Transcript of a2-l05
==========


_0:12_: This is lecture 5

_0:14_: and that the microphones are not place This lecture 5 we are on

_0:18_: schedule. This is good. So I aim to get through the rest of

_0:23_: Chapter 4 today, and we'll move on to Chapter 5, which is about

_0:27_: the Lorentz transformation

_0:30_: next time. And that is the in a sense the bit for quite a lot of

_0:36_: things come together, if you like. So the expressions for

_0:40_: length contraction and time dilation that we have seen

_0:45_: already

_0:46_: are tied up with the ball as it were in the transformation

_0:51_: but the before. I want to get onto that there's some

_0:56_: motor market. I want to make a big coffee diagrams

_0:59_: and about the sort of geometry first approach that I've taken

_1:03_: to special relativity in this course,

_1:08_: which is not the only way one could do it, but I think of

_1:11_: course a good way.

_1:14_: I talked about the Mickey diagram last time,

_1:20_: which which is

_1:23_: just

_1:25_: I say a slightly eccentric way of of plotting events. The the

_1:29_: one bit of eccentricity is the X&T actually being swapped

_1:33_: based on how you go in contrast to what you are more familiar

_1:38_: with. But that's not a big deal. It's just that the point

_1:42_: remains. Each event

_1:45_: has our

_1:47_: locatable on the mycotic diagram. And as I hope I showed

_1:51_: you last time talking about these the late flashes going

_1:55_: back and forth

_1:58_: from our moving train carriage,

_2:00_: you can think through

_2:04_: the process of what's happening on Minkowski diagram in a

_2:07_: useful, a constructive, very constructive way.

_2:11_: And I came. We produced that diagram in the frame of the

_2:14_: moving carriage, with the flashes moving forward and

_2:17_: backwards being reflected and coming back again,

_2:21_: and when we reasoned through

_2:23_: what this looked like in the

_2:27_: from from the station in the station frame.

_2:30_: Watching this and go past, we produced a different coffee

_2:34_: diagram

_2:35_: of the same events

_2:37_: could. Remember, events aren't in A-frame, they are in the

_2:40_: world if you like, and they are plotted

_2:43_: in A-frame

_2:44_: where it was completely obvious

_2:47_: that event in this frame

_2:49_: event. Well, whereas in the previous frame events two and

_2:53_: three were manifested simultaneous. Of course,

_2:57_: because of the same T prime coordinate in this frame, event

_3:01_: three was observed to happen before event two.

_3:05_: OK. So the the observers in the moving frame

_3:09_: would have, you know, watched the light being reflected from

_3:14_: the back and front of the carriage and noted down

_3:17_: different times in that sense observed.

_3:21_: So I'm going to give another example of A

_3:24_: and I think I I I think I did finish off with this quick

_3:27_: question where I asked you to

_3:30_: think through this, so in frame. So just what I've just said,

_3:35_: Event 3 happens unequivocally before event 2

_3:40_: because it had the allure

_3:43_: lower T coordinate even though they have the same T prime.

_3:51_: And I showed that that that diagram similarly I think

_3:56_: OK, let's go on now to look at

_4:01_: a different medical few diagram which I think is also

_4:04_: illustrative and again

_4:08_: plotting if you like our scenario we talked about in

_4:11_: previous weeks. So we're just doing something we've seen in

_4:14_: previous weeks but doing it in a slightly different way. So this

_4:18_: is going back to the specific, the specifics of the

_4:23_: that the the train moving again that that we see enough of the

_4:26_: tree.

_4:28_: So you see that, yes.

_4:33_: So again we're going to,

_4:35_: we're going to draw up medical tree diagram, there's we're

_4:39_: going to draw in the prime frame first

_4:43_: extreme heat frame and again a light flash at the centre.

_4:48_: And we're also going to draw on this the world lines of the

_4:51_: front and back of the tree.

_4:54_: The world lines are the set of events that that happen at the

_4:58_: front and back of the tree,

_5:00_: and the train isn't moving in the training frame.

_5:03_: So the world lane is very straightforward. It's just

_5:10_: that that's the

_5:11_: go to the front

_5:14_: in the back of the tree. So the world is are are not

_5:17_: complicated. OK,

_5:19_: the a set of events which happen at the front of the train will

_5:22_: all happen at the same X prime coordinate.

_5:25_: The

_5:27_: if we then see the

_5:30_: I like flash going forwards,

_5:34_: well I call that event two

_5:37_: and going backwards

_5:40_: event one

_5:41_: and I think complicated. OK, so I I I'm just in a sense

_5:47_: so if this were an exercise I would go I was going through. I

_5:50_: would at this point solemnly write down A-frame S prime. The

_5:53_: frame of the of the carriage frame S is the frame of the of

_5:56_: the of the station platform. Event one is blah, event two is

_5:59_: blah. And then draw them in Costa diagram

_6:02_: and I'll expect to see that sort of systematic approach.

