OK, this is lecture 3.
We managed to make excellent
progress last time and what we got
the right amount through Part 2,
what we're going to do?
Well, let's move forward to where we
did get to last time, the plan. Is.
We've got to vote there I think and there
and the idea that one way of visualizing,
mentally visualizing our our one form and
I think of planes or lines and the way of
because because that's a good visualization
because it makes the idea of contracting
A1 form with a vector very natural.
So the the contraction of these three vectors
with that one form field are all the same.
The team in each case
SO3 layers pointed even.
Although the one form fields varies
over the course of the the fields,
it varies in direction so that direction,
in that direction whatever,
and size magnitude.
We're just as if you're visualizing these
contour lines the the the gradient of that.
Landscape is larger when the
lines are close together,
so that's why it's it's a nice visualization.
Not too much hanging on that.
It's just we've got you can
have a picture in your head.
So this next section we're going to go
on to talk about if we find the right.
Page we'll talk about.
We'll start off in a second
225 talking about components.
Now this is where things start to get fiddly.
OK, so there's quite a lot of notation is
being going to be thrown at you in this.
In this lecture, it's not deep.
You have to think and go to mountain
top and meditate on on the ideas here.
But you do have to in since, you know,
go back over and concentrate, work out,
get everything straight in your head.
So don't expect to have any
wonderfully physically epiphanies here,
but there's, you know,
certain amount of apparatus and technology
that we're going to get through here.
So I am going to go
through it fairly briskly.
And show you if there are questions,
but I don't think there's a lot of.
Profundity, coming up is what I'm saying.
In, in,
in in in this neglect lecture.
Before I get going, are there any other
administrative things that I have to say?
I don't believe so.
Are there any questions of that
sort of sort that anyone has?
No good, homie. Let's get going, OK?
I said earlier last lecture that
the set of of vectors in one
forms is from the vector space.
That means that there is a basis
what from the the axioms of
vector space you can reduce.
We're not going to do it to talk
to mass department for that you
can get a basis for the set and
and what I mean by a basis is.
That you can't for any vector.
Yeah. A.
There's a set of basis vectors
which are going to call EI and any
any which which span the space,
so that in an N dimensional
space there are N basis vectors.
They're all in linearly independent
and I think there are basis.
We are saying that any vector
in that space A.
Can be expressed as a unique sum of
multiples of those basis vectors.
OK, that's that.
That's just the same as saying
that you have the the.
Any any vector in the plane is is
is too many I plus too many G's and
and and so you're familiar with that notion,
I trust.
And they and the the components are unique,
we can do the same thing.
For the vector space for
the for the one forms,
we have a set of basis one forms
which we originally call Omega with a
tilde above them to show the one forms,
and any one form in the space can
be expressed as a as a unique sum of
multiples of those pieces one forms.
Is anyone feeling surprised at that?
Official question.
Completely different space at this point.
Yes, there are different space.
We'll find out shortly that we're
going to make our very important
link between these two spaces,
but right now they're just different spaces.
OK.
Uh.
The other important thing is that
if you change the basis set.
Device, say I don't like those
I I'm going to have different,
IE then you change the components,
but you don't change the vector.
OK, so the same vector A can
be expressed as a different sum
of components of multiples of.
As a sum of components of
these basis vectors.
As a sum of different components
of these basis vectors.
Fairly obviously you just changed
that your choice of basis vectors
and the components that you have
to use to add add up to get the
the initial vector are different,
but the vector of the same.
That seems hardly worth saying at this point.
It seems so obvious because you're
you're starting off with with
A and you're discovering what
the components AI are,
but it very crucial,
it's crucial that you hold that
you remember that that the vector
A is what we're talking about here.
That's the important thing.
The way it happens to breakdown
in terms of this basis set,
of that basis set is a detail what
that what we'll learn that that's
the changing the basis is changing
the reference frame or changing
the natural reference frame.
So you change frame, you change the,
you change the coordinate system,
you change the components. That's all.
That's all I'm saying there.
I'm seeing some of you are familiar with,
I think in a complicated way.
And because it's useful to
for that for complication.
Now note the.
Conventional layout of the indexes.
Here,
all of the basis vectors have
lowered indexes and the components
of rate ones of the basis one forms
have raised indexes and their
components have lowered ones.
