This is like the six, which is at which point we go on to
Chapter 5 talking about the Lorentz transformation.
And this, if you like, is the central mechanical algebraic
mathematical tool of special relativity. It's where we learn
how to go from
events which have coordinates one frame
to events which have coordinates in another frame. I'd rather
abstract thing to to worry about, but it's the thing that
ties together the
the things we're learning about if you know more or less
qualitative way in the lectures up to now.
So what we learned about length contraction time relation first
in a in A in a as a qualitative way
with the argument about the trains going past and people
thinking that person's watch is going faster that that that
carriage is shorter.
Then we added some numbers to that and and worked out what the
length contraction and time relation equations would be,
that factor of gamma.
So in a sense we're not adding anything more to that, we're
just doing the same sort of thing
in different form. But it's a crucial form which and and and
the the number of consequences to doing it that way.
And before going on, I want to just
recall a little bit from the the the the last chapter of the last
lecture. Well, I introduced space-time. I talked about the
Minkowski diagram and so on and
think about it. I think that is one of the harder lectures of
this course.
I mean, it seemed to me, partly because it seems quite bitty,
seem to remember different things happening there, and
suddenly, magically these Minkowski diagrams are appearing
in the terribly important and we're talking about space-time
and geometry and all that stuff. And so there is quite a lot
happening there.
There's not a lot of maths and there's not a lot of astronomy,
but there are a lot of physics happening there,
which it does take time to it, which is a challenge and it's
not. It's a challenge for second years and you know, unless years
it would be easy. I think it would be a challenge for anyone
in any year because it is
intellectually
difficult to get your head around those things because
you're being asked to think about things in a way that you
fact wouldn't have been in in previous years. The this, this
is, this is where I think in the first lecture I said this wasn't
wasn't a mathematical course.
It's not a mathematical court, in the sense there isn't any
complicated algebra, there's not any complicated mathematics.
It is quite a mathematical course, in the sense that the
way you've been great to think about things is quite abstract.
And that's and that is in a nontrivial challenge
project, that's if you're looking at the last thing going,
I have no idea what's going on, then that's fine,
work hard. It just take a bit of effort. But it does pay off in
wonderful insights into the nature of being. Anyway,
this
one of the things I one of the important things which may have
seemed a curiosity last time
is the idea of the very interval.
And I said, if you recall, the point of the interval was that
it was like Pythagoras theorem
in the sense that if you turn round
distances don't change. If you change your coordinates from
that coordinate system to that coordinate system, the length of
things that you measure you know in the squared distance between
two points
is invariant. And that is a deep fact about the geometry of
Acadian space. Euclidean space is just nothing more than the
the space we're used to the the the space where expert plus y ^2
is an invariant of translations and rotations.
There's this species that you're used to.
So putting that Euclidean space, I've, I've been nothing more
exotic than than, than than that which you never had to give a
name to before.
So we're going to use the fact of the
invariance from invariance of the interval, the t ^2 -, X ^2
and see
if we start from that point of view
and and, and in a sense that's just a restatement of the second
axiom because it follows from the second, the the, the, the
2nd axiom that I introduced in the first or second lecture. So
there's in a sense nothing new, but this is a new way of
starting from that. If we start from that, what do we get?
Any questions of that or worries about from last time or or the
one thing that I do remember someone pointed out that in the
notes from lecture you know for for for, for chapter 5. At the
very end of it, some of the figures have ended up on the
wrong side of the page. There's a slightly cut off that's just I
didn't run later enough times so that the I have replaced the
the, the
at the file with the same the same content, just an
externality, so that should be prettier. Just now.
I think that's the only thing I could remember to see.
Something else may occur to me. OK,
what we
why I believing up to there
is that?
Well first of all but before going on I want to just make
clear what the the problem is that we are trying to solve
here. What? What? What we're trying to achieve here
is offering
the next direction and in that time direction and the why and
the Z suppressed. And there's an event and that has coordinates
T&X. It happens at a time and at a position,
an expedition and attending the clock. And what those numbers
are depends on where you choose your origin. You're you're
exactly as is the case in anything else. But there's also
and as you know, familiar with the idea of Mankowski diagram.