_6:07_: But as I mentioned last time,

_6:10_: the

_6:12_: most of the exercises in

_6:14_: this of course are basically the same. Exercise is, here's the

_6:17_: situation. Turn it into a sort of event. Use the right

_6:20_: transformation. Get yeah, get some other coordinates, The same

_6:23_: event

_6:24_: or all slightly artificial. But the point. The point is about

_6:27_: the transformation and that first step of turning things

_6:30_: into. Because your diagram is the process of thinking through

_6:33_: things

_6:36_: through, the same

_6:37_: set of events in our

_6:40_: in the

_6:42_: platform frame

_6:44_: are slightly different. Here the water lines of the front and the

_6:48_: back are also straight lines, but they are now slanted because

_6:52_: the train is moving.

_6:56_: So that's the

_6:57_: front

_6:59_: and back of the tree.

_7:01_: So I said revenge at flat, flat, flat, flat, flat, flat. Flash at

_7:05_: the front of the train will

_7:08_: appear at successively increasing X coordinates on the

_7:12_: platform,

_7:13_: but the the light flashes

_7:16_: still move it

_7:18_: 45 degrees.

_7:20_: The event one is still the event where the

_7:24_: back with the with the rear going light flashes hit the back

_7:28_: mirror. Event two is still where the front going light flash hits

_7:32_: that ohh and and the centre of the

_7:35_: carriage goes goes like that, which means we can also call

_7:38_: that

_7:39_: mark that is the T prime access

_7:42_: and again here recapitulating what we saw two lectures ago. I

_7:47_: think

_7:49_: this is is clear here that event one, the light getting to the

_7:52_: back of the carriage and event two right in the front carriage

_7:55_: are different times. They're no longer simultaneous in this

_7:58_: frame,

_7:60_: but we know that

_8:04_: events one and two

_8:06_: happened at the same time in that frame, so a line drawn

_8:09_: joining them is parallel to the X prime axis.

_8:12_: So similarly

_8:14_: the extreme axis

_8:17_: is that angle there. And again, this, as I've said a couple of

_8:22_: times, I mean of course diagram ends up looking like a a

_8:26_: Duncan spider mess when you when you get to the end of it. The

_8:30_: point is, it's all about going through and building the thing

_8:35_: up.

_8:37_: The things I want to emphasise there,

_8:43_: yeah, that that's,

_8:45_: that's basically it. But the,

_8:51_: the, the,

_8:53_: that's OK. I'm not going to do prattling on about that. The

_8:56_: point is that that's another example of the coffee diagram,

_8:59_: which recapitulates what we saw in previous previous lectures.

_9:04_: Any questions or puzzles about that question?

_9:08_: Prime access?

_9:12_: Yeah, yeah,

_9:18_: that's right. So this is the the first hint that that's very

_9:22_: important point. There's the first hint of the fact that the

_9:26_: the geometry of the of the space is strange and we're going to

_9:30_: come on to talk more about that.

_9:33_: But

_9:36_: you can think of it. I think it's useful to think of that as

_9:39_: a sort of perspective effect.

_9:41_: If you if if I

_9:44_: their pain if I you know at an angle it seems shorter to you if

_9:47_: your point of view because of the perspective you're looking

_9:50_: at it from.

_9:52_: And I think I'm being a bit vague there but woolly. But I

_9:55_: think if you think of that as a perspective effect when you

_9:58_: rotate

_9:59_: sort of in quotes from one frame to another, then the these items

_10:03_: are still have the relationship between them that they have in

_10:08_: this, but they look a bit funny from this in this frame.

_10:13_: But that the you you think it? Well, yeah,

_10:17_: that's an important point and and I think it allows me to see

_10:21_: geometry again.

_10:22_: OK.

_10:24_: And we're going to say a little bit more of that another

_10:26_: question there

_10:29_: you don't, you don't know between between there

_10:35_: and

_10:37_: I

_10:38_: and I think you you you wouldn't because not really meaningful

_10:43_: that angle and you know the angle between

_10:47_: the chief prime, the key and the key frame access

_10:50_: that is we're going to write that angle down. But in terms of

_10:55_: the gradient of that or the T prime

_10:59_: access,

_11:01_: is this the same as the gradient of these world lines here?

_11:05_: And that is just with the gradient of one over the speed.

_11:09_: So so if if it were a

_11:12_: TX

_11:14_: graph then the the the the the

_11:18_: X equals

_11:20_: Vt, the gradient of that line would be would be the speed. In

_11:24_: this case it's a T is equal to 1 over VX line. So the the the the

_11:28_: gradient of that is

_11:33_: it is 1 / 1 over the speed, which tells you that when the

_11:36_: speed is 1 speed of light it's going to be 45 degrees,

_11:40_: and with the speed is less than one,

_11:42_: then one of the is bigger than one so that the gradient is is

_11:46_: more nearly vertical. But there aren't. But

_11:50_: there aren't really any questions to which the answer is

_11:52_: the angle between those axes.

_11:55_: OK,

_11:57_: you have a good day putting it. You know it's irrelevant, just

_11:60_: that there aren't questions to which that's a useful thing to

_12:03_: work out.

