This is lecture 5
and that the microphones are not place This lecture 5 we are on
schedule. This is good. So I aim to get through the rest of
Chapter 4 today, and we'll move on to Chapter 5, which is about
the Lorentz transformation
next time. And that is the in a sense the bit for quite a lot of
things come together, if you like. So the expressions for
length contraction and time dilation that we have seen
already
are tied up with the ball as it were in the transformation
but the before. I want to get onto that there's some
motor market. I want to make a big coffee diagrams
and about the sort of geometry first approach that I've taken
to special relativity in this course,
which is not the only way one could do it, but I think of
course a good way.
I talked about the Mickey diagram last time,
which which is
just
I say a slightly eccentric way of of plotting events. The the
one bit of eccentricity is the X&T actually being swapped
based on how you go in contrast to what you are more familiar
with. But that's not a big deal. It's just that the point
remains. Each event
has our
locatable on the mycotic diagram. And as I hope I showed
you last time talking about these the late flashes going
back and forth
from our moving train carriage,
you can think through
the process of what's happening on Minkowski diagram in a
useful, a constructive, very constructive way.
And I came. We produced that diagram in the frame of the
moving carriage, with the flashes moving forward and
backwards being reflected and coming back again,
and when we reasoned through
what this looked like in the
from from the station in the station frame.
Watching this and go past, we produced a different coffee
diagram
of the same events
could. Remember, events aren't in A-frame, they are in the
world if you like, and they are plotted
in A-frame
where it was completely obvious
that event in this frame
event. Well, whereas in the previous frame events two and
three were manifested simultaneous. Of course,
because of the same T prime coordinate in this frame, event
three was observed to happen before event two.
OK. So the the observers in the moving frame
would have, you know, watched the light being reflected from
the back and front of the carriage and noted down
different times in that sense observed.
So I'm going to give another example of A
and I think I I I think I did finish off with this quick
question where I asked you to
think through this, so in frame. So just what I've just said,
Event 3 happens unequivocally before event 2
because it had the allure
lower T coordinate even though they have the same T prime.
And I showed that that that diagram similarly I think
OK, let's go on now to look at
a different medical few diagram which I think is also
illustrative and again
plotting if you like our scenario we talked about in
previous weeks. So we're just doing something we've seen in
previous weeks but doing it in a slightly different way. So this
is going back to the specific, the specifics of the
that the the train moving again that that we see enough of the
tree.
So you see that, yes.
So again we're going to,
we're going to draw up medical tree diagram, there's we're
going to draw in the prime frame first
extreme heat frame and again a light flash at the centre.
And we're also going to draw on this the world lines of the
front and back of the tree.
The world lines are the set of events that that happen at the
front and back of the tree,
and the train isn't moving in the training frame.
So the world lane is very straightforward. It's just
that that's the
go to the front
in the back of the tree. So the world is are are not
complicated. OK,
the a set of events which happen at the front of the train will
all happen at the same X prime coordinate.
The
if we then see the
I like flash going forwards,
well I call that event two
and going backwards
event one
and I think complicated. OK, so I I I'm just in a sense
so if this were an exercise I would go I was going through. I
would at this point solemnly write down A-frame S prime. The
frame of the of the carriage frame S is the frame of the of
the of the station platform. Event one is blah, event two is
blah. And then draw them in Costa diagram
and I'll expect to see that sort of systematic approach.
But as I mentioned last time,
the
most of the exercises in
this of course are basically the same. Exercise is, here's the
situation. Turn it into a sort of event. Use the right
transformation. Get yeah, get some other coordinates, The same
event
or all slightly artificial. But the point. The point is about
the transformation and that first step of turning things
into. Because your diagram is the process of thinking through
things
through, the same
set of events in our
in the
platform frame
are slightly different. Here the water lines of the front and the
back are also straight lines, but they are now slanted because
the train is moving.
So that's the
front
and back of the tree.
So I said revenge at flat, flat, flat, flat, flat, flat. Flash at
the front of the train will
appear at successively increasing X coordinates on the
platform,
but the the light flashes
still move it
45 degrees.
The event one is still the event where the
back with the with the rear going light flashes hit the back
mirror. Event two is still where the front going light flash hits
that ohh and and the centre of the
carriage goes goes like that, which means we can also call
that
mark that is the T prime access
and again here recapitulating what we saw two lectures ago. I
think
this is is clear here that event one, the light getting to the
back of the carriage and event two right in the front carriage
are different times. They're no longer simultaneous in this
frame,
but we know that
events one and two
happened at the same time in that frame, so a line drawn
joining them is parallel to the X prime axis.