If we stick by that convention.
Then there is a hugely useful additional
convention we can take advantage of,
and it's called the Einstein
summation convention.
Which, and unfortunately this is going to
be annoyingly slow to swap back and forth,
but there's a meeting eventually.
Is that if we have some? A i.e. Aye.
Then we can write that as just a i.e.
I, and if there is a.
Was there a repeated index?
One high, one low?
And in any expression,
then we assume a sum over it.
OK. Now.
And that is a repeated
and repeated index, so.
AIJ. The Newsroom.
There's no repeated index, and that doesn't.
If you've written that down,
you've probably made a mistake, OK?
If you write down something like.
AI. For EG, you've made your EI.
See see, see. You made two mistakes.
First, you probably didn't mean to write
a the the the I of the superscript.
Secondly, that's two raised indexes,
so there's a no sum,
and B you probably made a mistake.
OK, so this is it's.
It's quite quite useful in rotation
because it surfaces mistakes that you make,
notation mistakes that you
make quite quite promptly.
And if you've got 3 repeated index,
you know an index would be three
times you made a different mistake.
OK, so so all these mistakes
are ones you will make.
This course is.
And.
The the the little secret of on
of of advanced courses is the
harder courses of easy exams.
Because harder courses tend to
be harder to write exams for.
Which means that there are a
limited range of things one can
examine in a in a in a hard course,
but one of the things you can examine is
can you turn the handle on the algebra?
It's not a very exciting remember the
distinction made between English objectives?
I said aims with the point
objective with the party tricks.
Well, algebra is a party trick, OK, is it?
It's not a deep thing.
It's not the thing you'll remember,
you know in in years to come.
But it is it is nice and easy to examine.
So big hint the the exercises that are that,
where you exercise your,
your fluency at using this
notation are good ones.
Have a look at.
I'm not making any promises
but you know that that.
If you end up with if if the if in in
June you end up with an exam put paper
which is is clearly algebra heavy,
then go yes.
I'll try not to do that because
it's too easy to do it,
but it tends to be a binary that one,
so those who have done the exercises.
It's just a matter of not getting lost
those who haven't done the exercises.
I mean draw a literal blank.
I mean, it's, it's, it's.
It's not nice to see.
Sorry hint over there water.
Ohh yes so getting getting used
to getting good at this algebra at
this algebra this right this index
wrangling is important and the only
way of doing it is doing exercises.
The indexes are arbitrary here,
the other important thing so
that AI that and that a GE.
G's are the same thing.
So you can always swap the indexes
over as long as this the the
the the same pattern is there.
OK. Because in both cases is a
sum over the arbitrary index.
Are operating.
And So what happened if we do things
like applying one of the ohso?
So there's a.
Ohh yes,
Yep.
So what happens if we?
This is really annoying,
but I do want to rather swap
back and forth between these.
So so there's the.
Vector A is the sum of AI A1E1 plus
E2E2 or more complicated than that.
And they I'm not clearly not making
any assumptions about the relationship
between the the the basis vectors.
They're not the same length,
they're not the same.
The angle between the business
is 90 degrees because we haven't
don't have those concepts yet.
The idea of a length of a vector
and the angle 2 vectors doesn't
exist as far as we've got yet.
OK, so we're we're just talking about
two things which are not the same.
We are not just one E one is not a
multiple of E2 is the only thing that
that that's that's important here,
because if it was that I
wouldn't stand the space.
And submission convention.
So quick pause.
Which of the following is a valid
expression in the context of the
Einstein summation convention?
I'll go through quickly that your brief chat,
so one who said it was valid?
Two, who said those valid?
Three,
that was valid.
4:40 that was valid. That's AIG.
OK, have a brief chat and.
I mean, you're mostly right to say
one point, but what? The second one?
JEJ.
So with that.
With that reflection in mind.
Books with that reflection in mind.
Hello with that reflection in mind.
Who see that one was valid?
Yeah, it's it's it's two indexes,
the same, one up, one down.
We would say the second one valid.
Yes, on reflection it is because although
I've made a point of of writing one
forms as with lower case letters would
therefore have subscripts, there's no.
That's not an absolute rule,
so the fact that PG each is is out of one
form used letter doesn't change anything.
PII is is not valid purely because
there are other ₹2 indexes.