_12:04_: There are analogues

_12:06_: which we perhaps I can talk about later, we will talk later.

_12:09_: But it's not that angle

_12:12_: is a good questions. I like questions because it they they

_12:15_: reassure me that I'm going to ruffle the right speed.

_12:24_: OK. And and the in the notes there's another example of I

_12:27_: mean of course you diagram talking about the the the the

_12:31_: the light of our lighthouse tracking across a distant

_12:34_: coastline which is also allows me to some other remarks in the

_12:38_: notes more words just to back up what I'm saying in the

_12:43_: in the lecture and this is a good a good which is a good

_12:45_: prompt to say what I said in the first lecture

_12:48_: that

_12:49_: as far as I'm concerned the lectures of the main event

_12:52_: and the notes are extra words that might be useful to help you

_12:56_: process what wasn't in in in the lectures.

_12:60_: And

_13:01_: again to reiterate, the

_13:04_: aimed at the point and the objectives are the party tricks,

_13:07_: the the, the, the things that you will that that when you do

_13:11_: perform those in an exam reassure

_13:14_: be us that you have achieved the aims so that that's the

_13:17_: relationship between those, those, those various ideas.

_13:21_: So if you if, the if, if, if the lectures make perfect sense to

_13:24_: you in isolation and you can do all the exercises, you might

_13:27_: never read the notes. That's fine,

_13:30_: but do the. But the the, the, the, the measure of how much you

_13:32_: understand things is how, how quickly, how easily, how

_13:35_: comprehensively you do the exercises because the exercises

_13:37_: are very important, because you will actually understand

_13:40_: anything when you you've done it yourself.

_13:42_: I got I got an answer.

_13:46_: OK.

_13:47_: And

_13:49_: this is the bit we wanted to talk about what you said, what

_13:53_: that remark there,

_13:56_: we'll build that up.

_14:06_: OK. There's a diagram you have seen before I'm sure

_14:10_: is just that diagram of how you rotate axes and you've seen that

_14:14_: I'm sure.

_14:16_: And then we're going to see if they haven't seen that, that

_14:19_: that is a stunning innovation to them. OK you may not seen in

_14:23_: exactly those terms, but you are familiar with this because it's

_14:26_: just the the the setup of how you change coordinates through

_14:29_: rotation.

_14:32_: So that point P had coordinates X&Y

_14:37_: and in the different in the frame which is rotated

_14:40_: by an angle

_14:43_: offer theatre. Rather that same point has different coordinates

_14:46_: X prime and Y prime. They both have the same point, just in

_14:50_: different frames,

_14:53_: and the relation between them is.

_14:56_: Is that

_14:57_: again really well known?

_14:60_: The ex prime coordinate is X Cos Theta plus Y sin Theta. The Y

_15:03_: prime coordinate is minus X sin Theta plus Y Cos Theta.

_15:07_: OK,

_15:10_: with that, let me again

_15:13_: not banging.

_15:15_: I think this. I don't think it's hitting anything.

_15:21_: Umm,

_15:23_: who pretended to you

_15:26_: so the, the, the but the important points to make here

_15:30_: are but

_15:36_: is that me

_15:39_: not banging?

_15:46_: Let's not worry about it isn't happening. The the the important

_15:50_: point that about that transformation is that this

_15:53_: quantity expert plus y ^2

_15:57_: an X prime squared plus Y prime squared is the same in both

_16:01_: coordinate systems.

_16:03_: Now that's just a statement of Pythagoras theorem,

_16:06_: and what I'm seeing here is the Pythagorean theorem isn't

_16:10_: frame dependent.

_16:12_: Now again, this is a big surprise.

_16:21_: And

_16:22_: is that noise coming from this microphone? Yeah, yeah.

_16:27_: Well,

_16:30_: I'm not sure what where it's coming.

_16:34_: Ah, right. Broken. Broken.

_16:39_: OK, there's a there's a the cable slightly broken I think,

_16:43_: or something

_16:46_: which I probably need to report. Ohh, well, I'm talking about

_16:50_: misbehaving.

_16:53_: Yeah,

_16:56_: technology. Just parenthetically

_16:60_: the

_17:02_: echo. These sixty I have I think linked with the 360 recordings

_17:07_: to the Moodle.

_17:10_: Had anyone found those and looked at those?

_17:14_: Because I think I've done it right,

_17:16_: but I really have little confidence that I have. So if if

_17:19_: because I'm not completely sure what it is I'm supposed to do to

_17:23_: make that to me that link. So if anyone looks at that and they

_17:26_: are not what you expect from previous experience in the year

_17:29_: in other years with Equity 60, then please do someone tell me

_17:33_: because

_17:34_: yeah, it's it's not an entirely friendly system and I'm not sure

_17:37_: anything right.

_17:40_: OK) The point here is that I'm saying you're familiar with, but

_17:43_: I'm I'm making a big fuss about it because I'm seeing something

_17:46_: you already know but in a different language.

