Welcome, Richard. 7. This is the We'll finish off Chapter 5. This
time
would allow us to go into chapter six and seven, which are
the more dynamical bits of the of the material. And they will.
I hope to get them onto the Moodle today, but they'll appear
this afternoon. I expect
the couple of things I want to mention
which I I think I may have pointed out in the
in the in the forum posting. One is
in the middle. There's a link to a previous years I think a
couple of years ago,
and I I say with one,
so not this year. Previous years, good one to go to work
through because you can do everything in there you can do
at present. But the thing that I wanted to draw your attention to
was that although the notes of the bottom of that have fairly
discursive extra commentary on on the answer and so on, there's
also a handwritten version. Might I I did a I I wrote out by
hand what would be a great answer. The sort of thing I
would be happy at 2:00 in the morning to see popping up next
on my screen
and that would go, yeah, that's gonna be 16 or 16 but you've got
about reading it right. So that's what you want. That's the
sort of thing you want to to to be handing in. Now the important
points as I as I mentioned in in the model posting,
it's legible. It's OK. My handwriting isn't necessarily
the most beautiful, but it's it It. I can read it. If you if you
want to make a tight version, that's good too. What it's not
is photograph or scribbles on your notebook. It's not a mess.
It doesn't have to be beautiful. There's a couple of crossings
out in here. Bit of Tippex wouldn't have done any harm, but
it's not necessary. It is considered. It's a fair copy.
This is not your first grasping as a solution. It's a
is a a thoughtful description of what the answer is.
I I had a little comments to this. You know why such and such
is a good thing to see in the answer. I encourage you to look
through that. That's what the sort of thing that it is good to
see in assessment. If it's not, if it's a mess, then there will
marks off
because this is part. This is not just an assessment. This is
part of the trading for scientific communication,
writing papers. If you ever you end up in research, writing
academic people to put in by your colleagues.
Papers are there to communicate. They are there to show how an
idea goes from the basic idea of what the what. The point of this
is what the idea was, how it works out, what we discovered.
You know there's a structure, academic papers and you're
learning that in for example, your lab report. They are also
an exercise in scientific communication. And all this is
not a paper. It's the same rules apply. It's what I'm
communicating. Who am I communicating to and how do I
make it clear what I what? I think
Remember that the point of initial assessment on an exam is
to show that you are smart and you have a clue one way or
another. If you can communicate that, then then marks will be
found. If something's ambiguous in the question, then see. This
seems ambiguous to me because A or B and that's why you have a
clue. OK, so if some were ambiguous, you pick one of the
two and answer that question and that's fine.
They're not supposed to be ambiguous. But sometimes I think
it's to get you through that way. So study that
and remember the point is communication and and I think I
think I mentioned at the end of the
but her posting was
the main set to be in
writing an answer like this. I see one of your colleagues in
the class said, I have no idea what's going on with this
question. I'm I'm,
I don't know which to start.
What would you write to them? Someone who'd been to the
lectures, bit confused. How would you make clear to them how
you've understood it? They would help them. So that's the that's
the level you're for. You're not really talking to me, You're
talking to one of your colleagues and that's the amount
of detail you need to include and exclude. You don't have to
see what relativity is. Your colleagues know that you don't
have to, but it's not. It's not telegraphic.
I could go on and on about that because
could be resistant to writing things down. I think that in in
schools, people are encouraged filling the boxes.
Would that be right? Yeah, bad habit
to do that. Teachers should be banned for telling people to do
that. I mean, maybe it works for for schools have assessments,
but it doesn't work for higher education assessments.
Next thing I want to mention is
the padlet.
Now there is a link to
to this on the
on the middle
is a great this is a useful place, which I think I didn't I
I didn't emphasise the very first lecture because it didn't
seem to work in Chrome on this, on this machine, whatever you
could. I was a human.
Anyway, that The thing is, there is a very useful resource, a
very good place to ask.
Ask questions. I've created it. A couple of folk already mailed
me with questions, with good questions, excellent questions,
and I've created it with a couple of those questions and
the answers I gave. There's section for each of the
chapters, just double click somewhere in the right sort of
place, type in your question, and I don't get alerted from
something appears there, but I do try and remember to look
there from time to time. So it's not the fastest way of anything,
but has the advantage that everyone else sees. Oh, that's
good. I wish I'd asked that question.