They're both lowered, so that's probably a.
An algebraic mistake at some point,
and A i.e. G1 up, one down is.
Invalid because others one up,
one down is that it's not invalid.
It's not invalid.
It just doesn't mean very much.
So that there's no Einstein
summation convention.
OK, so.
The last two aren't really invalid.
They probably indicate mistakes
of some type, but they're not.
But they're simply not one to which
the Inspiration Convention applies.
OK? But probably valid.
There's probably a mystique has happened
leading up to that being written down.
Yeah, so bad isn't quite right there.
I mean it it's just probably a
mistake, but it the, the, the,
the the Commission doesn't apply.
OK, I'll come back to that moment.
Let's walk back here.
So what happens if we? And ask.
Apply around in one form P to one
of these basis vectors. Called EJ.
Well, as we know P as we know
know as of like 10 minutes ago,
as P can be broken down into its components.
So we can write Pi.
Well, we got i.e. Gee. And.
They would stop because we don't know
anything about that and what we got,
IE we don't know what that basis one
form applied to that basis vector does,
however, we can decide.
That will define the basis of one form Omega
in terms of the basis one basis vectors EI.
Such that. Omega i.e.
G is equal to delta.
IG. In other words.
And one if I equal to G and zero if I.
Is not equal.
So that's not we're not required to do that,
but it is foolish doctrine. OK,
we're defining the basis one form to be dual.
To the basis vectors. With that in mind.
We could no rate that this P tilde.
EG is Pi Omega I e.g. Is Pi delta?
IG. And we have a summation
convention we can we can do.
We can add that up.
What answer do we get?
In the sum over the the eyes.
The the Delta IJ is 0 except
when I is equal to J.
Therefore that is equal to.
PG. In other words, more clearly.
In other words, if we apply the one
form to one of the basis vectors,
what we end up with is we
pops out that Pops pops out,
there is the the corresponding component,
the JTH component of the one form P.
So that that's how we extract
components from an arbitrary one form.
Before and we can do exactly the same
by applying our vector to one of the.
Basically one forms that's a i.e.
I applied to Omega.
G which we will similarly.
It's similarly defined to be
a I delta IG which is a G.
So we we turn the handle and and
out pops the component and you can
see where the sort of algebraic
mistakes that can happen to you.
If I if I if I accidentally write
an instead of a G then I'm going
to end up with with with two eyes
in the top and and or three eyes
or something and I go Nope and
and I I step back to a bit and
work out where where I miss wrote.
So there are plenty of opportunities
to improve your handwriting and and
write very neat eyes and J's and
commerce and technical ones later on.
So that's that's something.
But mechanics,
how can we do the same thing for teachers?
Of course we can.
And if we now is just as we can write
could decompose our vector into.
The components. We can decompose
a tensor into its components.
LM. And and here I'm going
to use an E. L. He. And.
End.
I I think I said last week that I was
going to introduce out of public but not
actually use them for a little while.
I clearly had forgotten that I
was going to go through this.
So there were version with the top
we're writing the the components
that were expanding the tensor
in terms of components.
At times an outer product of.
All vectors 1 forms.
How do we in that case extract the
components of the tensor from that?
If we apply. Remember T what?
Breaking T as a AA21 tensor?
So that means it takes 21 forms
and one vector as arguments.
So if we drop in. Omega I.
We got a G&E. Key into that.
Then we. Discover that is T.
LMN remembering the definition
of the outer product.
El. We got I. Times on your
ordinary multiplication E.
And Omega. G Times Omega.
NE. Keith? OK, so so remember
that the applying the the
definition of the outer product.
Is that when that other product is
applied to boom boom boom 3? Arguments.
The three arguments are divided amongst
the three things in the product.
In this way, that's in the definition
that product mentioned last time.
If you if you go back to that,
you'll be reminded of of what
we talked about last time.
And at this point it's become
mechanical because we just replace
each of these by deltas. Tea.
LMN Delta Li, delta MG DD. And.
OK. We've got three repeat indexes.
We've got 3 repeated,
3 implicit summations here.
In each case the summation over.
L / M and over north.
The summation over N will be 0.
Except when a is equal to key,
the summation over M will be
0 except when M is equal to J,
and Speaking of L will be 0
except when L is equal to I.
So this will be TIG.
