Ohh, this is lecture 12, so we're heading in the home
straight.
A couple of things to mention. There's still a trickle of
things appearing on the paddle. That's good. I think I've
answered everything except the one that appeared yesterday on
the class test. I've been given this dude liberation. I think
I'm going to deem everything up to the end of chapter of Chapter
6
as being in school for the last Test. So that's up to the end of
the kinematics chapter, but not the dynamics chapter. That's not
because the dynamic chapter isn't important, it's extremely
important, but because I think the idea is in the end a little
more time to ferment in your head and the revision time is
excellent. Time for you to go back over that, start again and
let those things settle down. I think it would be. The
interesting question is because there are nice interesting
questions we can ask about the dynamics chapter.
But the interesting question is would I think, be a little bit
too forceful for just a couple of weeks after you've been
introduced the ideas. First of all, so I could only ask boring
questions in Chapter 7 if I deemed that in scope. So up
Chapter 6.
Other questions of that sort or technical can go in the padlet.
That's good. I like questions on the padlet partly because it
means everyone can see them and everything. It can be reassured
that that that that all someone has phrased my question better
than I could have or that's what I was going to ask for something
like that. So questions are all good.
Anything else?
Move that to say it
and do this.
We talked a bit about
the equivalence principle, last name
and how important it is. Remember that I said the way I
introduced this was talking about being in a box far away
from all gravitating sources, ocean space out of that,
and the idea that while you were in the box, just floating there,
Newton's laws work in the sense that things moving in a straight
line carry on moving the straight line until they hit
something, if it goes. May all those things work perfectly well
when you're in a box
far away from every.
I said then that if you put a rocket engine under that box and
accelerate the box gently at to pick a number at random 9.8
metres per second squared,
then what you would feel,
you you you the the box will accelerate toward the occupants
of the box. They would end up being pushed, pressed to to the
what suddenly has become the floor, and being accelerated at
9.8 metres per second squared.
And they wouldn't. They would think I might. Perhaps on Earth
I might have. I fallen asleep and I'm suddenly back on Earth.
And the equivalence principle.
You know, version one
says. Not only that would be tricky, but it would be
impossible to tell the difference because the
equivalence principle says
uniform gravitational fields, such as the uniform
gravitational field we feel standing on the on the ground on
Earth,
are equivalent
to frames the accelerate uniformly relative to inertial
frames. Not that it's hard to tell the difference, but they
are indistinguishable.
OK,
that's good. That's interesting. And that also explains
that
equivalence, that statement, these things are equivalent.
Then explains
why all things fall at the same rate, why there isn't a
difference between the inertial mass and the gravitational mass
that I mentioned briefly last time, only to dismiss them as as
as prompted dismissed them as being unimportant, different
from each.
OK, that's what that equilibrium. What explains why
that happens?
Now we can make another version of that, a stronger version of
that,
and see all local free falling, non rotating laboratories, which
we're going to call local inertial frames are fully
equivalent for the performance of physical experiments.
Now all the words in that are important.
Local
means
that there's a a boundary to to your laboratory.
We're not talking about the whole earth here. We're going to
talk about our our frame, which is
whatever size. Now it might be that size, or it might be 10s of
kilometres in size, but there's a boundary, and there's a
boundary in time as well, so we're not going to observe
forever.
So local means a bounded box. OK,
we're not talking about global features here.
Free falling means specifically that it's moving only under the
influence of gravity.
Now that might include not moving. So being way out in
space, far away from all gravitating matter, it's also
moving under the influence of gravity. You know there's none
to affect it, but it's still that. That's that's the only
thing that's going to to to to matter. If there's nothing else,
there's no, there's no engine on.
It also refers to something which is moving purely under the
influence of gravity on Earth, for example such as that.
So if I'm falling, if I'm jumping, if I'm leaping from
point to point in the gazelle like fashion, then I am moving
under under influence of gravity. And in that while I'm
doing that
I am in free fall
and this version and and I'm not twisting. I'm not doing
gymnastics and twisting at the same time. OK,
because rotation is is detectable you you you can tell
if you're rotating.
Ohh, your your ears can tell if you're rotating.
