Login lecture 4 and we are off to the races. Now before I get
going I'm going to mention a couple of things on the rural
page. One is the folder of lecture notes which I trust you
have found already. I really hope you have found that already
and that contains
the.
There's a four format notes
there as PDFs.
It contains
a note formatted nicely different way, which may or may
not be more suitable for reading on a tablet I have. I have no
idea how that works for you, entirely up to you what you use,
They're the same content. I will retrospectively post the slides
there as well. OK, one behind, but it doesn't matter. I will
say again that there's no difference between the content
in these various formats. The tablet format and the printable
format are the same, just would have margins basically. The
slides are just things on the notes I want to highlight here.
So there's nothing extra in the slides. Just
I, I, I do tend to end up with a lot of stuff here. And I I think
it sometimes happens that you end up going, Oh my God, there's
so much stuff and like going there, there's less there than
than their peers with a lot of the duplicated. OK. So we'll
make that really clear.
Also in the
little Peach,
if we can get to the right page,
get down to here,
back down to here. Um,
there are some some on
I thought it
the
right and
there are some things that are are from previous years
a I'm going to post but I haven't done so yet. My
recordings of the lectures,
the the the E360 does sort of work, but I can't work out how
to get it connect that to the Moodle. It's terribly hard. So
that might be easier to do it myself. We'll see what happens
there, but I will be posting the the the the, the audio
recordings. I just haven't done that yet,
but I do want to drink for your attention. And ohh, this is
quite important writing Greek letters because legibility is
important there. Have a look at that document it it contains
bitter experience. Well, bitter on previous students for I do
want to draw attention to this padlet.
I I really am. Honestly,
yes.
OK well I I think because I've rejected all cookies this is not
working at work anyway. The the, the, the padlet. Perhaps I have
to stand by. You know for the pilot is a good thing. It's
basically a collection of sticky notes on on a on a web page. Ask
questions you entirely anonymous. I can't say who you
are. Ask questions and I will you know when I remember, go
back there and look at what questions have been asked and
and and and and add some comments. Answer them or or add
further context. Different years. Sometimes
no one looks at this deserted wasteland, Sometimes it's
notable all over the place. Different classes seem to get
the hang of it in different ways. I don't know. Whatever.
We'll see what happens this year. I think it's generally a
good thing, but it provides a nice way of asking questions to
me. If you're shy and don't want to come and ask questions at the
end. Also everyone else gets to see the questions. Questions are
good questions because if you realise someone else has that
puddle as well, it's they may have expressed it really well in
a way that you had to go down to, or you may be reassuring you
just go. Oh, someone else didn't understand that either. So quite
all questions are good questions.
I think actually pretty much all questions are good questions.
Yes,
So we're turning to this
just to recap the end of last lecture, last chapter I talked
of the light clock, this idealised way of talking about
the passage of time and the passage of time in space.
I don't. I talked to the difference the light clock when
you were standing by it ping ping
and the light clock when you were watching it go past you on
for example a train where the the the point where the light is
reflected, it's little bit further down the track from the
point it was admitted and from your point of view on the
station platform and the point where it's received again is
further down again. And because in this context
you are seeing the light moved at the same speed in the two
different frames,
that allows you to come to some surprising conclusions. But the
time interval between emitted and being received in this frame
and then the other frame.
If you were doing this in a non relativistic context, say with a
a looking at the direction of a tennis ball being thrown back
and forth across a
train carriage,
and the thing that would allow you to go from one frame to
another is the fact that time goes the same in both cases and
to the tennis ball clock, the tennis ball would pick up some
speed from the from the train that's moving in and so there
will be discriminated at the speed but an agreement about the
time.
The point where we move into special relativity territory in
this case is the point where we see the thing that we can use to
make that bridge. The thing that we can hold on to in these two
cases, the 2 frames, is that the speed of the light is the same.