Key.
So just as with vectors.
We can extract the components of a.
Uproot enter by dropping in the
corresponding the right number of
basis vectors and basis one forms,
turning the handle and and getting a
number out, and so I mentioned this.
I worked through this in order to, you know,
show that this this whole process works
for tensions as well as vectors 1 forms,
and also to show the sort of of,
admittedly very fiddly sort of
algebra that is involved here.
The. This isn't fun algebra,
it's just does require concentration and
most people who do research, you know.
Could have written your research.
Tend to be very fond of computer
hardware packages because you
know all all the handle turning
can be done very reliably.
OK.
Are there any questions at that point?
You you see what I said about that?
There's quite a lot of detail here,
nothing very, not a very profound,
but it does require concentration.
Any questions? OK. And. Umm.
So just go back over over over to
emphasize a couple bits of rotation.
The set up the this set of.
So this is. This is a number.
OK, that's a number.
It's a set of N by N by N numbers.
In fact, it's that there are
in any dimensional space,
there are, that there's
T111T112T113 and so on.
And so this is N by N by N numbers.
But this thing here, TK, is just a number.
It's a real number on the real line.
There's nothing exotic about that.
But we will somewhat slangily.
Either talk about tea by writing down a tea,
or talk about the tensor by writing
down that and saying that's the tensor.
It's not tensor,
it's just important for tensor,
but we'll we'll sort of equivocate between
the between the two things, in what way
that will be be natural in context.
But remember that even if
we see the tensor T here.
Were being bad.
Not even that we're talking
the components of density.
OK, I I make a big fuss about that.
Again, that seems obvious now,
but it can trip you up later.
It's also useful to write it down that way,
because that makes it clear to 1 tensor.
It's just saying tea, you have to you have
to remember what was the rank of this tensor,
but here it's obvious it's a A21 tensor.
And hopefully there's two.
There's two intentions 2
upstairs and one downstairs.
Another thing to note is that um.
The.
If I write key IJ. And T. I. Gee.
They are not the same thing.
Because TIG. Is equal to
some tensor T. With the.
Had two documents filed in and the tensor
TIGIST. I.
In other words, these two tensors.
I could be different.
They're both 11 tensors in one.
In the first case,
the the the first argument is one form,
the second argument is a vector,
second case the first argument vector,
the second of the one form the
complete different things. Now.
In most cases, if we use the same
symbol we are for these two,
there's some implication that they
are related to each other just because
there are limited number of letters
in the alphabet and and and so on.
But and This is why the
indexes are staggered.
So we haven't written TI above G.
Written tea I.
G and the second case, TIG.
So it doesn't matter where these
things are vertically positioned.
There's a very dense notation.
There's a lot of,
there's a lot of information
in every last bit of of of
where the the notation is.
Lead out.
And and and and and. What this
also means is that we can.
But your basis vector E1C and we can
turn that into a set of components.
One you you, you. Whatever and so on.
So so so at this point we can write
down a set of components of the basis
vectors and make sure you you you see,
you think this is a bit more making
sure why you see it's obvious
that's 1000 and E2 would be 0100
and E3 B 001000 and and and so on.
So just make sure you see where
that's obvious in the mathematician
sense of obvious meaning going
think about it for a while you'll
see that you couldn't be otherwise.
But that question.
Another one for first, second, third.
Well yes because in this case
the the the two T's there
they are different sensors.
So the the the whole yes,
but it's for the second phase
of you know J on the left and
I on the ohh right and Yep so.
GIIST.
E.g. Oh my God, I.
There's not the same as the.
No, no, no, it's not, because in
the second line that is. And. And.
Is that with the same?
So so these these two things are
interchangeable because all we've done.
Is swapped I Ng. Not over. Alright.
And. About you exactly me.
But the in that case, the, the,
the, the tension right above.
Let's call that.
T1 and T2T1, that's attention which takes
is it attended with a different pattern.
It's a machine with a different set
of holes in the top one form shape
and a vector shape tool, as opposed
to a vector and one form shape tool,
and so it's carelessness on my part
to have written tea for both of them.
It is that we winning.
Because they must be one.
That makes sense.
Which place the holes are
which should change function,
but the second one you changed
the E has a is an I got the J.
OK, I I think it's possible.
Drug cross purposes, but no in this case.