_17:50_: OK,

_17:52_: I could have put that better, but I hope the point is made.

_18:00_: The what

_18:03_: of thought? Talking about

_18:07_: maybe

_18:10_: now the next section, what 4.4 point 4I.

_18:16_: It is not, I think

_18:18_: tragically well well explained in in in the notes. I keep

_18:22_: changing my mind about how best to explain this so I probably

_18:25_: could rewrite that again. So

_18:29_: which are on a bit, I think in the notes. I'll try not to in

_18:35_: verbally

_18:38_: in in this rotation.

_18:40_: Let's go back to here, this rotation.

_18:44_: We

_18:45_: have a. Let's draw this rotation *** prime YY

_18:51_: prime.

_18:54_: Imagine two

_18:57_: and

_18:59_: points in the plane.

_19:01_: If we

_19:05_: could, we do it X

_19:08_: Ohh. Thank you.

_19:13_: The two points in the play,

_19:15_: we can talk about the separation between them.

_19:18_: That's exactly why. And by that's theorem, delta X ^2 plus

_19:24_: Delta y ^2 equals. We'll call it

_19:30_: d ^2.

_19:33_: In the other frame,

_19:34_: we can instead talk about

_19:38_: That's something there that somebody does egg prime,

_19:42_: that's gotta Y prime, and we know that delta X prime squared

_19:46_: plus delta Y prime squared is also going to be d ^2.

_19:52_: In other words, this

_19:56_: calculated here Pythagoras theorem is an invariant of the

_19:58_: transformation.

_20:00_: When I change coordinates that that that distance, we're going

_20:04_: to call this, the distance between those two points is the

_20:07_: same.

_20:09_: And you could from that you could assert that, and you could

_20:13_: deduce all of Euclidean physics, all of your Euclidean geometry,

_20:17_: from that assertion. Because that that assertion that this

_20:21_: distance function is extremely independent

_20:25_: is in a sense generates Euclidean geometry. Is that

_20:28_: crucial? Is that fundamental?

_20:33_: So is there any analogue of that

_20:36_: for going from one

_20:40_: XT frame to another?

_20:44_: And the answer which the that section 4.4 attempts to justify

_20:48_: it in a rather in a way which I still think is somewhat

_20:55_: inadequate is that

_20:60_: if we talk about the

_21:03_: let's let's go back to here and for example talk about these as

_21:06_: the

_21:11_: the separation between two events

_21:14_: to the extreme

_21:16_: and

_21:18_: that at prime equals 0

_21:23_: delta X

_21:26_: there's a T so this separation between two events in and and

_21:30_: delta T

_21:32_: if the relationship between if there are a function involving

_21:37_: analogous to this

_21:44_: which we can write down which has some sort of properties, the

_21:48_: answer is yes. But it doesn't involve

_21:51_: plus thing

_21:53_: of the -

_21:55_: and

_21:59_: we can write we can write that

_22:03_: I sort of distance function.

_22:07_: For reasons which are not going to become apparent, we'll leave

_22:09_: it rather than rather than.

_22:11_: And just by the way, a bit of notational clarification. When I

_22:16_: write down delta X ^2, what I mean is delta X ^2, not delta X

_22:21_: ^2. That's that's the delta X opt me bracket squared as

_22:26_: opposed to the change in X ^2. And similarly here this is just

_22:31_: bracket squared X bracket squared rather than The change

_22:35_: in XI

_22:38_: will write but I won't be consistent about it doesn't

_22:40_: squared there, but I might something just right. Yes,

_22:42_: square, but it doesn't make any difference

_22:47_: delta

_22:48_: or and so this is the

_22:51_: and in for example this we're interested here in the set. In

_22:55_: how far apart are these two events? One and two and in. In

_22:60_: this case they are. There's no time difference between them,

_23:04_: they're time coordinate with the same. In this case, there's

_23:09_: that sort of time distance between the coordinates. So here

_23:13_: these two events are that are separated by that by delta X in

_23:17_: space and delta T in time.

_23:20_: And this, it turns out, is free and independent.

_23:31_: I don't know about you. I hope illustrate

_23:35_: and that when you're saying it's where the geometry

_23:39_: is different.

_23:41_: So when we see here in in the in the struggle on the right

_23:48_: the

_23:50_: the fact that the that that in

_23:58_: in the rotation diagram

_24:00_: the yeah so roughly the angles look reasonable and Steve

_24:04_: reasonable and this one

_24:06_: the angles don't look reasonable

_24:09_: that's because of that many say is because the the geometry of

_24:12_: that is different and that's a rather that you hand waving

_24:16_: thing to see but that is is a bit where I I brush aside you're

_24:20_: concerned about the angles looking a bit funny and worrying

_24:24_: about the angle between things.

_24:27_: So let's illustrate that if we go back to the light clock,

_24:36_: the like look.

_24:43_: So that this is, this is the

_24:45_: flash and the

_24:48_: like me coming back.