That's the point where the magic happens. If you like to drive
that home,
OK,
yeah,
I, I, I, we we allow that allowed us to deduce our
relationship between the time on the watch of the person of the
observers in the moving carriage and the times on the watches,
plural, of the observers in the on the station platform. With
that expression, we will see again and again
and I made some final remarks about the what I call the clock
hypothesis or the the clock clock hypothesis which just the
time that watches
clocks
measure, the measure, the passage of time in in, in a way
that we are going to assume was unsurprising
and an image that I think is very useful.
This idea that I think I mentioned, I think in the first
lecture of this thing called, you know Taffrail log, the thing
that if you're a yacht person, you've put you, you tow behind
you and it measures how much water you've has been pulled
through and that's how far you've gone. And a clock. Think
of a clock like that.
A watch measures how much time it's been pulled through,
and that's that's what that was recording. It sounds fussy to
have to meet these fine distinctions, but it turns out
with relativity, if you don't make careful distinctions you
end up confusing yourself.
But that's the recap of last time.
Are there any questions about that? Things have been puzzling
over
since last thing.
OK,
space-time and jump to this. We start with more exotic
terminology
and I and I plan to cover this in I think 2 lectures,
3
objectives, aims and so on, right?
No, we'll come to that in a moment.
There's a couple of things which are It's important to preface
this. I'll show you to report to preface this section with
in in the last chapter last when we're talking about lens and so
on.
What we were doing was
noting that different sets of observers in different reference
frames, the ones attached the platform, ones attached to the
station platform,
both make
observations of the positions and times of events.
And what we what we are aiming to do is relate the observations
of the people in the in the carriage to their observations
in the in the station platform. And that's not just because we
don't we want to know those numbers what that tells us is
the physics of the relationship between them. So that's
applicable in all sorts of other contexts all the way through the
rest of physics, just that's what's important. But we're
we're boiling that down to this question of of of of changing
coordinates with these two different frames which have
they're different. And it's not because we've chosen different
units for the on the, on the, on the, on the platform and on the
the train that's trivial. It's not because children are
different origin
because the whole business of special of standard
configuration resolves that problem. I mean that does make
that we sort of define that problem away. It's not because
we have to worry about the time of flight of the light that is
that is a I think that is a potential issue but the fact
that all our observations are taking place entirely locally
means we don't we we that that's to avoid having to even worry
about that even think about that. It's not terribly hard,
but you can just avoid thinking about buying the observation.
What it turns out
is that the difference between these different frames is
because of the finite speed of light and because we the 2nd
axiom says that the
speed related the same in all reference frames. That's where
the the change happens, as I, as I was seeing
remarks about the end of the last chapter.
And the point is that the
coordinates of the events
are measured by the moving observer are systematically
related to the coordinates of the same events as measured by
other observers. The notion of randomly different, they are
systematically differ
and the way we are going to. So we're going to explore more
about the the relationship between those events and
with the approach that there are a couple of different approaches
one could take to ratify the approach. I take it very
affirmative. Geometrical 1.
Now you grow up with an intuitive understanding of
Euclidean geometry.
In school you learn a little you you learn that word and you used
to systematise your your, your intuitions about it. But you you
understand Pythagoras theorem intuitively. You you know that
if if, if,
if I
stick or something.
OK, you know that if that's about a metre long,
but it's still a metre long when I turn it gasp, and that's not
surprising you. That's your intuitive understanding of
Euclidean geometry. OK,
now when you you, you, you, you, you, you hold down astronomy one
and you will recall you were talking about spherical
trigonometry. The distances in the sky, on on on the sky.
The geometry of the sky is not Euclidian
because the trigonometry you learn is different, because
things like the internal mechanism, angles of triangles
add up differently.
It's different and the same true on the surface of the Earth. If
you're talking about, you know, large, very large scale scale
maps, it's different. So you're you're used to the idea of
one step, you're understanding me accompanied by changing
geometry. And that's what we're going to discover is true in
ratifies will Just the geometry in question here, the geometry
of four dimensions in space and time. And the startling thing is
those end up being closely related to each other, in the
sense that going from one to the other isn't a isn't going from
talking about talking about watches, but our rotation in
that, in that geometry. Slightly mind-blowing, I thought, but you
will get used to it, I assert.
So this chapter is taking a first look at these ideas.