So similarly
the extreme axis
is that angle there. And again, this, as I've said a couple of
times, I mean of course diagram ends up looking like a a
Duncan spider mess when you when you get to the end of it. The
point is, it's all about going through and building the thing
up.
The things I want to emphasise there,
yeah, that that's,
that's basically it. But the,
the, the,
that's OK. I'm not going to do prattling on about that. The
point is that that's another example of the coffee diagram,
which recapitulates what we saw in previous previous lectures.
Any questions or puzzles about that question?
Prime access?
Yeah, yeah,
that's right. So this is the the first hint that that's very
important point. There's the first hint of the fact that the
the geometry of the of the space is strange and we're going to
come on to talk more about that.
But
you can think of it. I think it's useful to think of that as
a sort of perspective effect.
If you if if I
their pain if I you know at an angle it seems shorter to you if
your point of view because of the perspective you're looking
at it from.
And I think I'm being a bit vague there but woolly. But I
think if you think of that as a perspective effect when you
rotate
sort of in quotes from one frame to another, then the these items
are still have the relationship between them that they have in
this, but they look a bit funny from this in this frame.
But that the you you think it? Well, yeah,
that's an important point and and I think it allows me to see
geometry again.
OK.
And we're going to say a little bit more of that another
question there
you don't, you don't know between between there
and
I
and I think you you you wouldn't because not really meaningful
that angle and you know the angle between
the chief prime, the key and the key frame access
that is we're going to write that angle down. But in terms of
the gradient of that or the T prime
access,
is this the same as the gradient of these world lines here?
And that is just with the gradient of one over the speed.
So so if if it were a
TX
graph then the the the the the
X equals
Vt, the gradient of that line would be would be the speed. In
this case it's a T is equal to 1 over VX line. So the the the the
gradient of that is
it is 1 / 1 over the speed, which tells you that when the
speed is 1 speed of light it's going to be 45 degrees,
and with the speed is less than one,
then one of the is bigger than one so that the gradient is is
more nearly vertical. But there aren't. But
there aren't really any questions to which the answer is
the angle between those axes.
OK,
you have a good day putting it. You know it's irrelevant, just
that there aren't questions to which that's a useful thing to
work out.
There are analogues
which we perhaps I can talk about later, we will talk later.
But it's not that angle
is a good questions. I like questions because it they they
reassure me that I'm going to ruffle the right speed.
OK. And and the in the notes there's another example of I
mean of course you diagram talking about the the the the
the light of our lighthouse tracking across a distant
coastline which is also allows me to some other remarks in the
notes more words just to back up what I'm saying in the
in the lecture and this is a good a good which is a good
prompt to say what I said in the first lecture
that
as far as I'm concerned the lectures of the main event
and the notes are extra words that might be useful to help you
process what wasn't in in in the lectures.
And
again to reiterate, the
aimed at the point and the objectives are the party tricks,
the the, the, the things that you will that that when you do
perform those in an exam reassure
be us that you have achieved the aims so that that's the
relationship between those, those, those various ideas.
So if you if, the if, if, if the lectures make perfect sense to
you in isolation and you can do all the exercises, you might
never read the notes. That's fine,
but do the. But the the, the, the, the measure of how much you
understand things is how, how quickly, how easily, how
comprehensively you do the exercises because the exercises
are very important, because you will actually understand
anything when you you've done it yourself.
I got I got an answer.
OK.
And
this is the bit we wanted to talk about what you said, what
that remark there,
we'll build that up.
OK. There's a diagram you have seen before I'm sure
is just that diagram of how you rotate axes and you've seen that
I'm sure.
And then we're going to see if they haven't seen that, that
that is a stunning innovation to them. OK you may not seen in
exactly those terms, but you are familiar with this because it's
just the the the setup of how you change coordinates through
rotation.
So that point P had coordinates X&Y
and in the different in the frame which is rotated
by an angle
offer theatre. Rather that same point has different coordinates
X prime and Y prime. They both have the same point, just in
different frames,
and the relation between them is.
Is that
again really well known?
The ex prime coordinate is X Cos Theta plus Y sin Theta. The Y
prime coordinate is minus X sin Theta plus Y Cos Theta.
OK,
with that, let me again
not banging.
I think this. I don't think it's hitting anything.
Umm,
who pretended to you
so the, the, the but the important points to make here
are but
is that me
not banging?