_24:50_: I think there's two lectures ago. I think

_24:54_: you you might recall that the

_25:01_: between those two events and that this is in the

_25:05_: the pain free

_25:07_: separation in space between those two events

_25:11_: is 0 happen at the same place

_25:14_: and the separation in time

_25:17_: is

_25:19_: is is 2L. It's twice the distance of,

_25:25_: twice the distance from there to there. So the time separation is

_25:29_: just that same thing in units of of light metres in time units of

_25:33_: light metres. So if we're measuring this in seconds,

_25:36_: they've been extra factor of C in there, just as we type

_25:40_: so that. So is an extra factor of C but is equal to 1.

_25:44_: Is anyone uncomfortable with that statement?

_25:51_: And in the other case

_25:53_: where we saw

_25:56_: the late flash reflect and come back,

_26:01_: event one was here event two

_26:04_: was some distance down the station that down the track

_26:08_: we discovered that the the the separation

_26:14_: just X between these two events.

_26:17_: If this happens at time T2

_26:20_: position X2 and this happens at 00, we can just choose our axis.

_26:26_: So that's true, does X is going to be

_26:30_: two which is

_26:32_: the T2 where via the speed of the tree.

_26:35_: OK, so in time T2 the the the the train has moved Vt during

_26:39_: the in the platform

_26:41_: the separation delta T

_26:45_: is just is is T 2 -, t one but 10 is T it is delta T prime over

_26:53_: gamma. As I said last time, you can look back and and and

_26:56_: reassure you sort of that and let's put in some numbers to

_26:60_: that

_27:01_: actual numbers in whatever he could have the very thought. So

_27:04_: let's suppose that V is equal to

_27:07_: 3/5. So that's three fifths speed, right?

_27:11_: Thank you

_27:13_: to that.

_27:15_: Yeah, sure. Then that implies that gamma of

_27:22_: the 5th is at one 1 -, 3/5 ^2. What one square root is one

_27:29_: over? That is,

_27:33_: people before

_27:37_: then if

_27:40_: yeah and and and we'll see what it would say that all is going

_27:43_: to be

_27:44_: 1 metre.

_27:46_: Then in this prime frame.

_27:50_: This is not totally organised but just s ^2. You could do

_27:54_: delta T prime squared minus delta X prime squared which is

_28:00_: 2L squared.

_28:04_: My ex prime squared equal 0.

_28:07_: The duty IS2LL1 metre X is 0, so that it's not an area.

_28:14_: We can call this an area, but we we can put the numbers into this

_28:18_: expression here and get a number

_28:21_: in that I've calculated in that moving frame. If we do the same

_28:25_: thing in this from the platform frame to the same between the

_28:28_: same two events,

_28:30_: then we discover we get.

_28:34_: Let's just make sure this

_28:37_: delta squared equals Delta

_28:40_: t ^2 minus delta X ^2. Delta t ^2 we're saying is Delta

_28:47_: T primed

_28:51_: over Gamma. Delta T is one. One over gamma is.

_28:59_: Let's just be consistent here

_29:03_: and.

_29:08_: And

_29:09_: is over

_29:13_: gamma

_29:15_: squared minus

_29:18_: E ^2

_29:22_: 2.

_29:23_: It's weird.

_29:27_: I'm having difficulty working out how to determine what I've

_29:31_: written here into something that's readable. Hang on.

_29:40_: Right. Let's let's

_29:41_: go back, go back, go back.

_29:50_: Let's remember that little more readable run cramming into in,

_29:53_: in into the. The thing at the bottom end is 1 metre

_29:59_: dot T

_30:01_: The prime is 2L equals 2,

_30:07_: V is equal to 3/5. Gamma is equal to

_30:13_: 504,

_30:15_: so our

_30:16_: delta T

_30:19_: in the platform frame

_30:22_: go to T prime over. Gamma

_30:25_: is

_30:26_: going to be 2 / 4 fifths two or five four over,

_30:34_: which is one point.

_30:39_: You're not

_30:46_: and

_30:49_: that's 8 / 5. One point

_30:55_: this does work out. I've just

_31:06_: and

_31:11_: thank you

_31:14_: is

_31:21_: I'll be around the other T primed is a good delta T over

_31:26_: gamma.

_31:27_: So delta T here you could do gamma times

_31:31_: primed which is

_31:32_: like we were 4 * 2 which is equal to five half which is 2.5.

_31:40_: OK.

_31:42_: And our

_31:44_: or just X

_31:46_: prime ex brother is V dot AT

_31:51_: which is the fifths of 2.5 metres which is 1.5

_31:58_: metres.

_31:60_: So which means our delta squared delta t ^2 minus delta X ^2 is

_32:07_: equal to 2.5 metres

_32:09_: squared -1.5,

_32:12_: which is squared, which is 4.

_32:17_: It's a square if you look at it on your calculator.

_32:19_: I'm sorry, I've got I I confused myself by getting that upside

_32:24_: down.

_32:25_: You might want to step through the the, the, the, the notes

_32:29_: later. The point is that when I worked out what delta S ^2 was

_32:33_: in the

_32:34_: prime frame,

_32:37_: I got the same

_32:40_: the same figure.