That's the direction we're heading.
But first
we'll have to talk about units.
Now the speed of light is quite, quite fast. It's it's 303% to 8
metres per second
shift.
So
the the the things that we are, we're used to
crawl compared to this beautiful light. So it makes sense to
have. You might want me to try to discover that we have to have
a little think about units,
but you're astronomers, you know that we were talking about
distances within the solar system or within the Galaxy. We
have to talk about light, years and so on as a measure of
distance because we have to big distances
and you're familiar with that and that makes sense. And the
standard thing that confused about is that a light year is a
measure of time. It's not, of course. You know I like you as a
measure of distance. Is the distance that light travels in a
year.
We don't have and. Light second is also a distance
as the list that travels in a second. It's the and. And the
radius of the sun is I think 2 light seconds.
It takes 2 seconds for for for like to go back to that distance
and you could talk about the light nanosecond
the just the late travels in a nanosecond is but that that that
far there's not very far but it's it's it's a real time. So
we could if if we're talking about things moving at the speed
of light, talk about things moving at late nanoseconds per
second or something like that or late seconds per second, that
would make sense. There's nothing wrong with that. It's
not conventional
or can instead do is talk about the light metre
and then just a unit of time.
The same idea, it just so the other way up.
So a light metre is the time it takes for light to travel a
metre
and that's not, I mean that's a very small unit of time. It's I
think 3.3 nanoseconds,
but it's a but what that means is that light travels at one
metre
per light metre,
which is very convenient. So clearly the light is a useful
unit of time when talking about
about things moving. It's late, so we're just at this point, I'm
just talking about picking our units, right?
So what we could do then is, um,
you recall how to convert units between different,
of course you different units.
Have we got,
I think this is the one that's being recorded on.
OK. And
can we see that? OK,
OK, good.
So see, we had a quantity of
10 Joules.
OK. That's 10
kilogrammes, which is squared per second squared.
You know, this is first year stuff. OK,
now you you know how to convert between different units,
you multiply or divide by the conversion factor as
appropriate. It's also confusing. I mean, I'm supposed
to multiplying and dividing, but there's nothing nothing exotic
about it.
So what we could if we do is if one
light metre
is and I've I've gotta try work hard here trying to which we
update these things are is these things tend to the eight
second
OK so there's a very small amount of time then one
per second is these things tend to the
88 metres so one per second squared is the same as 10 to the
eight
like metres. It's nice too.
So 10 joules equal 10
kilogrammes metres squared,
3 * 10 to the eight
let me just squared, which is, um some number. I think I see
it's
1.1 times
grammes. You just squared,
black metre squared. OK now I haven't done anything at all
exotic there. I've just changed to to from seconds to a
different a different unit of time
I think.
OK,
it's the next bit that's confusing.
I'm going to look I I could carry on talking light metres,
but I'm going to stop talking about light metres and just talk
about metres,
OK? So when I say a metre
and it's going to be so ambiguous that I'm talking about
metre in distance or a metre of time, I'd like metre of time.
OK. And you think that's a really stupid thing to do
because that creates an ambiguity. It turns out it
doesn't. It's a bit confusing, but it turns out it doesn't. But
look what happens if we do that. We'll see. That's 1.1 * 10 to
the 16 kilogrammes of metre squared
per metre squared. Ohh. So we can cancel.
I think that's just a mistake, isn't it?
And it does look a bit like a mistake. But if you hold on to
that, that's what's happening here,
and this step here just a notational convenience. You
won't go far wrong,
OK,
but this does look weird. You just have to stay here for a
bit. But all that's happening
is East
a decision about the time units were choosing and BA bit of
notational trickery because what happens is the other advantage
of that
is that we if we ask what is the speed of light?
Well, this beautiful light we all know is 1.
It is 3 * 10 to the eight,
he says.
2nd
for a second but if we convert from seconds to lay metres,
discovered that one metre per
but like me here
and we make the second step of just writing metres for light
metres.
She got that one.
So speeds,
for example, if you're late, are all dimensionless. Are you? Are
all unitless. They're all unitless.