So
all we're gonna call things which are bounded in that sense
and moving under gravity local and national frames. Local
because the local inertial frames because the inertia that
Newton describes in Newton, Newton Newton's laws still works
OK and they match the inertial frames that we have been talking
about in the context of special relativity.
So just to
reiterate, the thing that's the different about the last five
lectures is that we're no longer talking to the special case of
no gravity, but we are explicitly bringing gravity into
the conversation. And the thing about the the equivalence
principle is that how we bring gravity in
the equivalent. The equivalence principle is the bit where we
make the connection between the physics we understand using
physics and the physics we don't gravitational physics.
Now, Newton had an answer for that, but we're not going to
follow Newton and and his law of universal gravitation. We're
going to use this, this, this principle, this statement about
the universe. To make that jump
through all such LAF are fully equivalent, but the performance
of all physical experiments.
That means you can't tell the difference between an experiment
done in one LF and experiment in another lab.
OK, not it's hard, But you can't. And that all physical
experiments. We're not talking about mechanics here. We're not
talking about juggling here. We're not talking about
electromagnetism. We're not talking about biology here.
We're talking about all things that you could do. There is no
experiment you could do could tell the difference.
OK, in other words this is saying something about the
structure of our universe.
There's a very important and and and the difference between this
and the previous version is that the previous version was the the
stepping stone to this. We're talking about uniform
gravitational fields and that uniform gravitational fields is
a statement about non locality is saying this gravitational
fields are the same everywhere. And the word local is saying
we're not going to talk everywhere. We're just going to
talk about a box,
OK, because global effects are are complicated in Geo.
There's another version of this that we'll come to later on. But
this is a a a key version
and and it's worth thinking about that and and where it lies
in the argument
possible times.
I've got two versions of that.
Strange. Anyway,
next thought experiment.
So you're back out in this
box floating in space,
and you have a late. You're just floating there
because there's no gravity. You're just moving purely and
then some influence of the gravity that isn't there.
You train the laser pointer.
You hold up up up to the up to the wall.
You showing it across the room.
It's horizontal because it's perpendicular wise, yes. But you
you just say it's horizontal,
it's going to hit the opposite wall
at the same height as it started off. If if if it's a right angle
to the wall and all that,
you can set that up so it's that that happens.
So the light crosses the the box and hits the other side at the
same height. Thank you. Nothing complicated with that at all.
And that's exactly as you'd expect.
No,
Let's say that you're doing this in a falling lift shaft.
Now, of course the objection is if you're in a falling lift off,
you have other things to worry about. But you might want to
distract yourself thinking about general activity.
If you do so, you will notice that if you did that in the lift
shaft, then
the question is where would the
laser pointer point on the opposite wall?
And the equivalence principle tells you the answer. It points
to the same place
because if you're falling,
if you're in this box, you're following.
That is a local national freedom if indistinguishable from our
frame, which is otherwise moving entirely under gravity. We have
in space and so the same logic would work.
So the equivalence which tells us this would be true in a
falling frame as well.
It would hit the other side
at the same point, because if it didn't you could tell you were
falling. This is another variant of of of the
the the the the first axiom of special relativity that you
can't tell you move.
I can't tell you falling anyway,
but how would that look from the point of view of someone who's
standing outside this left on in safety on the on the floor as
you see the, the, the, the, this lift accelerate away from them.
How would that look to them?
Well it wouldn't take long for the light to cross the the the
the following lift cabin, but it would take a a non zero amount
of time
and in that non zero entertain the lift was fallen slightly.
So that means that the point at which the light arrives at the
other side of the lift cabin,
it's going to be slightly lower
and the point we started off. So from the point of view of the
person standing inside here watching all this and thinking
should I call 999 is if the other things think about that
the light appears to what does not appears to, it does land
slightly lower than it started off. In other words, the
trajectory that they see for the light is curved. In other words,
they are seeing light bend in the gravitational field
at a direct and immediate consequence of the equivalence
principle.
OK.
And that's surprising.
It's a very direct consequence of the principle, but it's
surprising.
Any questions?
I see a lot of This is good. Smiles. That's good. Alex
smiles.