There's also another set of of
axes we could choose
and in that frame. That same event
has coordinates T frame and X prime, so the observers in that
other frame would measure that event, which is something like
that. Anything which happens a place and a time,
and they would scrap it different place and say to the
different coordinate.
That's what we're trying to do is given one of these pairs of
coordinates, what's the other one?
OK, that's the the mathematical problem we've got. We're trying
to solve
that. What we can hold on to
the point where we start
is seeing that whatever happens. Ohh yeah, so so so that If you
also imagine
another event which happens at the origin, that's at
position zero
from the special origin and times zero.
And because these two frames are in standard configuration
that origins coincide, the special origins coincide at the
temporal origin. So at X = 0, T equals 0X, prime is equal to 0,
and T prime is equal to 0. That's the definition of static
configuration.
What that means is that there is a different a distance, a
separation, an an interval between those two events and we
know that the
go to t ^2 method delta X ^2.
In both cases one end of it is the origin, so we know that
delta that t ^2 -, X ^2
t prime squared
my ex. Prime
square.
That's our starting point. So the goal is to find an actual
expression for T&XT prime and X prime in terms of T&X,
which makes that true. Now that looks pretty similar, something
we already know.
It looked pretty similar
to
X ^2 + y ^2 = X prime squared plus
my prime squared, the Python green one,
and we knew and
transformation
which preserves that. Can we turn one of these things into
the other thing? Yes we can if we instead of talking about T
we talk about
L which is just T * I, the the imaginary, the unit imaginary.
And I've also into the lecture root folder. On the middle I've
uploaded 2 extra documents, one on maths, one on. We've got our
recipe, which I'll come to in a moment, but the maths one is
just a little bit of of revision. So if you already know
about the actual numbers and hyperbolic trigonometry, which
we'll come to in a moment,
I think just to to to just for religion.
So
L is
T * I and L prime is equal to I times TT prime. Then
L ^2 equal minus t ^2
L prime squared is minus T prime squared and this
turns into
L ^2 + X ^2 = L prime squared plus
explain square
to have turned what we want. The problem we want to solve is the
problem we already know.
Because we knew how to, we have a a transformation
from XL to experiment and L prime which preserves that which
is just X prime is equal to
X Cos Theta
plus L sine Theta. L prime is equal to X sine minus X sine
Theta
plus L Cos beta for some angle Theta. If any angle Theta any
any value of Theta we turn, we take X&L, obtain X prime and
L prime, and the sum of the squares will be the same.
OK, that's fine. So that that's progress.
Now we're also then going to write.
Peter is equal to
I Phi.
So rather than an angle, we're going to see that this this is
the an an angle times the unit imaginary. And if you remember
well, if you you I I trust you, I have heard of hyperbolic
cosines, and hyperbolic you have here.
But what you may or may not remember is that the hyperbolic
trigonometric functions and the circular trig functions are
related by
sign.
I Theta is I
Saints.
I'm going to write that as say I Phi Phi and Cos
I Phi is equal to
koshi
and
we then
plug that into here.
I'm not gonna go through the the the the successive steps, but
the end result when we plug that replace Theta there with. If I
do this, swap this and then turn the handle a bit,
turn L&L primes back into T&T frames is we get
G prime is equal to
he cash Phi minus X,
sanctify
X prime is equal to minus T
Saints Phi plus X
and we're sort of done.
So this means that given a an X&AT coordinate for an event
in one frame,
we can work out what the
quadrant of the same event
and obtained by different observers in two frames of time
configuration are in such a way that preserves the invariance of
the interval at the top.
Job done.
And that basically is job done
in the sense that is the Makovsky, the the Romance
transformation derived
to the rest of this chapter.
You're looking at a couple of variants of that expression
that's that's not the most common expression for that and
it's not the expression of it you'll see most often.
And your used most often in doing all the exercises is ohh
that's. I think that's the other thing I meant to remember. I
meant to remember to say to you, we're gonna talk about the
exercises.
I mean, the exercise is at the back of my notes.
There's also a tutorial handbook I think would be part of the of
the you know up in the middle
and that has had exercises for the four sub part of of of a two
of operational astronomy starting the Spectra theoretical
astrophysics and relativity
and the exercise in there the relatively exercise and there
are good
I'm not going to refer to them because you know while they are
good that they're not keyed to my notes in the way that that
that that that that my now that they're they're not a hangover
but that they are being built up over the years for a two. So
just to be clear, when I talk about the exercise, I mean the
exercise and notes
as opposed to the exercises in the shoreline.