They're not dimensionless because the speed is still
a length divided by time.
But because their children same units were, both the units
cancel. Even the dimensions don't. I mean with speed. So you
can think of that either as a
as writing down the units where you've magically made units
disappear, or else just think of speeds as being all quoted in
fractions of the speed of light.
So any speed slower than that will be some number which is
less than one
in units of metre per light metre or less than one.
And
that is all there is to the idea of what we're calling natural
units. These are called natural units. This choice,
I'm not going to say more about that There there, there's more.
There are more words in the relevant section of the notes
which which go through this in a couple of different ways.
So I'm not gonna say that because if I see more about you,
you think there's much that you think there's a really big deal
happening here. There's not a big deal happening here. It is a
bit confusing, but it's just a change of units to convenient
set of units. And there's a couple of exercises which I
encourage you to do
just to settle this idea in your head. But hold on to the thought
is that there's not any particular topic here, right?
You will get used to these but it will take you go through a
couple of couple of exercises and it is just the standard
confusion of trajectory of of switching between units and I
noticed as well. Another thing you'll see is if I say something
like 1
inch
equal to 25.4
millimetres and I think in in the notes I got the decimal
point in the wrong place there because inches are weird weird
colonial unit. And
the other thing that you wouldn't see often is some it
writes some writing. One is equal to 2025.4
millimetres
per inch, and that's a weird way of writing about it, or without
a weird way of writing it,
but you're often not wrong. OK,
so if you see something like that and and see something like
one is equal to 3 * 10 to the eight
which is per second
to do
see C is equal to 1 is equal to that, then that looks that.
Again, that just looks looks wrong,
but that is all that that equality there is,
is that
it seems The thing is that it's just that's the conversion
factor between metres and seconds, and saying that, that's
the conversion factor between inches and millimetres. So it
looks wrong, but it's just a rotational, rotational fluke.
OK,
go in. Think about that for a bit. OK, you're all going,
ohh,
this is one of these things that is fundamentally not not deep,
it's just a bit confusing. You know? There are we will talk
about things which are fundamentally deep as well as
being confusing. This is just fairly trivial, but looks weird.
OK,
hold on to that thought and I again, there's more. There are
more words about that in the,
in, in the notes, but I don't want to spend too much time on
it because because I spend more time on it makes it sound
terribly complicated and sophisticated,
right.
And and that's that working, that working done.
OK. Well, just quickly,
umm,
OK. Well, you know that I've told you that this, but
I'll do it. Move this back to
and
as a quick question, which I've I've sort of spoiled by by
walking through the answer beforehand. And what would
remain nuisance
be in natural units?
Well, you know that the answer must be kilogrammes because I've
got I've worked this out. But you don't have any arithmetic
for that
because you can just think of the dimensions of
of force.
Ohh, no, no. What's the nuisance? I I talked to Jules
before,
so so OK,
hands up. Who thinks that they Newton's is going to be that. So
you're doing the arithmetic. Just look at the dimensions. Who
thinks it could be the first one? Hands up,
this is one of the second one.
With the third one,
we have a little chat to your colleagues
which one it is.
OK, who after conversation would think it was it was the first
one,
Second one,
third one,
right. Good that the majority of you got good second one. That's
sorry answer by the way. It's really useful to talk to people
about you know if I'm puddled explain why this answer can't be
wrong and it can't be right. And often just explaining to your to
the cat why they the the the reason for your incomprehension
that can somebody resolve problems. It's the second one
because
the conversion to natural units.
A conversion to units.
Well, we're not using seconds, but we're using light metres,
so the 9 kilogramme metres per second squared
would be converted into something kilogramme metres per
light metre squared.
If we turn the light metres into metres, just renovate them,
that's metres per metre squared, so it's per metre.
So that to kilogrammes per metre are the dimensions of the units
rather of
force. In these units, the dimensions are the same. They
are mass length T 2 -, 2.
But because of the units we've chosen some, some of the of the
units cancel.
Good.
We have the right speech,
right? No. How do we
visualise
motion?
This is a perfectly familiar sort of plot to you.