Let's not worry about it isn't happening. The the the important
point that about that transformation is that this
quantity expert plus y ^2
an X prime squared plus Y prime squared is the same in both
coordinate systems.
Now that's just a statement of Pythagoras theorem,
and what I'm seeing here is the Pythagorean theorem isn't
frame dependent.
Now again, this is a big surprise.
And
is that noise coming from this microphone? Yeah, yeah.
Well,
I'm not sure what where it's coming.
Ah, right. Broken. Broken.
OK, there's a there's a the cable slightly broken I think,
or something
which I probably need to report. Ohh, well, I'm talking about
misbehaving.
Yeah,
technology. Just parenthetically
the
echo. These sixty I have I think linked with the 360 recordings
to the Moodle.
Had anyone found those and looked at those?
Because I think I've done it right,
but I really have little confidence that I have. So if if
because I'm not completely sure what it is I'm supposed to do to
make that to me that link. So if anyone looks at that and they
are not what you expect from previous experience in the year
in other years with Equity 60, then please do someone tell me
because
yeah, it's it's not an entirely friendly system and I'm not sure
anything right.
OK) The point here is that I'm saying you're familiar with, but
I'm I'm making a big fuss about it because I'm seeing something
you already know but in a different language.
OK,
I could have put that better, but I hope the point is made.
The what
of thought? Talking about
maybe
now the next section, what 4.4 point 4I.
It is not, I think
tragically well well explained in in in the notes. I keep
changing my mind about how best to explain this so I probably
could rewrite that again. So
which are on a bit, I think in the notes. I'll try not to in
verbally
in in this rotation.
Let's go back to here, this rotation.
We
have a. Let's draw this rotation *** prime YY
prime.
Imagine two
and
points in the plane.
If we
could, we do it X
Ohh. Thank you.
The two points in the play,
we can talk about the separation between them.
That's exactly why. And by that's theorem, delta X ^2 plus
Delta y ^2 equals. We'll call it
d ^2.
In the other frame,
we can instead talk about
That's something there that somebody does egg prime,
that's gotta Y prime, and we know that delta X prime squared
plus delta Y prime squared is also going to be d ^2.
In other words, this
calculated here Pythagoras theorem is an invariant of the
transformation.
When I change coordinates that that that distance, we're going
to call this, the distance between those two points is the
same.
And you could from that you could assert that, and you could
deduce all of Euclidean physics, all of your Euclidean geometry,
from that assertion. Because that that assertion that this
distance function is extremely independent
is in a sense generates Euclidean geometry. Is that
crucial? Is that fundamental?
So is there any analogue of that
for going from one
XT frame to another?
And the answer which the that section 4.4 attempts to justify
it in a rather in a way which I still think is somewhat
inadequate is that
if we talk about the
let's let's go back to here and for example talk about these as
the
the separation between two events
to the extreme
and
that at prime equals 0
delta X
there's a T so this separation between two events in and and
delta T
if the relationship between if there are a function involving
analogous to this
which we can write down which has some sort of properties, the
answer is yes. But it doesn't involve
plus thing
of the -
and
we can write we can write that
I sort of distance function.
For reasons which are not going to become apparent, we'll leave
it rather than rather than.
And just by the way, a bit of notational clarification. When I
write down delta X ^2, what I mean is delta X ^2, not delta X
^2. That's that's the delta X opt me bracket squared as
opposed to the change in X ^2. And similarly here this is just
bracket squared X bracket squared rather than The change
in XI
will write but I won't be consistent about it doesn't
squared there, but I might something just right. Yes,
square, but it doesn't make any difference
delta
or and so this is the
and in for example this we're interested here in the set. In
how far apart are these two events? One and two and in. In
this case they are. There's no time difference between them,
they're time coordinate with the same. In this case, there's
that sort of time distance between the coordinates. So here
these two events are that are separated by that by delta X in
space and delta T in time.
And this, it turns out, is free and independent.
I don't know about you. I hope illustrate
and that when you're saying it's where the geometry
is different.
So when we see here in in the in the struggle on the right
the
the fact that the that that in
in the rotation diagram
the yeah so roughly the angles look reasonable and Steve
reasonable and this one
the angles don't look reasonable
that's because of that many say is because the the geometry of
that is different and that's a rather that you hand waving
thing to see but that is is a bit where I I brush aside you're
concerned about the angles looking a bit funny and worrying
about the angle between things.
So let's illustrate that if we go back to the light clock,
the like look.