OK.
Who was I?
Ohh, yes, so so the the, the form of the of the Lorentz
transformation that you
and
will use is not not that, not that. We'll come on to that in a
moment,
but the rest of this chapter
is all about using this to do these things and getting
thoroughly familiar with what it means about the the geometry of
the of of the space and time that we're looking at.
And I'll also mention that the, you know, OK, I'll, I'll, I'll
go on, go on a bit.
Um,
no this. See this event here
was happening
that there's a.
We would like to change the scenario and now the
the event we're talking about.
We'll choose always to have X print equal to 0.
In other words, it's at the origin of the moving frame,
which means that the position of that event in the platform
frame, the stationary whatever will always be at
X, equals
Vt. This P times times something which is happening at the origin
of the moving frame is always going to be at that position in
the stationary frame.
OK? Or in other words, OK tricky bit of algebra. Here V is equal
to X /
t If we look at this expression, here the second
the second one
for this event, then X prime we're saying is equal to 0.
So T sange Phi equals X cosh Phi.
Turn the handle a little bit, which over the V
is equal to
times Phi. In other words, we've filled in the remaining bit of
the remaining bit of the transformation, which says that
if these two frames are instant configuration and the second one
is moving at speed V,
then the value of Phi which is called the rapidity. The Phi
which corresponds to that is arctangent of of V and then you
plug that into the into this and get your
and get the answer.
It's we will not we we will in fact not end up using a lot of a
lot of
I've worked trigonometry you may or may not be relieved to hear.
Because
we're gonna get all that all our hyperbolic trig trig done up
ahead of time
and uh
use this to to rewrite
this transformation in a stage different form which I'm not
going to go through the steps. So I think one of the I think
exercise cycle one of 5.2 is encouraging you to go through
these these steps just to show you there isn't anything being
there's no slate of hand here.
But that turns into
the prime is equal to Gamma
t -, v ** prime is equal to Gamma
X -, v T where gamma is as usual. Now it's actually cosh
Phi as it turns out,
1 -, v ^2, the minus
and and and get it? One way of tangent squared the minus half,
blah blah blah. And correspondingly, T is equal to
Gamma T prime plus VX prime.
X is equal to gamma X prime plus Vt prime.
OK, so the all I want to stress that going from there to there
is just a matter of algebra. Go through set yourself nothing,
not the real fast one being pulled. But this is the form of
the transformation equations that you will use again and
again and again. And that predated Einstein.
A bit of bit of history, a bit of a bit of history of physics
here that predated Einstein. Einstein didn't invent that from
from scratch.
And remember I said that the the macro equations, the description
of electromagnetism and light? Where
it was clear at the end of the century,
Maxwell, I've done the right thing. This this worked as a
description of light, and the puzzle
that prompted
relatively true marriage
was that maximal equations didn't behave the right way.
They didn't behave in the way that a Galilean transformation
said they should.
And that was a puzzle and thought were aware that was, you
know, a terrible problem.
The ultimate resolution was special activity. Einstein,
clever chap. But there were other folk, including the
president, Lawrence and Pointer,
and Fitzgerald,
an Irish physicist and their third name, who was not coming
to me,
who said, well, migrations do sort of work.
If you use not the Galilean transformation to go from one
frame to the other, but you use this and this sort of plucked
out of the air as well, this works.
And you and you and you can. There's another route by which
you can get to that that answer, but it was sort of plucked out
of the air that this makes maximal equations work in the
sense that Maxwell equations work in the moving frame the
same way they work in the stationary frame.
But there was number reasons for that to be true. But so far this
works. But you know what was supposed to do here? And there's
some suggestion that.
It might be that if things were moving through the ether this
mysterious medium that that light was the waves in
that if things were moving through either they get squashed
a bit
you know sort of
hand waving way. But in this precise we would just right to
make length contraction work and and I'm actually equations all
happened but there was no
they'll get you just start again start again. From that point,
there was no real
motivation for that.
What was special about Einstein's approach was he
started from a different place.