Distance
and versus time. So as time. Yep, I'm teaching you about
truck X here. You understand this. But the way graphs work is
as time moves along. Here
you you plot the distance at that time. OK so this
first so this is imagine a a light bulb flashing. So it's
it's flashing as it as it moves about.
If the light bulb is just staying where it is,
then as time moves on, it's expedition just stays. Flash
flash flash flash flash flash flash along E
Nothing complicated there. If the light bulb starts
accelerating so it's moving faster and faster along a then
you'll get a graph like that of the flashes as the thing
accelerates along the X axis.
OK,
I'm really upset about that and The thing is moving well. We'll
choose the the units on the on the axes to be metres and light
metres.
If things are moving at the speed of light then it will move
at one metre
per light metre 1 metre per metre
I along the diagonal.
OK
and if you had an event, so all these are events. All these dots
are events
and and and if we join the dots in each of these cases then we
get what we call the warplane. The world lane is the set of
points in space and time that an object goes through.
OK, it's it's a join the dots of the all these and if an event we
call event one then that will happen at a place in the time
and we can plot that on the diagram. OK, at at at
event one has X coordinate X1 and time coordinate T1. Why am I
making so much of A fuss about this? There's nothing
complicated about that. OK, you have entirely familiar with.
That's the same diagram, but flipped so that
the X axis is along its horizontal and the same axis is
vertical.
Why? Because it's traditional,
OK. And it does make some sort of sense, but because
makes some sort of sense. But the point of this change of
slides is that is exactly the slide, the graph you're familiar
with but just flipped on its side and it's called the
Makovsky diagram. When it's done that way, it's quite useful just
to and you'll see these again and again and again and again
and again in this course. OK,
end up looking a bit confusing, but the point of the game is you
build it up slowly while thinking through the problem
in a way. The only so the two things that make up a graph of
the coffee diagram is this, this tick, this, this weirdness of
flipping the axes, and the fact that we always implicitly assume
that the that the units are are the same in the horizontal and
vertical
actually access. Which means
that things were to move at the speed of late, always move
at 45 degrees,
and things were moving less than speed of light
move at a gradient which is bigger than 45 degrees.
OK, that's OK. So you will
so so so they are the world line A.
Maybe something just resting at X = 0
it's a successive times B is accelerating and sees movement.
People like OK, but there's nothing weird there
that hand up or just we have the
right.
So let's use that, use the McCarthy die or let's build up
some other Mickey diagrams, describe what we've seen before.
Because the the point of when Gotti diagrams is that,
well, I did. I see this last time. I think it may have done
the
but but but the assessments in relativity with exams and
relativity and special relativity, there's basically
only one question that can be asked.
Now here are some events in this frame. What are the coordinates
event in the other frame? That's basically the only question that
gets. It is dolled up in a variety of ways, but that's
basically the question.
And it it in all cases the the thing that's hard or the thing
that makes it challenging. The question is there to test is for
some situation, it's described in words, you know this thing is
moving and this thing happens. How do you turn that into
a set of events
with coordinates? You can then turn into the other other frame.
So the way you do that, the way you go from a description of
this, this thing is moving in this way to assess events is you
focus on the idea of the machine diagrams. The McCaffrey diagram
is a way of you organising your thoughts
and organising the translation of something interesting events.
So what we're going to do is look at this
again, again, this idea of the of the, of the, of the flash
bulb, you know going off in the middle of the train carriage.
It flashes reflects from the mirrors at both ends, and the
light comes together again in the middle.
So how do we draw those? How do we draw that in the Minkowski
diagram?
OK.
And what we do first if we get some more paper,
so let's talk about the so. So I describe that in words. Light
flashes reflect come back. We're going to turn that into set of
events
through there are four events there.
Event one is
right
flashes.
Ohh no step no step 0
frames. The frame S prime will be the frame
of the
A tree
and the frame S will be the platform.
OK, if in one if the flash
event 2 will be a reflection
from around my flat from the front
event three will be reflection
from the back and don't fall will be the flashes light
arriving.
OK, that's eligible. But you know, I'm saying, so let's draw
those events on the musky diagram.