Or I'm glad someone else asked that quest.
You can do a service to your colleagues by asking clever
questions. No, no, no, no. Not by asking clever questions. By
asking a question that is a problem for you
because it's probably a problem for someone else as well,
I could talk on, I could blither on this in moderation
definitely. I will now close down that window. Just
illustrate, there are
more chapters to come
and I will close down that
and go to here and click window.
You.
Yeah. OK,
where we got to last time?
What we got last time was talking about the
that, the, the range transformation and a little more
beyond that
and it's taken us 5 chapters to get to this.
Now you will remember that you did a bit of special activity in
Physics one last year.
She doesn't. Vague, sort of slightly nervous noddings. There
you did, I think you did in two lectures.
You can get to this really promptly from a standing start
and you can you can see and the twins paradox exists, blah blah
blah. But I don't think you can actually understand much in just
two lectures. So what we've done in this, what I've done in in
this route I've taken you to get to
the Reformation is really roundabout route, which goes via
rethink about the axioms, goes to rethinking about lengths and
times,
goes to thinking about Mikulski diagrams,
goes to thinking about space-time and geometry. And
then
this pops out in a fairly natural way. I hope. I mean,
what would, the last time I hope made this make give this a
certain inevitability?
So that's why we're only getting to this now.
But the transformations are, in a sense, the core tool,
oh oh oh of of relativity. That expresses the whole, the
relationship between a moving and a stationary frame.
And the what we're going to talk about this time is two or three
of the peculiar consequences of of this
and we're going to go on to next time is how do we in the context
of relativity topic kinematics, how do we describe motion
And after that dynamics, how do we explain motion in relativity
context because at that point we can talk about those things in a
sensible way with this mathematical tool.
So this is not dilly dally. We talked about the velocity
transformation,
think Minkowski in 1908. So this was only three years after I
said thank you 5 paper. Very quickly this became ohh. That's
obviously the way of doing things.
Um, I talked briefly about I I didn't work through this example
because it's quite long.
We should have. But I I encourage you, I'll point you
towards the section of the notes to to really think through that.
It's a useful what example to go through step by step and make
things so the next with the watch talk about is kind of the
paradoxes or special relativity. Now the word paradox means a
couple of different things in different contexts,
say contradictory things.
But what I am taking paradox to mean that what it's useful sense
of of that is a scenario which appears to be wrong, appears to
be self contradictory but has not been working out why we're
the contract. By looking closely where the contraction appears to
be working, be precise for the contract appears to be, and
discovering why there isn't in fact a contradiction, you end up
with being propelled into thinking about the the structure
of what you're learning about. Paradox is in the sense in the
presence I'm referring to them are things which sound wrong but
aren't
OK, and the best known of them is called the twins paradox.
You probably heard about this from you in popular coach
relativity or on TV programmes or whatever. It's simple but it
has caused more not a noisier and more fundamental
disagreements about relativity over the last 100 years, but
people who sometimes ought to know better.
So first a non relativistic version.
But if she is
leaving, Troy battles, guard was to go home, get a cup of tea,
heads off toward Ithaca across the Mediterranean, right in a
straight line, because he has a clue.
He gets from trying to Ithaca.
Problem solved. That's what actually happens. He takes a
detour
and goes ticket dog leg route from Brightcove here. Yes, I
see. Now which takes longer? Which of these roots is the
longer, the longer route,
the dog? The dog later, Yes. So very obviously there's nothing
complicated here going going that the indirect route is
longer
if you if you don't on the on the same with Misery and touring
this Taffrail log behind him, this little propeller, it will
turn more times. If he goes the indirect route, then you go the
direct route.
No problem
right now as imagine
a species version of that we're at this time. Odyssey stayed at
home and Penelope heads off to a star which is 25 light years
away at half speed, right. Turns around, comes back,
ends up back at Earth.
OK, that's our dog legroom.
No
penalty. Is travelling at hospital late
and that's a gamma of I think 1.15, so you know, a bit more
than one.
So
Oh yes, yes, watching this or people or people in Odysseus
frame watching this will see Penelope's clocks run slow.