So you started from these two axioms from, you know, a
fundamental understanding of what was a fundamental statement
of what was going on here, and derive these in a very natural
way.
And that's what's important about special activity is much
more natural than the ad hoc way in which the the the the
peculiar peculiar behaviour of material equations were
different.
And another question there and the bottom line there. The one
take squared is that one as in the speed of light. Yes, that's
right. Yeah. Well or rather in units which which aren't we're
light metres are not our units of time. That would be v ^2 over
square.
So, so, so, so so it's it's one as a unit 1, but but here this
is,
there's sort of 1 / C ^2 that's invisible there because C ^2 in
the right units is one. Yeah, but yes, so. So yes, this is in
units where C is equal to 1 as as will always be the case.
And
there's a lot more one could see. But the history of special
relativity I I'm
with exceeding great restraint in not spending on the whole are
talking about that because quite exciting,
right.
That's important. I'll move on. I'll keep going. Keep going.
Keep
right now we're. As I said, we weren't going to use that
trigonometric version too much, but we can do one more useful
thing with it which is if we
adding substract are these together and subtract them from
each other
we then we get.
You pray my ex prime is equal to
E to the five
t -, X
T
is equal to each of the Mace by
another another. That's another version of the same
transformation
which
any particular useful except that
it lets us see.
But if we add,
if you ask OK,
the trains go through the station platform. On the train
someone is running,
so that means the three frames here, station platform, the
train and the person running.
And so we we we now know how to get from the vision platform to
the train frame and from the train frame to the the the the
the athletes frame.
Do we know how to get from the train frame direct to the
athletes frame?
Yes we do,
because if we have 3 frames
then the same thing will be true for getting from the.
There's a second one to the third, which will be T double
prime minus X double prime equals E
Fly two. Let's call it
T prime minus X prime which will be E by 2 * e Phi one t -, X
where? So here if if I wanted the rapidity with the speed of
the train and the platform frame and by two of the rapidity of
the athlete in the train frame. And that's very simple.
So now we have that
that's just E the Phi one plus Phi 2IN
other words,
And there's just a different range transformer.
So it's still, but it's still the right transformation as it
really has to be. Otherwise you could tell you were moving if
the if the transformations went all wrong. But typically simple
version of it
I would also means is that and I'm again I'm not going to
I I guess from hints for how to to to step through that but that
also tells us
but but but by telling us that the rapidity of this of the
athlete in the station frame is just the sum of the realities of
the train in the station frame and the athlete in the train
frame. We therefore have a relationship between the speeds
of the station of the of the of the train in the station frame
and the speed of the athlete in the train frame,
which is
that the speed
of the athlete in the train in the station stream is V1 plus V,
2 /
1 plus V1.
You too. And no, I'm not going to go through the steps of that.
You can do that in in in the exercise. But the important
thing to note there is it's not just V1 plus V2.
So we are used to
speech adding in the street in an uncomplicated way. If you are
going through a A station at 60 miles an hour and you are so
thuggish as to throw them out of the window at 20 miles an hour,
it'll hit someone on the station platform and you'll be arrested.
But it'll do at 60 + 80 at 60 + 20 miles an hour.
OK? And this tells us that they don't, that that isn't true when
things are moving at relativistic speeds. As things
are moving relativistic speeds,
you end up well. If you're still in that for a bit, you you you
convince yourself that
when thinking of moving at near the speed of light when when V1
and V2 are nearly one,
the combined speed is still less than the speed of light, still
less than one.
So no matter how fast you go relative to things, some of
which is going fast,
you still aren't going fast this prelate,
OK?
And that will keep me relevant. I'm not gonna make a big thing
about it, just that that does all hang together,
right?
Any questions about that
question there?
What does buy represent right? And I don't think it's anything
that's particularly visualizable really. Here
it's introduced here really just as the the the bridge
from the. Look this expression for the range transformation in
terms of the hyperbolic functions
to this version in terms of the things we are actually more
interested in. So
and I think you could think of it as just architecture the
speed
so it's a function of the speed which happens to to to make
things work out in that form. So I I don't I don't think that
anything any any physical intuition that really available
by by. I think with that but but.