So we'll draw the
the ex prime
and T frame axes. So these are the other the coordinate axes in
the in the train, in the train carriage.
Well see the the origin of these of of the of of of this
coordinate system is that is that the light is that at the
centre of of of the train carriage. OK, so that event one
happens at X prime coordinate equals 0. So it happens
along that that frame axis, and we'll see it and we'll see that
this carriage is 6 metres long,
quite short carriage, 6 minutes long. OK,
so it's 3 metres from the centre to each end. So we'll say that
the first event happens at
X1. Primed equals 0
T one prime equals -3 metres. Could we get to choose our
origin?
So we'll choose them so they're sensible. So that that means
that's event one there,
right? And that's and that's T1. Prime equals -3
metres
to the right. Flashes
should the light goes forward and backwards at the speed of
light. It being light,
what does that look like on the coffee diagram? Remember that
things which are this people like in the midfield diagram
move at an angle of 45 degrees, so the light moving forwards
when we plot it looks like that
and the light moving backwards
looks like that.
OK.
And the light moving forwards, we'll move three metres forward
and three metres of time
and so it'll get to the point where it reaches.
I thought frame equals zero,
that's the time when it got to event two.
So we can put event 2 on this. On this diagram,
event two
is there
and event three is there.
So we know that X2 primed is equal to 3 metres. We know that
anyway because the any event which happens at the front metre
at the front mirror is happening at an X coordinate of plus three
metres,
OK? And we know that any event that happens at the back mirror
is happening at an ex prime coordinate of -3 metres.
So we knew that X2 prime is equal to 3 metres,
X3 prime equal to -3 metres
and we've just worked out that because of our choice of origin,
two due primed is equal to zero.
D3 primed is equal to 0.
OK,
now this is just speed. This is the speed times time you learned
about this in school
and they reflect there and the light goes back in the opposite
direction.
In other words, it move, it reflects from three and goes
at the speed of light in the positive X direction. And
if you write the negative expression, and of course they
meet again at event four
and then 4X4
primed is equal to 0, it's happening at the origin again
the spatial origin.
And the time
of that
is 3 metres, because it takes 3 metres of time, 3 metres of time
to get from the mirror back to the centre.
And so we have completed.
And of course your diagram
in the
prime. The prime frame.
No, I'm doing this. Very sketchy. OK. I'm just scribbling
live in front of you,
the assessments.
Well, you'll be required with that question, yeah. So what
does the label label or #4?
Ohh
the that the light gets back-to-back to the centre so
that the the reflection we arrives at the back of the
light. So the
in the future like sizes and the in in the class test and the
exams, you'll be required to do things like draw that Drummond
coffee diagram.
You're right, if what I I get looks like that, you'll get
marks off. That's a mess, OK?
And they will take marks off if it's a mess because you, you,
you, it'll be a mess. The first version you do will be a mess.
Then you think, ohh, no end it. You draw a fair copy, right?
Because the communication here is, you know, part of the whole
part of the scientific communication thing. But it also
is about explaining clearly. You're working, explaining
clearly. You're the process of thought. And in a sense it's the
it's the explanation that you're getting marked on. You're
showing that you understand that you have just pick some numbers
out of the air,
So that's a little right, which I will probably repeat more than
once,
but they're supposed to be all 45 degrees. They're a bit
squashed there,
so we have assembled our recovery diagram. Note I was
explicit about the what the frames were. I wrote that down.
That's important. I was explicit about what the I about the
events that I identified and I thought about and thought these
are the events that happened, the things that have happened at
a place and time.
I wrote down what the events were in words
and I worked out one way or another. This is meeting time of
what the coordinates of those events were, and I plotted them
on the Minkowski diagram in a nice neat eventually way. Those
are all steps that you have to go through
and you'll get faster.
So far so good. Now let's just do the same thing in the other
frame, the frame of the platform,
and see what happens.
And this,
um,
so the frame, that other frame
that that would be a bit more thought, so that frame.
XT
No prime T, prime X&T.
And
there's one thing we can draw immediately, which is we know
that the train carriage is moving through this frame.