So
let's not talk about light
Apprentice
gonna take some sort of spring guys some some spring loaded
thing with which fired the ball bearing or something across the
lift shaft.
If it's set up like that
with three light bulbs AB&C
and I point this thing this spring loaded thing across that
in non relativistic spring load of thing across the lift shaft
and
fire the ball beating across, is it gonna touch hit AB or C Who's
A
which say B
who says C
put the hand up yet.
All right, we'll pick one there. I pick pick one at random and
who's The
Who say be
which they see.
Talk to your neighbours.
OK,
I hate to break into animated conversations, but that's right.
Again, should aim
me directly. So aim at B
should aim at sea,
right?
I should aim at B because if I were doing this entirely, you
know we out in space
with nothing around me, then Newton's laws say that something
we just fired goes in a straight line until it hits something
so we're out in space.
I should fire I aimed directly at B because the the the the the
ball bearings are going to go non relativistic speed in a
straight line
and if this is happening in a falling lift in contrast
then
they could responsible tells us the same logic. It's the case,
it's not. It's not special to light
it. It works for for everything.
OK, so I should aim directly opposite again. So saying that
if you are in space without the feeling of gravity, yeah be the
same as three fold, yes, there would not be the same as
standing on the ground, not on the ground, no. Yes. So if if
this were not falling but we're stationary on the on the ground
and you fired the thing across then yes from that in that frame
the the ball bearing would would would drop
and So what you would expect is is is
and the the the the the ball would drop. You have to aim
slightly above it in order to get further for it to to to to
to to hit the the the middle bulb.
But the point is that the that the the the government principle
doesn't just apply to light. It applies to to to to to
everything.
OK, next there's another good one.
See, I have a mask.
And so, as we learned a couple of weeks ago, that has a certain
amount of energy equals MC squared.
OK,
just by virtue of its mass,
I let that fall
and it falls through a distance Z
right?
So the energy, the total energy of this mass M is now
the
we must aim plus the
kinetic energy. It's it's picked up by virtue of having fallen
through a potential of of of ZZ. So the energy of this mass at
the bottom of this drop is bigger than it was before, as
you learned a couple of weeks ago, that that's a special
artistic remark that the energy of the mass increases. OK,
but but this this term is is is Newtonian. There's nothing
complicated about that.
So then
turn this mass entirely into energy, into into our an ultra
realistic photon and fired it straight upwards
and he gets to the top.
I turned it back into our particle. No, this is
kinematically impossible. There's no means of doing this.
But say you could. We're in the energy balance here.
You turn it back into
from from from a photon of energy primed to turn it back
into a particle of mass M
which will therefore have an energy of MC squared or M. Now
either we have
invented a perpetual motion machine, which we haven't,
or else
the energy of the photon at the top of its ascent E primed is
just M again.
In other words, for this to all the whole thing to hand
together, it must be that the the photon loses energy
as it comes up, as it climbs up through a gravitational field.
So a photon of energy E here
ends up being found energy primed at the top, which is less
by MGZ.
So photons are red shifted by ascent through a gravitational
field.
Then we can turn we can we can write that down a little more
mathematical detail.
What that means is
have
M
drop M + M GZ.
The photon
E primed equals
and
that is on on screen. Good.
No. That means
that
and
the E = M equals
and
is equal to E
minus MG Z.
E prime is equal to E minus
MGZ
and
uh
which is M plus.
I get this,
but I'm about to write a wrong thing here.
And
and.
And it has a ohh yeah,
I'm about to write something wrong here.
A second
and close
would write him trivial.
That's right. OK,
right.
This
that that the E primed
up. Here
it was just really trivially rearranging that E / 1 plus
GZ. So we can write we we We can write down what the
change in the
in in the energy is at the photon rises from the bottom of
that well to the top.
We can go out on a little further
again, get this right.
You will recall, or what you're about to be told, that photons.
The quantum mechanics tells us that photons have a certain
energy.
The energy of a photon is Planck's constant frequency.
Have you encountered that by this stage?
Something, something, some slightly slightly nervous nodes?
We're not gonna depend on anything, anything deep there.
So
the photo on when it starts off at the bottom here
will have frequency F
which is the energy divided by Planck's constant
and the fortune at the top
will have energy.