Except as maybe the the you know strictly in skateboards, the
rotation angle for for that. Because that is a a good point
that what we're looking at here isn't anything more than what
we're looking at when we see when we rotate our axes and get
different coordinate.
If I hold that up in front of you, I think I did mention this
last name, hold up that up in front of you and I rotated the
coordinates in your framework of the of the ends are different
but the the length is the same. You're not surprised of that
because all that's happened is has been a rotation.
What this is describing is a rotation and that's not strictly
a rotation but it but it it predicts it's close enough to
Puritan. That's a good thing to think about. And what that's
telling you is the and I'm repeating myself here that the
geometry of the space we're looking at is different from the
geometry the familiar with. But the thing I want to really drive
home is, because I'm saying it several times, is it is just a
matter of geometry.
The, the, the, the, the, the, the, the geometry, the, the, the
spatial rules of space-time are just like those of ordinary
cleaning space, but different.
And the way that things distort contract elite. When you're
looking at, I think a movie, different speech, it's sort of a
perspective effect in the same way that this seems to get
shorter when you look at it from a different angle.
And that requires going for a long walk and thinking about it
and and and and and, and and talk about just to let that
settle down in your head
because this is a big change. Because before,
before the 20th century,
the picture of the universe that we were left with promotion was
that there was a three-dimensional space which
was sort of marching through time.
We all shared that same 3 dimensional space and time went
tick tick, tick, tick tick tick tick for all of us at the same
time
and so. So space of three-dimensional time was
one-dimensional. End of story.
And what this is saying is that's just not true.
That is genuinely the whole thing is a four dimensional
thing, and one picture I find is quite useful, though I'm I'm not
sure I necessarily explained. I may not explain it very well,
but I invite you to think about it is. If you imagine two cars
heading across a desert,
see one of them's going.
No one's going in that direction
and when I was going
in that direction, so it seems to be different. Velocities
seems to be different direction
and that car
E
is heading in that NNN
NW it's going north northwest faster than carb is it's it's
it's heading in that direction as it more towards the north
northwest than carbo
so it's going faster.
Bicarb is heading to the north northeast
faster than car is,
So they're both going faster than the other one
really.
And that's not a contradiction because they're they're their
notion of straight ahead is different
at first. Car is concerned
N NW is IS is straight ahead and other ones going off at a
different direction. As far as club is concerned, N NE is
straight ahead, London different direction. That's why, it's
because of the change of frame
that they can be both, both of them be moving both faster and
slower than the other.
And that is what's happening
when you change frames in Russian context, you've changed
frames. So your your notion of what is straight ahead in time.
You know, I'm standing here looking at my watch. That's
straight ahead in time.
Your notion of of that is different from that of someone
who is looking at the same event in a different way
and there's nothing more happening that's happening
there. I think. I think I I think that's quite I I had a
little haha.
That picture occurred to me.
Any question about that? I like, I like questions.
I think there's a, there's a a quick question slide somewhere
in here, but the well, but I'll just stick with the, the, the,
the, the
at the moment. The other thing we're going to mention before we
move on to applications around transformation is the idea of
proper time.
If there's two of you, there's two events, one
here and one
here, right?
So the separate, the separate in space and in time
and any 2 frames I'm repeating myself a year will have
different ideas of what those the time, space and time
quadrants of those are. And we can now turn one turn to the
other. But there's one special frame that we can pick out. One
we can pick out special
who's the frame
which was present at both these where that event took place at
the origin in both cases. So it's if. If there's a frame
that's moving in the right direction at the right speed,
then this event will happen at it's already at its origin and
at its origin over over here. OK, so that that that frame is
sort of special.
It's special because the two events took place zero distance
apart
in that frame because they put you place the order,
so X prime
procedural there and and and the zero there.
And
if you had an observer
at those two those two events, then they could look at their
watch and write down what the time was.
But because we have no have the transformation
we can work out what that observers
coordinates, what got readings would be
and that observer, that special observer
who is Co located with both events
the the time on their watch is
sort of special. But also crucially, anyone in any frame
could work out what it was because of the transformer
and that time, that time interval between those two
things we refer to as the proper time,
then there's nothing fundamentally special about it.
It's just that there's a, there's a coordinate choice
which says that that that is a a nice, neat thing to refer to. So
that's when I talk about proper time. I don't mean it's
fundamentally distinguished time, but it is a I think
everyone can agree on
because it's happening allocated with two events
and.