The the letters I pick.
Will be arbitrary, so I could write T.
KKL. Will be tea.
The key. Oh my God.
And the same thing is being said so the
the the the the so so so pick a key,
pick an L and the the keys 1L3
then the the 1/3 component is
what you get when you E1 omega-3
in there and it doesn't matter
what in what letter you choose.
For, for denoting those.
Yes, yes, the the letter doesn't matter.
So sorry that, that that that that that.
I think which letter you pick doesn't
matter that the the fact that the pattern
on this side and the pattern on this side
is the same is is the key thing. So yeah,
so there's a famous arbitrariness here.
And and you could always change the.
You can always change the the the
the letters you you you pictures and
note the the arbitrary indexes as
long as it was anything both sides.
So that's that's an algebraic rule.
If you like, that's that's that's new.
OK, keeping going.
We do other things we could do
are what if we apply?
One form, actually, one form,
an operator vector.
Well, before that's going to be Pi Omega.
IAGE. Gee, the temptation there
was to me to write AIP IEI.
But that wouldn't be right, because then
would have at A4 indexes I, so I've got.
I've got to pick a random other letter,
so we're putting in the expanding
the A inside the brackets.
Pick some other. Other index,
but remember that the tense Omega I being
a tensor is linear in its arguments.
And remember that the
components are just numbers.
So because of that we can take the.
The output of the argument and write Pi.
EG Omega I e.g.
Which is equal to PIAJ delta IJ.
Some over the eyes or the J's.
In both cases is 0 unless I
equals J which is equal to Pi. AI.
In other words, the contraction of.
P and a.
P applied to A is P1A1 plus
P2A2 plus P3A3 plus P4.
How many up to which you will remember
from the definition of the inner
product between vector vector inner
product between the ground about school.
So that's the the vector inner
product we about school is
the same thing happening,
OK.
This and this is just a number, yeah.
Yes, so. So, so the that's of tension,
that's tends to, that's a tensor,
that's a, that's a tensor,
that's a tensor and because of our our
definition of the what basis one forms
has been dual to the basis vectors.
We decide that the.
Value of that is just the chronicle delta.
At which point we can sort
of turn the handle, you know,
make our way home up,
sum over G or I and get this,
which is a sum of numbers,
so π and AI numbers.
That's a based on the sum of numbers,
and end up with a number,
which is what we expected because the
which is what we should get because
remember that being a tensor, the.
A01 tensor Omega or P rather
is a move which takes a number.
I put takes a start game.
PRP is a 01 tensor,
which means it takes a single vector
as argument and turns it into a number.
So this does hang together. Question.
Yeah. Three, so with the.
Yeah, yes. So that's what we.
Yeah, so so so they are we we that
that that that's not. No, I suspect.
Given that we chose that to be the case,
I suspect we couldn't not
choose that to be the case.
I I think I I suspect that implies
that implies we'll put it up.
Put it right, it implies that,
but the article. But you're right,
the are two separate things.
Looking at.
This because we are applying
A1 form to a vector here.
We also end up applying one
form to a vector here.
Just want to break it down into applying
a basis one form into a basis vector.
OK.
So what we're doing here is we're
going back and forth between the
geometrical objects one forms and
vectors and calculations in terms
of numbers which are components,
all remembering that if we changed
the the the the the the the the bases,
then we changed the components.
Any other questions for that?
OK, we're making, we're, we're on. Good.
We're timing this well so far. Right, so.
Ohh yes and also. And. Here.
I haven't mentioned bases that,
that that's just we don't,
that's just a number, OK.
Here on this side.
If we change the basis basis vectors and
that's changed the the basis one forms,
then these numbers Pi and AI would change.
But the sum of them would not change.
Which is interesting. So the the
the components are basis dependent.
But this inner product,
or this contraction, rather, is not.
Which is surprising.
I mean that the that you might not
have guessed that would gonna be
the case before we demonstrated
that it was true here. Umm.
Similarly, if we were to do the
same thing with a tensor TP.
Q. E would end up with
PIQIQJAK. T. Um, IG.
Where as you can see I've got two
of each index. One up, one down,
and I have carefully staggered.
The indexes of the T to match the.
One for one form vector,
1 from 1 from 1 vector arguments of the.
Of the tensor. Um.