Let's take more bass going out and coming back. So in both
cases Penelope clocks are running slow and and so Penelope
is younger
when they're just use where did you get the twins? That's what
they call the twin paradox is younger than Odysseus. When she
gets back back home to earth
and V equals half Gamma is 1.15. She is 87 years old when
Odysseus has aged 100 years because it takes 100 years going
hospital like 25 years out, 25 years back, whole thing 50 years
there, 50 years back, 100 years. So that 100 years is why that's
that's why that's 100 years.
OK, no problem. That's nice and straightforward. Now you
understand about relativity and time relation, that's not a
problem,
but then some breaks, Park says. Ohh, but from from Penelope's
point of view,
it's audacity is moving.
He's moving half of late in in in between, which is true.
So this is a clock will be running slow
on the way out and the same is true on the way back
through this year. It's got running slow so but then finally
gets back home. It'll be he'll be 100 years old and it'll be or
some some age. And it's
Odysseus who were younger than her.
And that's a problem because while it's while two people in
different frames can measure the length of a of a clock of of of
of a of a rod,
and they mutually measure each other's roads to be shorter,
like let's length contracted. And we discussed in chapter
three, I think it was whenever it was
that that that's not a problem big partly because you're
talking about extended things. And in that same scenario we
discovered that the two observers on the trains could
look at each other's watches and discover and work out the time
we've been deleted. The time was moving slower symmetrically in
the two things. That doesn't isn't a problem
here.
We're looking at two people with two clocks arriving at the same
time, and that, and those two clocks they arrive at the same
point at the same time. They can't both be both.
What? What one can't be behind
and ahead of the of the other that that can happen
because you're talking about two things are happening to
temptation happen at the same place at the same time
because we are talking about the time dilation in the trades
moving past each other scenario.
There,
the observations being made are are, if you go through it
carefully, be observations of different clocks
at
at different times.
So you're making a statement about the passage of time in the
other frame, as opposed to making a statement with passing
time on a particular clock. OK,
that's a point. That's the point which is important, but which
bears a second thought. So, so we through the notes, read
through notes on this.
So what's happening here that this seems to be just wrong?
That there seems to be a paradox, I think, which can't be
true here.
And this has caused us see this has caused people who were
professionally engaged in thinking about relativity to say
overall, who must be wrong, Oh my God, what's happening here.
That there is a deep problem. It's not, because
the thing that we haven't
stressed is that
Odysseus Penelope are not vague about who's been out and who's
turned back there. The situation is not symmetric because only
one of them has turned around
at the remote star.
In order to go and come back,
Penelope had to go to half up to hospital right? And then slow
down and then and then turn round. So she is in no doubt
that her who is slow would slow down and and and and and and and
and and and and turned round.
And that in a way
breaks the resolve. The paradox the the the paradox of it comes
because we think that the two observers are symmetric and so
and so and so the whole time relation thing has to happen in
the same way for both of them. It doesn't, because only one of
them
to actually turns around. There's no. There's no ambiguity
there.
And
another way you can think of this.
Well, so, so, so the the and the key point here is that there is
no inertial frame that Penelope stays in throughout the entire
adventure.
Odysseus is standing still in his inertial frame and stays in
that inertial frame the entire time. But now P is an inertial
frame on the outward journey
and in an inertial frame on the inward journey. But they're not
the same inertial frame. And that's where in a sense, the
problem fails to appear.
And you don't even have to worry about the the slowing down and
turning round. Because what we could do
is would you guess to the star she knows there's someone is
heading inward back toward earth at the same speed you
conveniently And she and she says, OK, here's my log book
could you take that back to Earth or this is the time
showing my clock You have to send a radio wave or something
to see this time a mic lock. Can you reset your clock to this and
keep track of how much time there?
So the fact that there are three inertial frames happening here
is for Brexit, Brexit. There's symmetry. Another way we can see
that
is by
if we're looking at the relevant Minkowski diagrams.
So
I'm not sure what the.
I'm not sure which one of these is the one that is
right. I think that's the one that appears on
what do you call it E360 rather than this one. So not on that
basis, that's one will be recorded in the OHH, just just
parenthetically. And the other thing I want to mention,
which I think I did mention in posting, is that the
sound recordings of these lectures are available in two
places, 1 via E360 and there is a link from the Moodle page to
that collection and also my own recordings. The other ones are
also available at a podcast linked to there. I think the
the equity 60 ones have video attached to them. You know of
the slides or or or this. I think the my ones are better
audio quality because I don't really know which microphone
that was supposed to be using. So whichever one works for you
is fine. They have the same content. I would be interested
in any problems with either either of them. So I'd be
interested if you which one works, because if the 61 works
fine then I didn't bother making my my own recordings
anyway. So I believe the the the the pad
is
is that vegetable? Yes, yes Good. OK,
so in the course diagram there is.