_32:43_: OK, that's that was a long and I apologise. Slightly confusing

_32:48_: explanation of a very simple calculation.

_32:51_: OK.

_32:52_: Going back to the, the, the, the, the, the what? What we

_32:55_: calculated

_32:56_: 2 years ago,

_33:02_: yes,

_33:04_: going back to going back to numbers, to the expressions we

_33:08_: we we we worked out in the two electrical and this time putting

_33:12_: in numbers we find that we can calculate the separation in time

_33:17_: and to expression and space

_33:19_: between the these two events one,

_33:24_: one and two

_33:26_: in this frame and in this frame. And we discover that this

_33:29_: quantity,

_33:32_: the interval so-called, is the same in both in both things. In

_33:35_: other words, that is an invariant of the transformation

_33:39_: in the same way that

_33:42_: Pythagoras equation, Pythagoras formula is an invariant of the

_33:45_: transformation

_33:47_: in rotation.

_33:49_: And that's telling us that the geometry of this Minkowski

_33:52_: space, of this Minkowski space

_33:55_: is different from Euclidean geometry.

_34:02_: The point of all this

_34:05_: is,

_34:08_: I mean, I am an actual ruler,

_34:13_: the line there. So I mean I I mentioned in passing last year,

_34:17_: yesterday

_34:19_: if I

_34:21_: would that ruler up that's 15 centimetres long,

_34:24_: hold it up in front of you and you and you you can look at it

_34:27_: and and you know you see well do the angle calculations and you

_34:30_: see that that's 15 centimetres long.

_34:32_: If I turn it like that,

_34:34_: then from your point of view it's shorter,

_34:38_: it's got shorter the the the the angular size is is less.

_34:42_: But you're not puzzled by this where, where, why The rules are

_34:46_: no shorter because you know that

_34:48_: the the way

_34:51_: extent of this ruler is now non zero. And if you take that

_34:57_: horizontal or perpendicular, extend

_35:02_: that

_35:03_: longitudinal extent, square them and add them, you'll get 15

_35:06_: centimetres. And that doesn't surprise you. OK, because that's

_35:09_: that that that that the invariant of particular theorem

_35:11_: is part of your intuition about you about about Euclidean

_35:14_: physics.

_35:15_: Because in geometry

_35:17_: geometry.

_35:18_: And what I want to suggest is that the same thing is happening

_35:22_: in this diagram here

_35:25_: you're having a. This is a. This is just an exact. It's. It's a

_35:29_: very close analogy

_35:35_: here. You're switching between frames, but Pythagoras theorem

_35:38_: reassures you that there is an invariant. Here

_35:42_: you're switching between frames, but

_35:45_: this is an invariant,

_35:48_: OK, and the difference is just a matter of which geometry you

_35:52_: choose.

_35:53_: And this is what I what what I mean when I say that I that this

_35:57_: is a a geometry first account with special activity. Partly

_36:00_: because it's quite elegant,

_36:02_: but also a very significant Part of why I'm introducing it this

_36:07_: way is because this gives a very natural

_36:11_: path to talk about general activity and to understanding

_36:15_: general activity. Because the geometry I'm talking about here

_36:19_: is is is is the the, the the the simple version of the geometry

_36:22_: of of general relativity.

_36:25_: General the geometry of general relativity

_36:29_: hard to see is this basic idea with our distance function,

_36:35_: which had a - in a crucial place.

_36:38_: But

_36:41_: with the restriction, the specialness of talking a little

_36:45_: bit, constant speed has been dropped

_36:48_: and that's why I'm making a fuss about geometry just now.

_36:55_: Question about that.

_36:57_: OK. So that would just be

_36:59_: taking obtained to make a fuss rather than go through any

_37:03_: specific calculations Here

_37:07_: there is a section 4.6. If you've still got a bit puzzled

_37:10_: at at at McAfee diagrams then but perhaps have a look through

_37:14_: that because that goes through goes through how you build up

_37:18_: Minkowski diagram another it's yet another report. It's a

_37:22_: worked example of how you build up because your diagram from

_37:26_: from an argument because diagrams at the end again looked

_37:29_: like a a complete mess. But once you see it through them, sort of

_37:34_: drawing them out in parallel, they should

_37:36_: make some sort of sense

_37:38_: I hope.

_37:42_: But if you work through the exercises then you should get

_37:45_: all the all the practise you need.

_37:49_: Key points there.

_37:57_: And that's the point of the rule.

_38:02_: That's the thing.

_38:04_: Well Sir,

_38:11_: so the what this means that we we now start to talk about the

_38:17_: overall

_38:18_: overall shape of things on the in the Miskovsky plane, in

_38:22_: Miskovsky diagram

_38:23_: in Minkowski space. Because when I talk about Nikki diagram, I

_38:27_: mean this X&T diagram when it when I talk about Minkowski

_38:31_: space, I am talking about a geometrical space, just like

_38:35_: Euclidean space, but with a different rule, different rules

_38:38_: for the invariant, different rules for how you go from one

_38:42_: frame to another.