A remark there about how you can
define contraction in a slightly
more general general sense.
I'm not going to go into that
I I almost want to put that
in a dangerous bend section,
but it's it's it's useful to
have to mentioned in here,
but it it's not something we
depend on greatly greatly.
Just to summarize all this.
Well, I'm not gonna read out the the.
There's somebody at the
end of that section 225.
You you can look at yourself
and and and and and just go,
go go back to what we've covered here.
So we've covered a lot
of fiddling around here.
Nothing very profound but fiddly.
And you will get very used to it if you
what's your bucking exercises. And.
Yeah and there's other remarks.
I I think that section has grown
somewhat over the years as I
thought ohh nothing you say,
nothing to say another you can say
appended to the bottom of that section.
So there's lots of extra remark one
can make without necessary without
taking up a lot of time in the
lecture talking over the over them.
OK, moving on. And.
One thing we haven't done yet,
and I make sure only in passing,
is we haven't said have.
They don't have any notion
of how how long a vector is.
Or how long one form is?
So so the the this space of things
we're talking about here has no
lengths in itself at this point.
And you can do a lot without
talking about lengths.
So adding lengths and the definition
is a thing, you is a an extra.
It's vitally important for the
use of different geometry in GR,
but it is an extra thing.
You can do all sorts of maths
without worrying with this here.
We're gonna have the notion of.
The metric center.
Now we pick our tensor,
which the way we introduce lengths is.
We pick a tensor, we pick a A20 tensor,
so tensor that takes a.
It takes 2 vectors of arguments. And.
We use that as our definition of length,
and we call this metric.
This centre the metric.
Metric meaning measurement or or
ecological relative to, to to measure.
The tensor is the idea of it's where
length enters the geometry here.
So if one says that's a
centimeter rather than a parsec,
you know it's what gives definition
to the developer length.
The point we introduce we introduce length
is the idea that if you take a vector A.
And you stick it into both
slots of the metric tensor.
Then the number you get out is.
We're going to call this that
the square of the length of the
of the of the of the vector.
OK, that's our definition of length.
That's definitely the length of a vector.
Um, and you can?
Right. And if you um.
You can do other things.
Like talk about the angle between 2
vectors in this way which? If we had,
the notion of angle could come in here,
but we're not going to worry
about that at present because,
you know, it's something that we
we need to know at this point.
So the vector. So what do I say?
Yeah, OK, and let's stick
with the slides for more.
I've as usual. We can partially apply the.
The arguments to the vector, the tensor.
So if we apply just drop in one.
Vector argument to the tensor.
What we have is a tensor with one free slot,
one vector shaped hole.
In other words, I want form.
So what this does is it defines A1 form a
tilde which is associated with the vector.
A. Via the metric tensor.
So this isn't dual in the same way that
the basis one forms were dual to the.
Basic vectors. Well, not quite.
This is a dual like thing.
Push, which is via,
specifically the metric,
the metric tensor. OK.
And and we'll we'll see
a bit with that anymore.
Umm.
That that these quick questions are the
the the selection of the of the exercises
which are sort of just are you awake?
Take questions so that they're they're
notated in the in the exercises.
Then at the end of the of of the part
as the notice has been just just quick
things just to keep I'm not going.
With that for the moment. Um.
No, it is a useful thing to to to
to go through because it just if
it further illustrates the the.
Uh.
And.
The limitation working so.
G. A/B.
It's useful thing illustrated.
I introduced this by talking about
the the length of a vector being
what what you get what length squared
be what you get when you drop both,
when you feel both of the of the
metrics arguments with the same vector,
but if you apply different. And.
Vectors to those two things,
then what do you get?
And before you get something
like G applies to
A i.e. IBGE. Gee, again choosing
different indexes in the cases and
again AI and BJ are just numbers,
so because the tensor with the tensor
is linear in each argument, so the.
The IBA BJU could come out,
pop out here, and that gives us a I.
Be GG. Well, GG.
And um. OK. What can we do with that?
Nothing quite yet. But if we go
back and and and and look at this.
And tensor there's one form A
which we get when we apply a.
And to just one of the.
Arguments of the tensor.
Ohh and and by the way, what do we get?
What we get is. And a.
How do I how do I write this?
How do I how do I freeze this?
Well.