Odysseus is free
X&T
and Jesus goes from
origin where the the two twins separate.
To the endpoint, just straight that world. His water line is
along there. Nice and simple very simple. Water line stage at
equal 0 as time moves on. So that's the sequence of points
that that creates the world. Like
Penelope heads off
in this sort of direction,
so her
if we draw her.
So that's the world line. Hope not the world line.
And if we if she's stationary in her frame the the prime frame,
then she is moving along the T frame axis of her frame.
Nice and simple. And so that's what the the the the the mycotic
diagram of her frame on.
On the
Ohh, this is free. Looks like
we get to that that that's I think let's call that event one
when she does the turn around.
OK.
And at this point
she entered a different frame,
should change his frame into A-frame which is not moving in
that direction to hospital light, but moving in that
direction hospital light. How do we draw that on the screen
diagram we draw? Our world line looks like this
in the in the in the in the frame. So she's moving at a
constant speed in the negative X direction. As time goes
increases, she's moving the negative X direction. So in that
other frame that's the T double framed
frame and there will be a a
X double primed axis there. But the the right
now
one thing. So I I what I'm getting at here is that
I have sort of resolved the paradox by seeing that she
changes streams and that they're not symmetric. What I'm doing
now is is is trying to give you a bit of a clue to where this
extra time went.
Why is it that there's thirteen years or whatever? It is
certainly missing from Penelope or from only one of the two
other participants.
So at the point where she turns around,
that time there is simultaneous
in Penelope's freedom
with an event which happens
atrocious at event two.
Why have I drawn it at that angle? Because that's parallel
with X prime axis. The X prime axis is the
the collection of events which all happened at T frame equals
0.
OK,
so if I were to draw the
diagram
X primed, T primed.
All of the events which happened along the extreme axis happened
at frame equals 0. All the events which happen along a line
parallel to the extreme axis happen at the same
with the same T frame.
So there
the events which happen
event one and two
because the panels the extreme actually happened the same T
frame, so they are simultaneous in penelope's frame
but not simultaneous in
or disease free.
And then in the new frame the Penelope jumped to
the X prime. X double prime axis
comes along here and through the event 3
which is simultaneous with event one in Penelope's new frame.
So event one,
it's simultaneous with event 280 Penelopes Outward Bound frame
and simultaneous with event three in her inbound frame.
And so that's sort of where some of the times gone missing, if
you're looking for where's the time gone, missing where the 13
years gone, it's sort of there.
If you I'll be thinking about this
is if every birthday
Odysseus sent out a happy birthday message to his his twin
sister.
Then they would these by radio, and these ***** greetings
would be sent out at regular intervals at the funeral, 8:00,
so at 45 degrees,
and would
but the same benefits halfway there. She's only receiving the
***** greetings from the first half
or just 100 years
so. So from her point of view they appear to be slow, they to
be to be to be late
on the way back. However
I you have to sort of think I think this through is not
completely obvious. Audacity is still sending austerity at
regular intervals, but on the way back
to help you seeing them at faster
than than than once a year.
So she sees the busted busted meetings being slower than once
a year on the way out, and see the buffering being faster than
once a year on the way on the way in.
So you're still seeing a hundred of them,
but they don't add up to 100 of her years. The I haven't shown
that that that nod not adding up on the on the diagram, but this
is the way you can think about how the different perceptions of
the passage of time go in addition to frame and penalties.
If that last bit doesn't make a lot of sense, don't worry, I
think I mentioned it in the notes. This is the. The point
here is just that I'm
showing that you that again building up a rather messy,
rather complicated and costly diagram step by step.
I'm sort of want to trying to suggest you where the
with the. With the extra missing years have gone.
I could keep talking about that,
of course, more or less indefinitely. But
other other things that you know this requires you know, going
and thinking about the things that are puzzling right off the
top of your head.
What things you're holding a little bit shell shocked
or or or big size or or or or something are the things that
our seem obviously amiss.
Walk it through again you hate ohh yeah question.