So that this is, this is the
flash and the
like me coming back.
I think there's two lectures ago. I think
you you might recall that the
between those two events and that this is in the
the pain free
separation in space between those two events
is 0 happen at the same place
and the separation in time
is
is is 2L. It's twice the distance of,
twice the distance from there to there. So the time separation is
just that same thing in units of of light metres in time units of
light metres. So if we're measuring this in seconds,
they've been extra factor of C in there, just as we type
so that. So is an extra factor of C but is equal to 1.
Is anyone uncomfortable with that statement?
And in the other case
where we saw
the late flash reflect and come back,
event one was here event two
was some distance down the station that down the track
we discovered that the the the separation
just X between these two events.
If this happens at time T2
position X2 and this happens at 00, we can just choose our axis.
So that's true, does X is going to be
two which is
the T2 where via the speed of the tree.
OK, so in time T2 the the the the train has moved Vt during
the in the platform
the separation delta T
is just is is T 2 -, t one but 10 is T it is delta T prime over
gamma. As I said last time, you can look back and and and
reassure you sort of that and let's put in some numbers to
that
actual numbers in whatever he could have the very thought. So
let's suppose that V is equal to
3/5. So that's three fifths speed, right?
Thank you
to that.
Yeah, sure. Then that implies that gamma of
the 5th is at one 1 -, 3/5 ^2. What one square root is one
over? That is,
people before
then if
yeah and and and we'll see what it would say that all is going
to be
1 metre.
Then in this prime frame.
This is not totally organised but just s ^2. You could do
delta T prime squared minus delta X prime squared which is
2L squared.
My ex prime squared equal 0.
The duty IS2LL1 metre X is 0, so that it's not an area.
We can call this an area, but we we can put the numbers into this
expression here and get a number
in that I've calculated in that moving frame. If we do the same
thing in this from the platform frame to the same between the
same two events,
then we discover we get.
Let's just make sure this
delta squared equals Delta
t ^2 minus delta X ^2. Delta t ^2 we're saying is Delta
T primed
over Gamma. Delta T is one. One over gamma is.
Let's just be consistent here
and.
And
is over
gamma
squared minus
E ^2
2.
It's weird.
I'm having difficulty working out how to determine what I've
written here into something that's readable. Hang on.
Right. Let's let's
go back, go back, go back.
Let's remember that little more readable run cramming into in,
in into the. The thing at the bottom end is 1 metre
dot T
The prime is 2L equals 2,
V is equal to 3/5. Gamma is equal to
504,
so our
delta T
in the platform frame
go to T prime over. Gamma
is
going to be 2 / 4 fifths two or five four over,
which is one point.
You're not
and
that's 8 / 5. One point
this does work out. I've just
and
thank you
is
I'll be around the other T primed is a good delta T over
gamma.
So delta T here you could do gamma times
primed which is
like we were 4 * 2 which is equal to five half which is 2.5.
OK.
And our
or just X
prime ex brother is V dot AT
which is the fifths of 2.5 metres which is 1.5
metres.
So which means our delta squared delta t ^2 minus delta X ^2 is
equal to 2.5 metres
squared -1.5,
which is squared, which is 4.
It's a square if you look at it on your calculator.
I'm sorry, I've got I I confused myself by getting that upside
down.
You might want to step through the the, the, the, the notes
later. The point is that when I worked out what delta S ^2 was
in the
prime frame,
I got the same
the same figure.
OK, that's that was a long and I apologise. Slightly confusing
explanation of a very simple calculation.
OK.
Going back to the, the, the, the, the, the what? What we
calculated
2 years ago,
yes,
going back to going back to numbers, to the expressions we
we we we worked out in the two electrical and this time putting
in numbers we find that we can calculate the separation in time
and to expression and space
between the these two events one,
one and two
in this frame and in this frame. And we discover that this
quantity,
the interval so-called, is the same in both in both things. In
other words, that is an invariant of the transformation
in the same way that
Pythagoras equation, Pythagoras formula is an invariant of the
transformation
in rotation.
And that's telling us that the geometry of this Minkowski
space, of this Minkowski space
is different from Euclidean geometry.
The point of all this
is,
I mean, I am an actual ruler,
the line there. So I mean I I mentioned in passing last year,
yesterday
if I
would that ruler up that's 15 centimetres long,
hold it up in front of you and you and you you can look at it
and and you know you see well do the angle calculations and you
see that that's 15 centimetres long.
If I turn it like that,
then from your point of view it's shorter,
it's got shorter the the the the angular size is is less.