OK,
that's, you know, the basic setup
and we know that the world line of the centre of the carriage
is moving at speed VV. It's some number
which means we can draw the world line of the centre of the
carriage, the world of the length of the light flash of the
late bull
immediately.
That's what we go to the middle there
at the water line of the train.
But if we look at the first diagram,
we can see that the.
OK, hold on to that, we'll talk about that.
Welcome. We will come back to that later. So that's the water
line of the, of the, of the, of the train carriage
of the centre of the train garage.
But we know
from looking at the top diagram the events one and four happened
on that world lane.
Who they happened on that world line. It happened if they
happened at at the centre of the train carriage in the 1st frame
and they happened at the centre of the train carriage and the
2nd frame. So events one and a four must be on this model line.
So events one and four are something like there
and
there.
This is not a scale drawing. Minkowski diagram has never
scaled, but there's something like that.
But also
we know that event one was a light flash.
The late moves forwards and backwards
and move forward and backwards. At what speed?
Beautiful light so that the water line of that of that light
flash has moved? At what angle? The majority diagram
145 reads yeah, at speed one or angle 45 degrees. OK, so it
looks like that.
It looks like that.
Good.
And when it reflects and comes back and it comes back and they
both arrive together at
the at event four
up there, but we think reflect back they are also moving at the
speed of light. So they are also on this diagram moving at 45
degrees
that the late flash.
So when they arrive back at event four,
they are moving at
45 degrees.
OK, so so that's good. But we also know
that the point where the looking at the top diagram, the point
where the light flashes turned round
it events to an event three.
So from that drawing we can see where it would plot events 2 and
event 3,
namely
2
and three.
From the top diagram we can see that the that that
events one in four happen on the on the on the on the T prime
axis.
So we can draw the location of the T frame axis on this new
diagram.
It's just that.
And we also just can see that events
two and three happened on the X prime axis.
So we can draw the position of the expert Maxis
of this diagram as well.
So we've worked out what the most country diagram in the
UM station plane frame is.
Rather messily, I reiterate rather messily.
And it looks a bit different.
Why does it look different? It looks different because in both
cases the speed of light is the same. In other words, the speed
of light the the the the world line of light flash is 45 region
both cases. That's the second axiom again.
But look at what we've discovered.
The
time coordinates who events two and three. So the time
coordinate of of of event two is about there T is, T is plus
something,
and the time coordinates in this frame of van three is
T equals minus something, and they're not the same.
In other words, in the top frame event is 2 and three with
simultaneous,
unsurprisingly because if you look at the picture of what's
happening, the the lights get to both ends at the same time. But
in the moving frame,
the light gets to the is reflected from the from the back
earlier, it reflects from the front. And that's the same thing
that we reason through
in the last lecture,
just did it in words. We're doing the same thing here, but
in a diagrammatic form.
So that is the relativity of simultaneity,
this phrase that I used last time. Two things being sipped
being simultaneous is framed dependent.
If it's two or three are simultaneous in that frame.
They're not simultaneous in another frame
because they are physically separate.
To vent a repeat which are which are simultaneous at the same
place.
You know two car crashing
are simultaneous in every frame. That's that is absolute. That's
unequivocal to eventually simultaneous but specially
separate.
That it was unity is relative,
and as you can see that made coffee diagram looks a bit of a
mess. Not just because of, I've just drawn it quickly, but
there's a lot of lines happening there.
If you look at that, you look at the end result, you go. That's
terrible, confusing mess.
But it's not if it's the end result of you working it out.
So if you look forward in the notes, you'll see lots of
McCaffrey diagrams and they all look very intimidating. There's
lines and curves and all sorts of other space. That's because
they are the end result of a thought process.
So I've I've spent quite a lot of time on that.
I will pick that up next time
because I just want to go quite carefully through the idea, but
it's because it's so important. I'll have to speak up a little
bit next to to to catch up myself. I aim to get to the end
of Chapter 4. Next time I'm going to move on to Chapter 5,
which is the main, the main deal. We'll see you next time,
which is tomorrow.