What we we get that if we know ask what are
the are the components of that one form.
We what we then what we we know how to
do that we we write a I applied to.
Basis vector e.g. But this is he.
G is that which is G applied to a. E. Gee.
We can expand that as usual G to a. I.
GE i.e. G which is equal to EI. G.
IG. So again, I'd be I'd be several
important steps here, which you will,
which will confluent to you eventually.
I've just. The component is what we get.
We apply one of the basis vectors
to the one form. From this.
The definition of that is that
we can expand a into a i.e.
G and take the I out.
We end up with G applied to two
basis vectors, which is just the
component IG components of that that.
Tensor. To what we get is that AJ?
Is equal to AIGIG.
In other words,
the component version of this
thing here is saying that the.
Viewed this way,
the metric tensor is a thing which
turned vectors into into one forms.
View this way, the metric tensor
is a thing which lowers the index.
In the sense that it turns.
And EI.
Or which again we're equivocating between AI,
the components and a vector,
and the vector A into a J the
components of a corresponding one form.
And so we're talking raising Lord indexes.
What we mean is turning.
Using the the metric to turn.
A vector intercourse one form
or a thing with the right index
into a thing with the Lord index.
OK.
And you'll be that would become a familiar
operation you'll see again and again.
Um, so looking at this expression here.
We just we we can write that as a IB.
I.
And discover that this operation of
applying the metric tensor to two vectors.
Produces something which is looks
a lot like the inner product
that you're familiar with.
In the sense that it's a 1B1 plus A2,
B 2 + 2, AB three and and so on.
But where the the,
the the the the the ones and twos are from?
Alternately a vector and a
corresponding one form? OK.
And the.
I don't expect you really to be having vague,
fluent pictures of these in your heads,
yet the the thing here is to get
practiced with the the the the
technology of of components.
Um, rach? Um, and similarly, uh.
Another thing that's important is this.
So this GIG remember is just
number is the IG component of the
the the the matrix of components.
So G is a tensor.
GIG is actually a matrix,
but not all, not all.
10 old rank 2 tensors can be
represented as a, as a, as,
as a square matrix, and by matrix.
Not all matrices correspond to, to, to, to.
To tension.
Anyway,
that might become a little clearer
in a moment. The other thing is that.
This fix this this this tensor GAB.
Is. But we can imagine also a tensor which.
Takes a vector in one form of
argument. A tensor which takes.
Also one forms documents and they
are completely different answers.
But we will choose to link them together,
and we'll actually think of them as
all as all facets of the same tensor,
although that's not mathematically.
Correct, because we will decide.
The the this tensor here which has
components I GG raised because
these two arguments are one form.
We will decide that that
matrix is the matrix inverse.
Of G so that so that that that's us
defining brings a little bit lower,
so I can actually point to it rather
than leaping about the well we'll define.
This tensor here G applied to two one forms.
Via its components, we'll see that the the
components of that tensor are the matrix
inverse of the components of that tensor.
And what that means is that if we apply them.
GIGG. GK we.
Contract over one of these indexes.
This. Operation this this.
We discovered here that this process
of a much of finished momently this
process of almost playing by the
the the tensor with lowered indexes
lowered an index so that is GI. K.
And if these two things are matrix inverses,
then that is the unit
that the identity matrix.
Equals delta IK.
In other words, so if if if that matrix
and that matrix matrix inverses,
then when you matrix multiply them together,
which you which you realize looking at it
is what what's happening here this this,
this idea of adding up that.
And if you say that for a bit,
you will reassure yourself that
that is the the the what you do
when you do missions application.
So that matrix multiply that matrix we.
If we're seeing this inverse,
then this must be identity matrix,
which is that matrix.
So in other words,
defining this means this potential
here also by its components.
We're just components.
The identity matrix, so that's.
So these are all different different tensors,
but we have chosen that the
the the tensors of different.
Passive index indexes have this
relationship to each other,
and what that also means is,
and I'll stop, I'll stop here,
is that the the metric tensor is also a.
Can also be an index raising.
Offer operation by that.
And that was a little bit garbled at the end.
Go through the roots to, you know,
make sure all hangs together.
I'd hoped to get slightly further than that,
but it means that we will have,
we'll make a really bit of a root
March through the last section
of this next time and aim to to
finish the this part.