You prime equals H,
F prime
and if
and
right if E then here
is equal to M which is E primed
one plus
G and I'm going to write that as the I'm going to change change
rotation here rather than talking about height there. And
we're talking about radial distance R away from the centre
of the Earth, for example
GTR.
Where that's there for that.
Hey there Delta R
and switching from from
E to F that's going to be
each F is equal to HF primed 1 + U Delta R
And I'm then going to say OK
if that's true for our height delta R that is true for an
infinitesimal height Dr
and
F prime is equal to F plus
some infinitesimal mode
PDF. So the FF will be equal to
F plus
TF
1 + g
Dr.
I can expand that
F plus
PDF
plus
FGDR plus DF
yeah,
but because these are infinite and quantities I'm going I can
ignore
2nd order in them
and get a DCF
over
R
radio DF
over F
if you could minus G
Dr
which it looks good nice. It's. It has a a simple form
then I'm going to see
that
and again make sure that it's step.
And
if we
rate the gravitational force in terms of the
gravitational potential
then the
I'm missing out some some minor steps that are in the in the
notes will be the Phi by Dr.
So the the the gravitational force we feel is the radical
change of the gravitational potential as we we ascend
through a A
height
and take down trust for the moment
which means this turns into DF by the DF by over F is equal to
minus.
Define
so the change in the frequency of this photon as it rises from
from below to above is proportional to the change in
the gravitational potential. And that's just nothing exotic there
about the gravitational potential. That's just basically
nuisance gravitational potential,
OK,
And we could integrate that
again. I'm gonna miss it the the intermediate steps and get that
the
change in frequency
he's
make sure I've got this right. We up
that if I go from somewhere with the gravitational potential of
F0
movement some of the gravitational potential of F1 of
Phi 1,
then the frequency of the of the light when it goes from the
bottom to the top will be. We'll go from F naughty to F1
Question.
So the the, the, the the the expanding the brackets. OK and
there I'm just expanding the brackets so in the usual
fashion. So F * 1 plus D, F * 1 plus
F * 3,
but F times GDR plus DF times GDR.
We're just expanding that
just
or and
so I'm I'm discarding that because because the
infrastructure quantities in two two small things multiplied
together are small are ignorable small.
So that that that that's the importance of going to of of
talking about of going from Delta R as of macroscopic
difference in our that would be
yes. So the FA council. So zero is equal to 2DF plus FDR.
So
address
cancel.
Ohh yeah so. So yeah, do manoeuvring there at once.
So I'm not gonna do anything more with that, but the I think
the point of
mentioning that is to negative concretise that this difference
in frequency is
directly linked to the difference in the gravitational
potential at the two heights. And you can measure that in a
laboratory,
one of the early collaborations of this in 1950s using the the
most power effect, which is an effect where our photon is
absorbed by a crystal with insane precision, Insane
resolution in terms of the crystal will absorb photons of 1
frequency only with incredibly narrowly.
And what folk did
was the fired X-rays donor tower which was 22.8 metres high, so
not high, I mean a couple of 10s of metres. And found that in
order for the photons to be absorbed in the crystal of the
bottom, they had to be tuned extremely precisely at the top
so that the change in frequency was exactly matched. This so
that is directly experimentally measurable
in a beautiful experiment called based on the most part effect
and it was pounded into Ribka. We did this in 1959
rather beautifully.
OK.
Any questions about that?
So
a couple of mathematical steps,
I think. Quite instructive mathematical steps, but you
know, not something to test you on. OK, I I mentioned that
because I wanted to be very clear that there's no slate of
hand here. You can get to an experimental measure measurable
outcome here really quite promptly.
OK,
good.
Thank you.
And
the next thing I'm going to mention because I think it's
important, but I'm not going to go through in detail because the
out, the, the, the end result of this argument is more important
than the than the details. The details are interesting and I
encourage you to go and look at the relevant section of the
notes. But this the the the point here, and they're referred
to Shields. Photons
is what? If you do this and you do it twice. So you fire a
photon from one place
up, and then a little while later you do the exactly the
same thing again, you fire a another photon from that same
place up.
