving object note down where they are

_38:46_: and what time and and and what the pre arranged time is. And

_38:49_: the difference between the coordinates of those two

_38:52_: observers is the length of the thing and that is defined to be.

_38:56_: That's what we mean by the length of the thing in the

_38:59_: moving frame. So let's set that up in in, in this, in, in, in

_39:02_: this form.

_39:04_: So we say that there are going to be two events, like two

_39:08_: people pulling, pulling bangers at at the moment when they see

_39:12_: the end of the road moving past them.

_39:16_: We'll call those tonight

_39:20_: and

_39:24_: what we'll have one happening at, even one will happen at the

_39:28_: origin of the platform frame, the stationary frame. In other

_39:33_: words,

_39:35_: X1

_39:36_: and T1

_39:38_: will both be 0.

_39:39_: You know, we're we're shooting our coordinates, our freedoms.

_39:42_: So that's true.

_39:45_: And the

_39:46_: and

_39:52_: and

_39:53_: other

_39:55_: event

_39:58_: it will be it will happen at X2 equals something.

_40:02_: And T 2

_40:05_: = 0.

_40:06_: In other words, it happens simultaneously with other event.

_40:09_: That's what we mean by saying we make observations of the two

_40:12_: ends of the, of the, of, of of the of the rod at the same time

_40:16_: same time coordinate. And we'll call we've chosen 1 to be

_40:19_: happening at the temporal and spatial origin equals 0. T prime

_40:22_: equals 0,

_40:23_: the other one will happen at the same time

_40:26_: and the

_40:29_: what we're seeing is that that that the X coordinate of that

_40:33_: second event is

_40:35_: the this expression for the length that we're looking for

_40:38_: here.

_40:42_: OK, So what do we know? How do we link that to the other frame?

_40:47_: We said that these events happen at the

_40:53_: and both ends of the

_40:56_: road.

_40:58_: That means that if the event at the front of the of the rod

_41:01_: happens at the front of the rod,

_41:04_: they will happen at the front of the road in both frames.

_41:07_: Now we've got that so that, so that event too will happen at in

_41:12_: both frames. We've got

_41:15_: a value for that that that that event, that the court, the X

_41:18_: coordinate, that second event.

_41:20_: But we don't know. That's we're trying to find out. But we also

_41:23_: know that X2 primed

_41:25_: equal to L Naughty, because it's happening at the front of the

_41:28_: road. And so in that frame

_41:30_: anything anything that happens at the front of the road happens

_41:33_: at coordinate X2 frame.

_41:38_: And we also know that X1 primed equals 0 and X1 and T1 prime

_41:42_: equals 0 because there is some configuration.

_41:48_: Configuration means we can write down that and that OK if if

_41:54_: event one is happening at the at the

_41:60_: um order.

_42:05_: But what this means is we can then use the right

_42:07_: transformation equations

_42:09_: and write down that X2

_42:14_: is equal to gamma X2 primed plus

_42:18_: the

_42:19_: two friend

_42:22_: and that

_42:23_: T2

_42:25_: which is equal to 0

_42:29_: gamma

_42:31_: T1T2 prime 3 plus

_42:35_: BX2 framed

_42:39_: um.

_42:41_: And we can

_42:43_: yeah

_42:45_: that being zero gives an expression for two primary terms

_42:49_: of of the experiment of nothing complicated distance B times

_42:52_: time. We substitute that into the other one and get X2, which

_42:56_: remember is equal to L

_42:59_: is equal to gamma

_43:01_: 1 -,

_43:03_: v ^2.

_43:05_: They're not,

_43:07_: Remember is 1 -, v ^2 to minus 1/2.

_43:11_: So all, or in other words, one way of record is one over gamma

_43:15_: squared.

_43:16_: That's equal to L naughty

_43:19_: over gamma,

_43:21_: which is what we got last time for the length contraction

_43:24_: expression.

_43:26_: But we've got it this way by a much more systematic route,

_43:32_: and it's just a matic route that you will get used to

_43:35_: because.

_43:38_: So I should have this up already and let's go back to here,

_43:49_: right?

_43:53_: I trust all of you are using two factor authentication

_43:56_: and for your

_44:04_: whatever.

_44:14_: Oh,

_44:20_: oh,

_44:23_: yes. Whatever

_44:25_: now gotta find where that going to and the ohh no here it is.