Just wondering from the diagram as when the
like back and forth, are they meant to be a specific angle
relative to there are 45 degrees, OK, but they're all
radio signals so so they tried to speak light which is 1 light
metre per metre. So, yes, they're all 45 degrees.
Yeah.
Yes, that that's not, I think, completely differently. That
would be clearer
question. This double prime of X double prime frame, is that like
that's some that's the fear of the way back. I'll be back,
yeah. And it's the point it starts at is like when they turn
around. That's right. Yes. So that's a good point. Yes. So. So
if the frames if the frame S and frame S primed are in standard
configuration,
frame double frame is not in standard configuration with
others, because why? What? What's? What is it that makes it
not in standard configuration?
Yes,
so the the origins of of of of the of S&S prime coincide
at at the origin. The origin of the of's double prime, which is
is. It is event one which is doesn't coincide with anything.
So you could you could use the range transformation to get from
frame to frame S prime,
but you could not without further algebra,
you the rest transformation to get naively from frame XS primed
to double framed because they're understanding figure.
That's a very good point.
OK.
And the other diagram drawn somewhat
politically,
it's in the note.
The other point is worth mentioning here
is that
the dog leg in the original Mediterranean version, Penelope
Odysseus in this version is clearly takes a longer route
because he goes a dog leg root and as we all know, a straight
line, the shortest distance between two points.
We all know that, don't we?
But this is a good point to
mention
that
in the in the geometry of the
of Minkowski space,
a straight line is the longest distance between two points,
It turns out
and there's an exercise in the. One of the exercises in attached
to this section allows you to work through that and and and
and part reassure yourself that that that that that particular
case. So in the Makovsky space, with the right intuition
it. Well, I don't know whether intuition but at some point it
will become it becomes obvious that a dog like group is going
to be shorter because it's not the straight line.
OK
and and I I emphasise that when I talk about the exercises, I
mean the exercises attached to to to to to these notes which
are in the direction notes folder. The tutorial handbook
has exercised attached relativity. They are good
exercises
going through those, but not tightly keyed particular
sections as my exercises are. So just to avoid ambiguity there.
OK, moving on
the other famous
and
I would draw that. I'll blow that up. The other famous
paradox is the Polebarn paradox, or the ladder in the barn
paradox, or rather the variance of that. And you have a barn
which is 10 metres
in length.
Are you a farmer?
The pool which is 20 metres in length
and the agile young farmer who would run at speed where gamma
is 2.
So the farmer is running through the farm
at whatever that's a .86 of the speed of light. Gamma is 2 and
so the pole is length contracted from 20 metres to 10 metres.
So getting into the, into the, into the barn, the farmer's wife
slammed the door shut and to the poles entirely inside the barn
because length contracted
with athletic and all that. But you're sort of familiar with
that idea from other things about land contracts.
But then you look at it from the point of view of the farmer
running.
In the farmer stream,
the farm is moving at .863 late in the direction, so the pole.
So the barn now has its length contracted from 10 metres to
five metres.
So it's ridiculous to suggest that the pool is going to be
able to get into the barn because it's 20 metres long in
the farmer frame and the barn 25 metres long in in in that frame.
So what's happening? Does the poor get into the barn
or not?
So you see the problem.
What do we how do we draw that
in Minkowski diagram?
Let's have
the
so I'm not
and the world lines of the front and back of the barn
are
nice and simple because the barn isn't moving in the barn stream.
The barn isn't moving in the barnyard
so the the world lines are nice and simple. The the front of the
barn so the the the the pole is going to be going in this
direction in front of the barn.
It lined up along the keyframe access, but we assume that it
could be at X = 0, the back of the barn. Another nice simple
water line.
OK,
now let's have the water lines of the
of of the front and back of the pool in the running through
that.
Let's have the word line of the back of the pole, something like
that.
And that's the back of the pool. And we've chosen our origins so
that the
at time equals 0. The back of the pole is at the front of the
barn.
OK,
so this is back in the pool. This is the
front of the barn,
at the back of the barn,
and we're the world line of the
front
of of the poll. In this case, well, it's going to be at A
and
it's going to be moving at the same speed at the back of the
pool, so it could be at the same angle. So it'll be a line
more like that
and I'm going to sort of
but by thinking through the consequences of this I, I, I, I
know that it we're going to be talking this be length
contracted. So I'm going to draw it at an angle somewhat like
this, like this,
and uh,
no, not not. No, I'm not.