You can draw that Anand Koski diagram.
So the starting point of this zaxis is pointing upwards. The
time axis is pointing upwards on the graph. So this is going from
a low point to a high point, a photo on going from,
you know, below to above, moving at 45 degrees in the usual
fashion, and then a short while later the same thing happening
again.
And you can see the
time difference. Let's say the time difference between these
two
emissions at the bottom
is some number of
Ohh of of periods.
So at no over FN over that frequency.
And
as we've just established, when the Photon gets to the top, the
the freaking ball have changed.
But everything will still be in feet,
so that the difference in time between B&B primed will also
be a whole number of periods of frequency.
But because the frequency has changed,
a whole number of periods of the light of the light at
height A and a whole number of periods of the light at FB
will be different.
In other words, the time difference between A and A
primed
will be different the time difference between B&B
frame.
So that looks like a parallelogram,
but it's not.
It looks like a prime and B prime should be the same length
and the other same length and periods because the frequency of
different they're not the same length in time.
And that is the first hint
that the geometry of Minkowski diagram in the present of
gravity is a bit a bit a bit different from what you expect.
It's not just flat
because what you you saw in the midfield diagram the the
geometry of the of Minkowski space in special relativity. OK,
it was a bit strange with this, with this odd idea of the
metric, but it was flat in the sense that a parallel
parallelogram would a parallelogram,
and this construction is telling you that's not actually true. So
this is where Cover Show has come in here already.
Non Euclidean geometry has come in here already.
We're only you. I know We're into instruction to to general
activity.
One can go, one can step through that more slowly and I encourage
you to to to look at the relevant bits of the notes, but
that that that punch line is the important point there
and so we quick question similarly on that.
Umm,
a couple of quick questions to help you. You think it through.
The next thing, and this is getting us right to the the the
heart of the problem,
is what if you have two things high above the earth to objects
high above the earth,
and you let them fall?
They're going to fall directly toward the centre of the Earth.
Straightforward.
Not really. I'm not taking anything that exotic
through the start off
a distance XI apart from each other,
OK?
And as you get closer to the centre of the earth,
because they are converging another point, that side of T is
going to get smaller.
So the separation between the two of these two objects will
decrease. It will decrease
faster and faster. Are these things accelerate towards the
earth?
OK,
but so so so the second derivative of PSI
will be known 0
and and the the notes you walk through the step by step. It's
not complicated, but you know I'm not gonna do it.
So the second derivative of X is going to be non zero D2 XI by DT
squared equal not equal to 0.
Does that mean we're accelerating? This is the second
derivative of of of of a separation is we're not zero?
OK, they're accelerating. They're not accelerating
because these two things are freefall.
So if if these two things were in free fall, then they're not
going to feel
push it. You know if you were inside a box
falling like that, you wouldn't feel any acceleration
you've been free for.
So this is a case where the 2nd derivative of a of a distance is
non 0 but there is no acceleration.
What is going on here?
The difference is that we have to be careful what we mean by
the word acceleration.
And by acceleration I do not mean
merely second derivative, opposition being 0 being non 0.
What I mean by acceleration is a push.
So if I if I push you, you are accelerated. If a car pushes me
then the the seat pushes pushes my back and and we move forward.
If I hit the ground, I'm accelerated and there's I I can
feel it. If I turn round, if I rotate,
I can. I can feel it.
So acceleration is force applied
that you can feel and there's and there's nothing relative
about acceleration you can you can definitely feel
acceleration. It's not frame dependent thing. If there's
acceleration is enough to to break something then it breaks.
There's no, there's no frame in which that thing won't break.
OK. So when I say due to what acceleration, I mean
that
but this the two objects A and B are not accelerating towards
each other.
In that sense, because they're both in freefall,
they don't feel any push.
All that there is is a difference in the second
derivative of the separation of separation.
So that is also. That's another thing that's that's that's
pushing us toward an interesting thing here that that that the
coordinates here are not straightforward or linked to the
the, the, the physics. We expect
that that this coordinate of of of physical separation is not
quite what we expect. So we have to be careful what we mean by
the difference in two coordinates
and that have gone. Very important.