_38:44_: But it's it's just as much a question of geometry, the

_38:47_: geometry you learned about in in school.

_38:51_: So we can therefore plot out a variety of points on the

_38:55_: Mikowski diagram and ask

_38:57_: and and talk about the interval between them and. And by the

_39:00_: way, I'm going to talk about the interval that when they talk

_39:03_: about the interval I mean the squared interval, I mean squared

_39:07_: interval when I'd be saying interval but it and we're always

_39:10_: talking about this thing squared there isn't. Although the

_39:13_: Patagonian distance function d ^2 = X ^2 + y ^2, we can take

_39:16_: the square root of that and find and talk about that as a

_39:19_: distance. You don't have to take square root of it to have

_39:22_: meaningful thing and we don't take the square root of this of

_39:25_: this interval. We just talked about

_39:28_: this weird thing.

_39:29_: It's great aspect. So this means that there's a couple of a

_39:33_: variety of points,

_39:35_: and if you remember, the interval is Delta t ^2.

_39:39_: My name is Doctor Expert.

_39:41_: If you look at at event five,

_39:44_: they are the separation of event five from the origin

_39:50_: is more in time than it is in In space just is bigger than delta

_39:54_: X

_39:55_: just t ^2 X ^2. Just t ^2 -, X ^2 is positive.

_40:01_: And that's true of everything above the

_40:05_: these two diagonal lines.

_40:08_: The everything in that quadrant is

_40:11_: had had an interval which is just s ^2 greater than 0.

_40:17_: Event three

_40:19_: they are the separation from the origin in time is less than the

_40:23_: separation of the from the origin in space DD T is less

_40:27_: than delta X,

_40:29_: so the interval

_40:31_: in this area and in this area

_40:34_: is negative.

_40:37_: Similarly for event one that's also

_40:41_: in,

_40:43_: it has a delta T which is bigger than delta X ^2,

_40:47_: so it it. It also has positive intervals, so this is also

_40:52_: with that

_40:55_: and

_40:56_: venture event 4 Justine dot X are

_41:00_: both are the same. With the interval is 0, so the distance

_41:05_: between the origin and event four and of origin and event two

_41:09_: is 0

_41:10_: and that's weird so that this is because of the - in that

_41:13_: distance funk

_41:15_: in Euclidean geometry for Pythagoras theorem because

_41:19_: there's a + in it

_41:20_: if two events. If two things two points are zero distance apart,

_41:24_: then they are the same point,

_41:26_: and that's just not true in this geometry.

_41:30_: Now what we can also do is we can make a version of that

_41:37_: diagram

_41:40_: and we'll we'll draw

_41:45_: T

_41:47_: and

_41:48_: will draw in the transformed X prime

_41:54_: straight line

_41:56_: T prime and we'll look at what

_42:01_: that that.

_42:02_: Let's look at event.

_42:03_: Ohh

_42:06_: seems to reset itself

_42:09_: Event 5

_42:12_: an event

_42:14_: three.

_42:15_: Now in this

_42:19_: we saw that event five was had had DD T bigger than delta X. So

_42:26_: for red 5

_42:28_: ^2, event 5 is greater than 0 and prevent 3

_42:35_: ^2 with less than 0.

_42:38_: But if we now look at the

_42:42_: dot X primed

_42:44_: and the

_42:46_: delta T primed coordinates of event five,

_42:50_: we see that again the delta T is bigger or still delta T is

_42:55_: bigger than delta X. So the interval in this case will be

_42:59_: also be positive, which we knew already because we we we we know

_43:04_: that the interval between these events is going to be

_43:09_: the seam in the different coordinates.

_43:12_: So event five will still have

_43:19_: and in the other coordinates will still be

_43:22_: positive.

_43:25_: This is still above. It's still above the X frame axis,

_43:31_: but

_43:33_: and

_43:35_: for event three

_43:39_: the time coordinate

_43:41_: ohh then 33 in those in that prime coordinate in the that

_43:44_: prime frame

_43:46_: is no, is no below the extreme axis.

_43:49_: So event three has

_43:55_: well I've got I've got the delta. Event three has will have

_44:00_: a a negative time coordinate.

_44:04_: In other words, what that is seeing is that

_44:08_: in the XT frame

_44:12_: you say the stationary frame whatever

_44:15_: event 5 happens in the future and event 3 happens in the

_44:18_: future.

_44:20_: But in the frame frame,

_44:23_: the time coordinate of of event 5 is still going to be positive

_44:26_: that there's no way you can see that transformation happening

_44:30_: where the where the X prime axis ends up passing by

_44:33_: event event 5. So event five will always have a positive time

_44:37_: coordina.

_44:39_: But

_44:40_: in the case of the transmission down there, where the ex prime

_44:44_: has squashed in this way to go past event three, event three

_44:48_: now has a -10

_44:49_: coordinate. In other words, in that frame, event three happened

_44:53_: before the event at the origin.