But you're not puzzled by this where, where, why The rules are
no shorter because you know that
the the way
extent of this ruler is now non zero. And if you take that
horizontal or perpendicular, extend
that
longitudinal extent, square them and add them, you'll get 15
centimetres. And that doesn't surprise you. OK, because that's
that that that that the invariant of particular theorem
is part of your intuition about you about about Euclidean
physics.
Because in geometry
geometry.
And what I want to suggest is that the same thing is happening
in this diagram here
you're having a. This is a. This is just an exact. It's. It's a
very close analogy
here. You're switching between frames, but Pythagoras theorem
reassures you that there is an invariant. Here
you're switching between frames, but
this is an invariant,
OK, and the difference is just a matter of which geometry you
choose.
And this is what I what what I mean when I say that I that this
is a a geometry first account with special activity. Partly
because it's quite elegant,
but also a very significant Part of why I'm introducing it this
way is because this gives a very natural
path to talk about general activity and to understanding
general activity. Because the geometry I'm talking about here
is is is is the the, the the the simple version of the geometry
of of general relativity.
General the geometry of general relativity
hard to see is this basic idea with our distance function,
which had a - in a crucial place.
But
with the restriction, the specialness of talking a little
bit, constant speed has been dropped
and that's why I'm making a fuss about geometry just now.
Question about that.
OK. So that would just be
taking obtained to make a fuss rather than go through any
specific calculations Here
there is a section 4.6. If you've still got a bit puzzled
at at at McAfee diagrams then but perhaps have a look through
that because that goes through goes through how you build up
Minkowski diagram another it's yet another report. It's a
worked example of how you build up because your diagram from
from an argument because diagrams at the end again looked
like a a complete mess. But once you see it through them, sort of
drawing them out in parallel, they should
make some sort of sense
I hope.
But if you work through the exercises then you should get
all the all the practise you need.
Key points there.
And that's the point of the rule.
That's the thing.
Well Sir,
so the what this means that we we now start to talk about the
overall
overall shape of things on the in the Miskovsky plane, in
Miskovsky diagram
in Minkowski space. Because when I talk about Nikki diagram, I
mean this X&T diagram when it when I talk about Minkowski
space, I am talking about a geometrical space, just like
Euclidean space, but with a different rule, different rules
for the invariant, different rules for how you go from one
frame to another.
But it's it's just as much a question of geometry, the
geometry you learned about in in school.
So we can therefore plot out a variety of points on the
Mikowski diagram and ask
and and talk about the interval between them and. And by the
way, I'm going to talk about the interval that when they talk
about the interval I mean the squared interval, I mean squared
interval when I'd be saying interval but it and we're always
talking about this thing squared there isn't. Although the
Patagonian distance function d ^2 = X ^2 + y ^2, we can take
the square root of that and find and talk about that as a
distance. You don't have to take square root of it to have
meaningful thing and we don't take the square root of this of
this interval. We just talked about
this weird thing.
It's great aspect. So this means that there's a couple of a
variety of points,
and if you remember, the interval is Delta t ^2.
My name is Doctor Expert.
If you look at at event five,
they are the separation of event five from the origin
is more in time than it is in In space just is bigger than delta
X
just t ^2 X ^2. Just t ^2 -, X ^2 is positive.
And that's true of everything above the
these two diagonal lines.
The everything in that quadrant is
had had an interval which is just s ^2 greater than 0.
Event three
they are the separation from the origin in time is less than the
separation of the from the origin in space DD T is less
than delta X,
so the interval
in this area and in this area
is negative.
Similarly for event one that's also
in,
it has a delta T which is bigger than delta X ^2,
so it it. It also has positive intervals, so this is also
with that
and
venture event 4 Justine dot X are
both are the same. With the interval is 0, so the distance
between the origin and event four and of origin and event two
is 0
and that's weird so that this is because of the - in that
distance funk
in Euclidean geometry for Pythagoras theorem because
there's a + in it
if two events. If two things two points are zero distance apart,
then they are the same point,
and that's just not true in this geometry.
Now what we can also do is we can make a version of that
diagram
and we'll we'll draw
T
and
will draw in the transformed X prime
straight line
T prime and we'll look at what
that that.
Let's look at event.
Ohh
seems to reset itself
Event 5
an event
three.
Now in this
we saw that event five was had had DD T bigger than delta X. So
for red 5
^2, event 5 is greater than 0 and prevent 3
^2 with less than 0.