_44:29_: Right

_44:30_: when the documents in the in the actual folder is ohh

_44:44_: is this

_44:48_: race these recipes document

_44:52_: and I want you to to to put it under your pillow

_44:56_: and absorb it during the outer darkness and

_45:00_: it's a recipe for doing

_45:03_: these calculations and and I went through that recipe when I

_45:07_: was doing the calculation there

_45:12_: standing for creation

_45:13_: it's almost done configuration the retransmission works only

_45:16_: works if you're things are starting variation and you have

_45:18_: to write and your

_45:20_: and

_45:22_: the assessment should you exercise you do the assessment

_45:25_: deliver in the exam you'll have to write these steps down to

_45:28_: just to to be very clear that you're following everything

_45:31_: explicitly. And also because if you go through these steps, if

_45:35_: you are forced to think what you know, answer this question at

_45:38_: this time, then you end up with things in systematic enough way

_45:42_: that you don't confuse yourself. There's a way of not of avoiding

_45:45_: confusion.

_45:47_: So what did I do?

_45:49_: OK I didn't draw Mickey diagram that I'm a I'm a bad person I I

_45:53_: I would expect you to to to to It will almost always be useful

_45:57_: to draw Minkowski diagram. The very few cases where it is just

_46:00_: to fix just turn the story into into numbers. What I did was I

_46:04_: said there's the story I was telling you was a rod at rest in

_46:08_: one frame. What the location of the coordinates? What are the

_46:12_: coordinates of the these two events? In the other one

_46:18_: identify the frames

_46:19_: frame of the, of, of of the, of the road frame of the station,

_46:23_: whatever. And I would write that down if we're if we're writing

_46:26_: that down carefully that this is not

_46:29_: I submittable that one quick

_46:34_: and this is not a submittable version of the of of of the

_46:39_: answer. Yeah what a fair copy there.

_46:44_: Identify the origins of the frames. What is what is at the

_46:47_: origin of the frames. There will always be some sort of event,

_46:51_: almost always be some event at one of the origins. In this case

_46:54_: the one of the firecrackers, one of the one of the bangers was at

_46:57_: one end of the of the of the of the road which we decided we're

_47:00_: going to be at the origin.

_47:03_: These are all simple steps but but they're all, they're all

_47:06_: important. Identify the events. There will be events

_47:10_: and often the key to solving these these exercises is is

_47:14_: thinking straight about what the events are. I said the two

_47:17_: events were going to be two events which were at each end of

_47:21_: the station of the of the road and at the same time.

_47:26_: And I would write that down, explain what those two events

_47:28_: were and a number of them one. Event one and event two,

_47:33_: identify the world lanes. OK, they weren't very complicated in

_47:36_: this case. Didn't have to do very much there.

_47:38_: Write down what you knew, what it did in this other thing. Here

_47:42_: it wrote down

_47:44_: the quarters of X1 and T, the coordinates of one

_47:48_: in the 2 frames. Because that was not easy because I've chosen

_47:52_: that meet the origin, I wrote down that the that the that the

_47:56_: the story I was telling were the events were meant that two is

_47:60_: equal to 0 at all. That done,

_48:02_: I wrote down a. The link to the other other frame is that the

_48:05_: event happens at the end of the road. Where does that mean it

_48:09_: happens at X2 prime? Is equal to

_48:11_: is equal to L nought.

_48:13_: What am I trying to find out? I'm trying to find out the

_48:16_: length of the rod in the boom frame. That's L. So what I would

_48:20_: do,

_48:22_: it would, I don't know, you know, put the arrow or

_48:24_: something, you know this, this is what we don't. This is what

_48:27_: we don't know

_48:30_: and and right. So write down what you have to find out.

_48:33_: At that point. Your Minkowski diagram is a mess. If you redraw

_48:37_: it

_48:39_: and at that point it it's downhill. At that point it's

_48:42_: just algebra from from the the they're on

_48:46_: and I and and you know so this bit of algebra blah blah got I I

_48:50_: got that job done.

_48:53_: So if you are systematic about it and I will want to see

_48:56_: demonstrations of viewing systematic about it in the sort

_48:59_: of things,

_49:00_: they're actually really easy

_49:03_: choosing the events or what the events are. It tends to the

_49:07_: hardest bit now in the assessment that's due. Well the

_49:10_: the, the assessment number one, I thought it was due this week.

_49:16_: So I therefore thought, well, it's not really fair for me to

_49:19_: give them a I think about the musket, the transformation, so

_49:22_: that so it doesn't involve transformation

_49:25_: and. But it should have done. So what?

_49:31_: So the the, the, the, the bit locked off at the end of it. But

_49:36_: further assessments will

_49:38_: involved with recipe more carefully and the other exercise

_49:41_: in the note will encourage you to do that. Enough talking about

_49:44_: the recipe, I say. I will see. You know tomorrow is the. I

_49:48_: think some of you have supervisions already have

_49:51_: the provisions, as I'll see some of you in the divisions. I'll

_49:56_: see you next. Next lecture will be next.