_44:56_: In the untransformed frame, event three happened later. In

_44:59_: the transformed frame it happened before.

_45:02_: True, there isn't that the the the question of which came first

_45:07_: is

_45:08_: frame dependent for event 3,

_45:11_: but it is unequivocal for event Phi

_45:15_: and that is to see.

_45:19_: That's why I'll leave one of those.

_45:23_: That's why that top quadrant is labelled the future,

_45:28_: because everything that happens in that top quadrant is

_45:31_: unequivocally in the future, and there isn't A-frame. You can

_45:34_: find where

_45:35_: event 5 happens in the past,

_45:40_: and similarly for this past quadrant.

_45:42_: Events that happened in the past

_45:46_: were always happened in the past. There isn't A-frame. You

_45:49_: can find where that happens with the time coordinate of event

_45:52_: one. The T frame coordinate event one is positive, but for

_45:55_: things in this other space

_45:57_: which called elsewhere,

_45:59_: whether event 3 happens before or after an event of the origin,

_46:04_: it's frame dependent.

_46:07_: And what that also means is

_46:10_: the the cannot be a causal link between anything event that

_46:14_: happens

_46:15_: at the origin

_46:16_: and an event which happens at event 3. Because if it's aware,

_46:21_: or if you thought there might be,

_46:24_: then you when you discovered the event Three could happen in the

_46:28_: past,

_46:29_: in in one frame or other. That would mean that an event was

_46:32_: going from the future, from something that happened later in

_46:34_: time to if something happens

_46:36_: earlier in time which can't happen.

_46:39_: So another way of of of thinking about the various things in this

_46:43_: diagram

_46:44_: is that

_46:46_: event five

_46:48_: you could always get a message a six and a signal to event five

_46:51_: from the origin.

_46:54_: By some

_46:55_: slower than late

_46:57_: means,

_46:58_: you could You could send something

_47:01_: possibly going quite fast. Fast. You could send something the

_47:04_: order to event five. You could send something from event one to

_47:07_: the origin,

_47:08_: but it is not possible to get from the origin to event 3

_47:12_: by anything travelling slower than the speed of light, so they

_47:15_: cannot be causally connected.

_47:18_: Is the important thing here.

_47:20_: So that's the

_47:23_: and and and and things that are on on these diagonals. As you

_47:26_: can see from the from this some of this on the diagonal will

_47:30_: always be on the diagonal

_47:33_: there frame so something so two events that are separated by a

_47:36_: null or yeah so so so so so names

_47:39_: the interval between the origin and event five

_47:43_: option one and the origin

_47:45_: with interval being positive are called time like

_47:49_: because the you can you can you can find a frame in which the

_47:53_: the time that the time axis

_47:56_: join them up.

_47:58_: An interval for an event separation would event three and

_48:02_: the origin

_48:04_: where the interval is negative. It's called space like

_48:08_: because you could find A-frame in which they are both on the

_48:11_: same

_48:12_: X prime axis

_48:15_: and the interval between event two and the origin or the origin

_48:18_: event four is called null or light like

_48:21_: and those are invariants. If true events are are separated by

_48:25_: a time like interval,

_48:27_: a non interval or a space like interval, then they'll be

_48:30_: separated by a time like null or space like interval in any

_48:33_: frame. So those these are other things you can hold on to. Part

_48:36_: of the thing was especial activity, you know the

_48:39_: relativity bit causes people some, some, some, some, some,

_48:42_: some. Some

_48:44_: unease because they always everything's relative what we

_48:47_: hold on to. Oh my God, the world is ending. But

_48:50_: I think Einstein didn't like the fact that relatively became

_48:53_: called relativity,

_48:55_: that that sort of missing the point as far as you were

_48:57_: concerned, because the point of the whole endeavour was to find

_49:00_: absolute. What can you hold on to that is absolutely true

_49:05_: and the separation between two events is frame independent.

_49:08_: It's absolute. The the fact that that an event is a separation of

_49:13_: like like Swift and all like

_49:16_: time like Late Lake or Swiss Lake is an absolute. These are

_49:19_: all the things you can hold on to and these are things you can

_49:22_: build on. And the last remark I'll make about this before

_49:26_: finishing off this chapter is this is an XT diagram.

_49:29_: If you imagined AY coordinate sticking out here and then

_49:34_: rotated that then this

_49:37_: these these these triangles will turn into a cone because someone

_49:41_: called the and. And if you imagine as a ZZ axis pointing in

_49:44_: in in a fourth direction into the same thing that then the

_49:48_: light cone would be a a section of a sphere basically. And that

_49:52_: idea of the light cone,

_49:54_: the the, the, the, the future of the possible future of of things

_49:57_: which will stay in the the future pointing like one is a

_50:00_: thing that will come back to later on.

_50:03_: So that it got to the end on time, I have hastened to add. I

_50:07_: will appease myself of chapter 4. We'll go on to the range

_50:11_: transformation next time, which I think is next is next

_50:14_: Wednesday.