But if we now look at the
dot X primed
and the
delta T primed coordinates of event five,
we see that again the delta T is bigger or still delta T is
bigger than delta X. So the interval in this case will be
also be positive, which we knew already because we we we we know
that the interval between these events is going to be
the seam in the different coordinates.
So event five will still have
and in the other coordinates will still be
positive.
This is still above. It's still above the X frame axis,
but
and
for event three
the time coordinate
ohh then 33 in those in that prime coordinate in the that
prime frame
is no, is no below the extreme axis.
So event three has
well I've got I've got the delta. Event three has will have
a a negative time coordinate.
In other words, what that is seeing is that
in the XT frame
you say the stationary frame whatever
event 5 happens in the future and event 3 happens in the
future.
But in the frame frame,
the time coordinate of of event 5 is still going to be positive
that there's no way you can see that transformation happening
where the where the X prime axis ends up passing by
event event 5. So event five will always have a positive time
coordina.
But
in the case of the transmission down there, where the ex prime
has squashed in this way to go past event three, event three
now has a -10
coordinate. In other words, in that frame, event three happened
before the event at the origin.
In the untransformed frame, event three happened later. In
the transformed frame it happened before.
True, there isn't that the the the question of which came first
is
frame dependent for event 3,
but it is unequivocal for event Phi
and that is to see.
That's why I'll leave one of those.
That's why that top quadrant is labelled the future,
because everything that happens in that top quadrant is
unequivocally in the future, and there isn't A-frame. You can
find where
event 5 happens in the past,
and similarly for this past quadrant.
Events that happened in the past
were always happened in the past. There isn't A-frame. You
can find where that happens with the time coordinate of event
one. The T frame coordinate event one is positive, but for
things in this other space
which called elsewhere,
whether event 3 happens before or after an event of the origin,
it's frame dependent.
And what that also means is
the the cannot be a causal link between anything event that
happens
at the origin
and an event which happens at event 3. Because if it's aware,
or if you thought there might be,
then you when you discovered the event Three could happen in the
past,
in in one frame or other. That would mean that an event was
going from the future, from something that happened later in
time to if something happens
earlier in time which can't happen.
So another way of of of thinking about the various things in this
diagram
is that
event five
you could always get a message a six and a signal to event five
from the origin.
By some
slower than late
means,
you could You could send something
possibly going quite fast. Fast. You could send something the
order to event five. You could send something from event one to
the origin,
but it is not possible to get from the origin to event 3
by anything travelling slower than the speed of light, so they
cannot be causally connected.
Is the important thing here.
So that's the
and and and and things that are on on these diagonals. As you
can see from the from this some of this on the diagonal will
always be on the diagonal
there frame so something so two events that are separated by a
null or yeah so so so so so names
the interval between the origin and event five
option one and the origin
with interval being positive are called time like
because the you can you can you can find a frame in which the
the time that the time axis
join them up.
An interval for an event separation would event three and
the origin
where the interval is negative. It's called space like
because you could find A-frame in which they are both on the
same
X prime axis
and the interval between event two and the origin or the origin
event four is called null or light like
and those are invariants. If true events are are separated by
a time like interval,
a non interval or a space like interval, then they'll be
separated by a time like null or space like interval in any
frame. So those these are other things you can hold on to. Part
of the thing was especial activity, you know the
relativity bit causes people some, some, some, some, some,
some. Some
unease because they always everything's relative what we
hold on to. Oh my God, the world is ending. But
I think Einstein didn't like the fact that relatively became
called relativity,
that that sort of missing the point as far as you were
concerned, because the point of the whole endeavour was to find
absolute. What can you hold on to that is absolutely true
and the separation between two events is frame independent.
It's absolute. The the fact that that an event is a separation of
like like Swift and all like
time like Late Lake or Swiss Lake is an absolute. These are
all the things you can hold on to and these are things you can
build on. And the last remark I'll make about this before
finishing off this chapter is this is an XT diagram.
If you imagined AY coordinate sticking out here and then
rotated that then this
these these these triangles will turn into a cone because someone
called the and. And if you imagine as a ZZ axis pointing in
in in a fourth direction into the same thing that then the
light cone would be a a section of a sphere basically. And that
idea of the light cone,
the the, the, the, the future of the possible future of of things
which will stay in the the future pointing like one is a
thing that will come back to later on.
So that it got to the end on time, I have hastened to add. I
will appease myself of chapter 4. We'll go on to the range
transformation next time, which I think is next is next
Wednesday.