You know that
the way this works out
somewhat like that,
so that's the
front of the pool.
So this is going to be our prime axis.
So it means our X prime axis is at the same angle in the other
direction.
And this is a Makovsky diagram drawn for the right speed. How
do I know that? Because in this diagram
the
the pool fits exactly into
the the barn. So the angles aren't right for for for for
that speed and and and and that gamma. But the the the the the
relationship is right. So this this shows
the case where the
pull it length contracted
to just the right length of the barn.
What does it mean to say the pool is then contracted to the
right side of the barn? What What it means it's when we
measure the length of the pole, it comes out to be the same
length of the bar.
What does when we measure the length of the pool mean? It
means having two observations of the front and the back of the
pole at the same time in the farmyard frame. In other words,
at the same time t = 0 in the farmyard frame, there's an
observation of the back of the pole being at the front of the
barn, and at the same time an observation of the front of the
pole
be at the back of the barn
and they are simultaneous in that frame. Therefore that is
the length of the pole in that frame. So when we talk about
length contraction, what we mean is when you do that sort of
observation, when you measure the length of the moving object
by that means you get
that distance, the separation between those two worlds being
the same with the length of the pool.
But
notice
that in the farmer stream,
who's the farmers is stationary moving along the the T frame
axis. So for the farmer I'm covered for these chalk. For the
farmer,
lines parallel to the X prime axis are at the same T prime
coordinate.
So for the farmer, the event there of the front of the pole
hidden in the back of the barn
happened
had a negative tapering.
So the back of the pole hitting reaching the front of the barn
happened at time T frame equals zero
happened at the origin that we will set this up.
But that event there happened at negative press,
so it's already happened
by the time this the back of the back of the pool reaches the
front of the barn.
So for the so those two events that we the separation which was
our length of the pole in the form in the form of frame,
I don't do each other
and from the point of view of the farmer
because this event happened before. So, so the point where
the farmer very reasonably
from their point of view the back of the port,
the front of the pool rather were beyond
the back at the back of the barn at the time when the back of the
pool
was.
So that that's that that event there is simultaneous in the
farmer's frame with that event there and that event there is
happening beyond the back of the of the barn.
So that makes sense from the farmers point of view
because at the point where the
back with the pool hits the front of the beaches, the front
of the barn,
the front of the of the pool is well beyond the back of the
barn, as it should be of course, because it's longer than the
barn.
So the so the, the the, the the the. Paradox disappears when you
think carefully about what kinds of simultaneous.
And when you
are talking about the length of something, you are implicitly
talking about simultaneous events, because that's what we
mean. When we talk about the length of something and we talk
about simultaneous events then that's A-frame dependent thing.
So if ever you are looking at our a description of relative
doesn't make sense because
online or whatever
what you look for to see why someone has confused themselves
is we have the implicitly thought, so we're the implicitly
thought. Something, I thought Simpson 80 doesn't actually
apply
civil society. That is where it all breaks down in in the
and we can draw that
paper. Actually
yeah
that's that threat thing just drew and that's the
the same the same scenario drawn in the
farmers stream where
going to there and the event of the is very clear that the event
of the front of the pole hitting the back of the barn happens at
keyframe negative it's already happened but the same the back
the back of the pool hits the front of the.
Another way you can think about this
is that if you were to try that, if if there's bound didn't have
a
had a, you would vote against a a mountain or something. So
there's a solid back wall. So the whole thing is just 10, you
know, 10 metres, then rock,
then what would happen? Then
the poor can go beyond the end at the end of the the the the
the end of the barn.
In that case the front of the pole would hit the back of the
barn and would stop. Say it's an immovable mount or some,
but the back of the pool doesn't know that's happened yet because
the information
that the front of the pool has stopped
has to get to the back of the pool
by a shockwave.
Which can travel faster? The speed of light.
You work this all out the information. Even if the the
shockwave were to travel through all the atoms in the in the pool
at the speed of light, it couldn't get to the back of the
poll before the
back of the of the of the pole got into the barn.
So that's two different explanations of why this all
makes sense.
And that's also an important point, because in the two
different frames,
the two sets of observers have different physical explanations
of what's going on in terms of what the sequence of events is,
what the physical things are happening are. But they have to
come to the same overall conclusions in terms of things,
of what things happen at the same place at the same time.