Ohh, yeah, and. And that means that if we have two events which
are
separated by
there are two squared
and delta X of prime squared,
then the
interval between them is.
Is that exactly as before?
But if we're talking about about proper time, then separate the
physical separation between those two events will be
such that X prime does X prime is equal to 0
that you could delta D prime squared
zero and that time would refer to as
the proper time tour
To to I I will give you refer to the proper time and write down
as tall or taller or how we want to pronounce it. And it's just
another way of writing it. It's fundamentally just another way
of writing the invariant interval S,
but but it makes it clear that that that,
yeah, it's just another way of writing that. I'm not going to
bang on about that,
yeah,
but I want to lodge that idea with you.
Any questions?
Thank you.
Right
at
that that that that's as I said the that's the main bit of of of
mathematical machinery
for this for this chapter and for the chapter chapter to come.
The rest is applications, so think over that. Make sure you
it's all it's all the right way up in your head
and we're not going to applications and and way in
which you can use this to do things, to do other things.
One is to imagine.
Thanks
fine.
A rod
lead out along the X frame axis, so this is the axis of the
quartz moving free.
And the distance in that frame is going to be L0. And I'm going
to refer to that as the proper length
because in the that also is our to some extent a free moving
quantity in the sense that everyone could work out what it
is. It's the length of the rod in the frame in which it's not
moving.
OK, so if you're standing by the the, the, the, the, the, the
redness, not move, you have to do anything complicated about,
you know, multiple observers watching the that the ends at
the right time. It's just nice and simple
to an effect of the proper length. That's what I'm
referring to,
I think. I've I think I have used the term proper length
before part of this for sure,
right?
What we want to do is work out what is its length in the other
frame in the T frame in frame in frame rather than frame S prime.
And we want to rederive the length contraction formula.
What we can do
is
we'll have two observers in
South if you if you remember how this works.
I hope I measure the length works. I said that you have
something moving past your station platform and at a
certain time in your frame,
the observers who are at the
ends of that moving moving object note down where they are
and what time and and and what the pre arranged time is. And
the difference between the coordinates of those two
observers is the length of the thing and that is defined to be.
That's what we mean by the length of the thing in the
moving frame. So let's set that up in in, in this, in, in, in
this form.
So we say that there are going to be two events, like two
people pulling, pulling bangers at at the moment when they see
the end of the road moving past them.
We'll call those tonight
and
what we'll have one happening at, even one will happen at the
origin of the platform frame, the stationary frame. In other
words,
X1
and T1
will both be 0.
You know, we're we're shooting our coordinates, our freedoms.
So that's true.
And the
and
other
event
it will be it will happen at X2 equals something.
And T 2
= 0.
In other words, it happens simultaneously with other event.
That's what we mean by saying we make observations of the two
ends of the, of the, of, of of the of the rod at the same time
same time coordinate. And we'll call we've chosen 1 to be
happening at the temporal and spatial origin equals 0. T prime
equals 0,
the other one will happen at the same time
and the
what we're seeing is that that that the X coordinate of that
second event is
the this expression for the length that we're looking for
here.
OK, So what do we know? How do we link that to the other frame?
We said that these events happen at the
and both ends of the
road.
That means that if the event at the front of the of the rod
happens at the front of the rod,
they will happen at the front of the road in both frames.
Now we've got that so that, so that event too will happen at in
both frames. We've got
a value for that that that that event, that the court, the X
coordinate, that second event.
But we don't know. That's we're trying to find out. But we also
know that X2 primed
equal to L Naughty, because it's happening at the front of the
road. And so in that frame
anything anything that happens at the front of the road happens
at coordinate X2 frame.
And we also know that X1 primed equals 0 and X1 and T1 prime
equals 0 because there is some configuration.
Configuration means we can write down that and that OK if if
event one is happening at the at the
um order.
But what this means is we can then use the right
transformation equations
and write down that X2
is equal to gamma X2 primed plus
the
two friend
and that
T2
which is equal to 0
gamma
T1T2 prime 3 plus
BX2 framed
um.