So this is all rather
abstract and it's rather a bit of a of a toy setup.
But what if the thing was happening here was not
two things falling towards Earth, but two particles in the
accretion discount of black hole somewhere you know the other
side of the universe.
What we want to do is we want to understand what's happening
there,
understand the physics of what's happening there
and we know that what's happening there is in A-frame,
which is highly governed by the gravity of the the black hole
the seclusion disc is is moving around. So what we want to do is
understand how the how the physics works.
And one way in which that gets complicated is say
I'm in box A
and someone else is, I know it's in box B.
They have some coordinate attached to the frame, they're
framed, some coordinate attached to the box. You know the they've
drawn lines
on the on the side of the box and they've got, they've got a
watch. I want to understand the physics of what they're
measuring
in their books
based on what I'm measuring in my box and my knowledge of the
relationship between these two things.
Because see they were on the other side of the earth, then we
both be quote accelerating towards each other in the sense
that secretive below 0. We've both been 3-4 and each
understand that.
And that brings us to the key thing here,
which is
how do we go from geometry to general activity.
And this is the plan.
First, what do we mean by the geometry of space? I think you
you right, you know roughly what I mean,
but we need to be a little more precise about that. We're going
to, we're going to be in a moment.
How do we describe geometry mathematically?
You know, how do we do calculations with it? And that's
that's not easy.
But that's a separate, separate step.
And how do we make the link from geometry to to gravity?
So what? What we can perhaps describe the the strange shape
of, for example, the Minkowski space in that shows photon
setup. But how do we
talk about that? Step 2 and step three make the link between that
and gravity. And that's the plan for GR.
For general activity
and
step one involves stepping back a bit and think. Having to think
Step 2 is mathematically hard.
Step three is Einstein being clever.
OK,
so we can do step one. We can do steps two. We can do step one
and Step 3
here.
We can't do Step 2 because it's just too hard.
If you carry on your study of of massive astronomy or whatever,
then the generativity course
which does 10 weeks just on Step 2 there
into one and then another 10 weeks in semester 2. Talking
about the consequences of that,
because it's quite challenge,
excellent fund, but a bit, but a bit, a bit, a bit much for a
second year.
So we're going to gesture toward Step 2 and concentrate on step
one and Step 3
because and and and and and and and in a sense, this is the is
the is the payoff for the way I've I've been talking about
special relativity because I have been stressing general
geometry. And where is the physics? All the way through the
discussion of or special activity. So that this section
of GR is we can we can talk about the important ideas in GR,
the important technical ideas in GR, even if we can't go into any
of the details. So this is the payoff for the route I've taken
into special activity.
OK.
Umm,
that is the important bit.
OK,
geometry. I'll just start on this just now.
And
so geometry is the mathematical description of shapes.
And you've you've learned a bit about geometry. You you learn
about angles and sines and cosines and Pythagoras theorem
and parallel lines and all that stuff. And in the 19 century, if
you were an undergraduate, you'd be given Euclid vaccines and be
chased up and down the steps of the Academy,
this, that, and the other, using Euclid's axioms over how to
construct this construction, this using rules and straight
edge, and so on. So there's a lot of intricate details you can
learn about geometry
and because there's a lot of it, and I started off in classical
Greece and it was developed and made more intricate through over
the course of
of my of mathematics from classical race through multiple
different cultures to 13th century and it was thought to be
essentially done.
What can you say? What you say geometry, that hasn't been said
for 1000 years
and it was in the 19th century with Riemann who
discovered that.
Well, one of the of the of the standing puddles of that branch
of of geometry, mathematical geometry was how to prove
Euclid's parallel postulate.
Nucleus Panel postulate said that if you have two lines which
are parallel
and then if I meet.
So that's sort of the definition of parallel means you know you
have two lanes to start off the point of them direction you
could go along those lines and they've just never meet.
And that's perfectly true as far as Euclid is concerned and it
was thought this was hopefully feeling a bit trivial. So there
was a there was like 1000 years of folk trying to say, well
obviously this must just be a consequence of the other axioms.
And so people trying extremely hard you dying poppers because
they were cranks and they said the eye finally proved Euclid
parallel postulate
and but eventually Riemann.