With that introduction from me, it's probably a good idea, as I
say, to go through that step by step in the notes. Just let
Reese resettle.
We've we've ploughed the ground
again. Other puzzles that are immediate there.
The other clever, you know, insightful, thoughtful, or just
frankly baffles questions.
All good.
OK,
that there is a lot of stick in there.
The last thing I would mention I'm not going to go through in
much detail is Belle's Spaceship Paradox.
And John Bell was a
you looked at sound for most of his working life. He was very
thoughtful about quantum mechanics and other other things
that you know of varieties. And the let's talk about John Bell
this, he raised this this puddle in the CERN canteen one time
and and said ohh, what would happen? And there's a big
argument and the majority of the question go whatever be the
wrong answer. So this confused them.
They're only particle physicists, but yeah,
two spaceships
to to to rocket launchpads A kilometre apart.
The spaceships take off and head off along the X axis with the
same
navigation programme
for the accelerate in the same way.
The engines are designed to prove the right mouth thrust and
they celebrate in the same in the same way, so they follow
the same trajectory, just displaced A kilometre
don't range,
so the speed up after a rough for short time, roughly after
after quite a short time the good it doesn't take a lot of
time going at sea accelerating about G to get up to fraction of
the speed of light. It remarkably short time it takes
to work it out, but
to the moving at Russell's speed at some point,
so that's fine.
Before they take off, we attach a bit of string to the two of
them.
One kilometre long piece of string,
the kickoff and the the move along move rustic speed. So
after after a point, this string is moving at a realistic speed,
so at length contracts,
so it can no longer stretch between the two rockets.
So it's not
fairly obvious,
but
in the rocket stream
they're stationary
and the length of string is kilometre long and they're
kilometre apart, so the
there's no problem
so the string wouldn't snap. But there's another case where you
could have the same answer in both frames. Either string snaps
and it destroyed and the bits of string of fragments of string
flying around or it doesn't snap. So which is it?
And as I say this caused an argument in the quarantine and
some people go the wrong conclusion
and we
the the place where I have up where I went wrong in telling
you that story
the place where
you should have got ah you know you've jumped a step there
the police were I smuggled in something when I said
in the rocket stream
in seeing in the rocket stream there's still a kilometre apart
assumes that there is a rocket stream. But is there.
This is the main coffee diagram of of the two trajectories of of
the new rockets R1 and R2
and P1
is
this one year anniversary party
of the people in the rocket the the one year out from earth and
they have a a celebration
and the
but in their frame at that point
they are moving in in that direction with tangent to the
the the water lane. So in their frame and that frame also
instant configuration with respect to,
that's their cheaper time and that's their extreme axis.
But
look at what they're telling us. It's telling us that the event
which is simultaneous in their frame
with the one your party P1 is not P2
but P3.
In other words, it's an event which happens later on,
after the one year party in the other rockets frame,
at which point the rocket will have, you know, moved, moved
further,
so
P1 and P2.
Those two events are simultaneous in the
launchpad stream
and so because of these trajectories are the same, they
will be a kilometre apart.
But the
the
events P1 and P3 are further apart than that,
more than a kilometre apart to the street and the string if
it's to be simultaneously at this at this
when your party and it and it P3 is having to stretch more than a
kilometre. So it breaks.
And I give a couple of other things explanation of how you
can think about this in the notes. But the point is there
isn't A-frame
where P1 and P2 which are simultaneous in the Earth frame
where they are simultaneous in the rocket stream.
So there isn't our frame,
which is the rocket string.
There are those all the way along that that that water lane.
There's a sequence of of frames
which, because the Rockets are accelerating, are not all In
Sync configuration with respect to each other. And there's an
infinite number of frames there you're talking about. And so
there isn't one where the
rocket,
which is the rocket frame, where the two vans are still 1
kilometre apart,
that also needs going through, again settling down in your in
your head. And I think that's one of the cases where the
utility Makovsky diagram in getting things straight in your
head on paper is, is cleanest because this is a really
perplexing thing. And of course, time lets you write, draw it,
draw it down.
That's the end of Chapter 5. We'll go on to Chapter 6
kinematics next time, which I think is tomorrow. I will get
the notes up this afternoon and I will see you then.