And we can
yeah
that being zero gives an expression for two primary terms
of of the experiment of nothing complicated distance B times
time. We substitute that into the other one and get X2, which
remember is equal to L
is equal to gamma
1 -,
v ^2.
They're not,
Remember is 1 -, v ^2 to minus 1/2.
So all, or in other words, one way of record is one over gamma
squared.
That's equal to L naughty
over gamma,
which is what we got last time for the length contraction
expression.
But we've got it this way by a much more systematic route,
and it's just a matic route that you will get used to
because.
So I should have this up already and let's go back to here,
right?
I trust all of you are using two factor authentication
and for your
whatever.
Oh,
oh,
yes. Whatever
now gotta find where that going to and the ohh no here it is.
Right
when the documents in the in the actual folder is ohh
is this
race these recipes document
and I want you to to to put it under your pillow
and absorb it during the outer darkness and
it's a recipe for doing
these calculations and and I went through that recipe when I
was doing the calculation there
standing for creation
it's almost done configuration the retransmission works only
works if you're things are starting variation and you have
to write and your
and
the assessment should you exercise you do the assessment
deliver in the exam you'll have to write these steps down to
just to to be very clear that you're following everything
explicitly. And also because if you go through these steps, if
you are forced to think what you know, answer this question at
this time, then you end up with things in systematic enough way
that you don't confuse yourself. There's a way of not of avoiding
confusion.
So what did I do?
OK I didn't draw Mickey diagram that I'm a I'm a bad person I I
I would expect you to to to to It will almost always be useful
to draw Minkowski diagram. The very few cases where it is just
to fix just turn the story into into numbers. What I did was I
said there's the story I was telling you was a rod at rest in
one frame. What the location of the coordinates? What are the
coordinates of the these two events? In the other one
identify the frames
frame of the, of, of of the, of the road frame of the station,
whatever. And I would write that down if we're if we're writing
that down carefully that this is not
I submittable that one quick
and this is not a submittable version of the of of of the
answer. Yeah what a fair copy there.
Identify the origins of the frames. What is what is at the
origin of the frames. There will always be some sort of event,
almost always be some event at one of the origins. In this case
the one of the firecrackers, one of the one of the bangers was at
one end of the of the of the of the road which we decided we're
going to be at the origin.
These are all simple steps but but they're all, they're all
important. Identify the events. There will be events
and often the key to solving these these exercises is is
thinking straight about what the events are. I said the two
events were going to be two events which were at each end of
the station of the of the road and at the same time.
And I would write that down, explain what those two events
were and a number of them one. Event one and event two,
identify the world lanes. OK, they weren't very complicated in
this case. Didn't have to do very much there.
Write down what you knew, what it did in this other thing. Here
it wrote down
the quarters of X1 and T, the coordinates of one
in the 2 frames. Because that was not easy because I've chosen
that meet the origin, I wrote down that the that the that the
the story I was telling were the events were meant that two is
equal to 0 at all. That done,
I wrote down a. The link to the other other frame is that the
event happens at the end of the road. Where does that mean it
happens at X2 prime? Is equal to
is equal to L nought.
What am I trying to find out? I'm trying to find out the
length of the rod in the boom frame. That's L. So what I would
do,
it would, I don't know, you know, put the arrow or
something, you know this, this is what we don't. This is what
we don't know
and and right. So write down what you have to find out.
At that point. Your Minkowski diagram is a mess. If you redraw
it
and at that point it it's downhill. At that point it's
just algebra from from the the they're on
and I and and you know so this bit of algebra blah blah got I I
got that job done.
So if you are systematic about it and I will want to see
demonstrations of viewing systematic about it in the sort
of things,
they're actually really easy
choosing the events or what the events are. It tends to the
hardest bit now in the assessment that's due. Well the
the, the assessment number one, I thought it was due this week.
So I therefore thought, well, it's not really fair for me to
give them a I think about the musket, the transformation, so
that so it doesn't involve transformation
and. But it should have done. So what?
So the the, the, the, the bit locked off at the end of it. But
further assessments will
involved with recipe more carefully and the other exercise
in the note will encourage you to do that. Enough talking about
the recipe, I say. I will see. You know tomorrow is the. I
think some of you have supervisions already have
the provisions, as I'll see some of you in the divisions. I'll
see you next. Next lecture will be next.