I think we're well,
I think that's slightly unfair, but it wasn't really. We remain
but
the market associated with them. So there's more history of
mathematics there which we'll talk about. But he said,
actually you can't, because it's an independent postulate it it.
It is something you can decide to be true or not to say to be
true.
And if you just say it's true, then what you get is you could
geometry, Euclidean space, the flat space we're used to on
paper and in three-dimensional space.
If you don't say that's true. If you say that you can have two
lanes which startup in parallel and which you could produce in a
straight line and the end up meeting
or or not meeting or something, then you can get a consistent
geometry that way.
And thus was non euclidean geometry born.
And I shouldn't spend much time talking about but it is
interesting.
But what is important is I talk about what geometry is, and what
that means is I talk about what spaces
this is. This is good stuff.
So a space is
space to speak. I mean it's a thing you can move around
and
you can always attach coordinates to that space. So
you can see OK in this space here I I can draw a line on on
on on the ground mark it off
like the walls and and I could identify a coordinate for
everything in that space.
And I can ask questions like well, how far apart is that
point there? From that point there,
and I know how to do that in in Euclidean space, I take the
difference in the X coordinate,
the difference in the Y coordinate, difference in the
coordinate
XL plus y ^2 + ^2 = r ^2, the 3D version of Pythagoras theorem.
And that worked. That's that's a definition of space
in including space in terms of Euclidian or Cartesian
coordinates, the XYZ.
And we've discovered that you could do the same in Mickey's
face, but the only difference is there's a there's a - in there
as well. So it all gets a bit, you know, a little bit
confusing, but the same general idea.
So what the way I can write that down
is I can see
that have not been used too much space here. Alright, that was
12.2.
Is you can see
that in Euclidean space. Whoops,
the
unit of distance EA squared is equal to DX squared plus
y ^2.
And that's not surprising. That's a differential form of
Pythagoras theorem
and then you can ask questions like say we have
and
this is X.
Why?
See, we have two points. P Naughty
And
P1 and that's that
point. X Naughty.
Next one.
What's the difference between those two? What's the distance
between those two points?
Let's do it the hard way. OK.
The distance between those two points,
imitation rate call
is the integral
from
be naughty to P1
of the
differential
different
step from one point to the next.
What is that? That's going to be the integral from X naughty to
X1 of
DX squared plus the Y ^2 just just using that definition of
our our our differential version of Pythagoras theorem.
But in this part here do you dy is 0,
so that becomes just integral from X naughty to X1 of
square root of DX squared, which is the X which is equal to X
makes not to X1, which is of course X1 minus
X0, which you which you could just said right from the right
from the outset.
So that seemed to be part of a long way of getting to it to an
obvious answer,
but the point is, it shows that the length between between two
points is the integral of the of the differential lengths as you
step along that path. So as you step along the path from one
place to another, the total distance distance you go is the
sum of all the steps you take
and and and this is just a. You know that. That's all I'm seeing
there.
It looks quite exotic. We have written it, but it's an exotic
we're writing something that makes that should, I hope makes
sense to you.
And that's how the metric gets
is used. So the metric is is is defined in terms of of of the
separation between 2 points and infinitesimal difference
distance apart in terms of the coordinates.
Now I can.
That's not the only way I can talk about, I could talk about
distance. I and I I will say this quickly. I know you're all
rushing off to get back, and I could see that get the end of
the next bit quite, very quickly.
In polar coordinates I can see that that distance is equal to
the r ^2 plus
and outward.
Do you think the squared for two things which are separated but
separated in polar coordinates by different distances and
different angles? So
E, Theta,
Dr
and I can do, I can ask what's the the, the, the, the length,
the distance between these two things in the in the same way
you know P1. Yes.
And I end up seeing it X 1 -, X. Naughty
by a long by security route which comes to the same answer.
The point being and I will shut up right after I finish. Saying
this, that although these are two different ways of writing
the same
distance, two different ways of writing what the difference is
between 2222 coordinates, they end up with the same answer for
the total distance.
Met you in that I'll, I'll just touch on that very briefly next
time to use that as the launch party. We're going to the next
the next part.