OK folks. Hello. Welcome back.
The division two.
Thank you for those who have put
notes on the ballot and there were a
couple of other ones who added one,
quite detailed ones by e-mail,
but both a couple of general ones
which have transferred to this.
So I think we have a number of things
we can usefully talk about. And.
I think there's a fairly natural order here.
Which is I think I can talk about
this the time reduced first. Umm.
I think this question is what one of detail.
I'll make you that a moment these,
this one and this one.
Think are linked.
I think I'll do,
I'll talk about that that that
that this question 12 that is.
Are fairly say you're turning
the handle exercise that part of
the the the the peel off of that
exercise is more familiarity with
with export manipulation and so on,
and the idea of of components
in different frames.
It's there aren't some any magnificent
epiphanies available there but
it is really useful practice.
However it is fiddly and it's easy
to get lost and the notes that
there's of answer slash notes that
I have in the in all exercise
compendium goes you says and as
you will see the answer so it is it
is a little bit vague so I think
I do need to someone else pointed
out as well I think I do need to
go and expand that some point so.
Of or postpone.
I'll defer discussion of that question
there until I get around to making
an expanded version of that note.
And so I think therefore the useful
thing would be to talk about that,
that I had trouble that for that first.
This business of the time like so.
So where we were there was the
timelike geodesics of the local
space-time and the picture.
Here's this,
yes.
If we. We've said to here the.
Alright. Well. The picture I
want you to have in mind there.
Is. This one here, a particular
restaurant in a national frame.
They are used in Minkowski space. Now uh.
So, so where did that?
So that that sort of makes sense,
but I hope but this question of timely
geodesics and mikovsky space sort of
leaps out from nowhere to some extent.
So where did that come from?
If you think back to your
study of special relativity,
you will don't look at some point.
Remember the twins paradox?
That's where the where someone
stayed on Earth and someone else
heads off to another planet,
comes and comes back and they end up.
Younger than the person who stayed
put you you remember that that that
that's familiar is it, Wallace?
At that point it's just that, that,
that, that, that, that diagram there.
So so all it is is, is the idea of,
I mean configure diagram. And and.
The idea that of of a world line
which is going along the, the,
the, the, the, the the time axis.
And if you think of the.
The twins paradox. Then if you draw the.
That story if you're like ohh
mccottry diagram. Then someone stays.
On earth.
So they they they they they they
just stay at X = 0 for the entire.
Duration.
And someone else heads off.
That's some.
That's some speech to to turn
around point and then comes back.
And the traveller?
And the stay at home person
end up being different ages.
Why? And there's a page where you can
you you you you can. Talk about this.
One of them is the to to to note. That.
Because of the way that the interval
works out, you know the, the, the,
the the the invariant intervals.
DS squared equals D T ^2 -,
d X squared, the interval along the.
Time like but not time axis.
The these paths here is shorter.
It's smaller than the interval
along the the direct route.
In other words, the less proper time elapses.
Therefore in other words,
that person ends up younger.
That that that's one way of of
just slicing right through the
twins paradox and seeing why the
the person moving is younger.
We're just talking special relativity here.
But you can make it a very so So what?
What's happened here is that
this is straight line here.
It's longer than the dog leg lane,
which is the opposite way
around to Euclidean space,
including Euclidean space.
The straight line between points is the
shortest distance between two points.
And if you look at that a bit more,
and and and and and think through it,
you can deduce fairly straightforwardly
that in Minkowski space,
A straight line is the longest distance
between two points is a straight line of
makowsky space is the longest distance.
In other words,
it's the extremal distance.
And if you recall what I
said about you desiccate,
one of the problems of eugenics is they are
the extremal distance between two points.
That's the the the definition of of
one of the definitions of geodesic.
In creating space,
that's demonstrably true,
because the if you if you could
get string and you pull it tot,
you have discovered the shortest
distance between those two points,
and that's a straight line.
If the surface of a sphere
you pull the string taught,
then it forms a great circle,
the shortest distance in miskovsky space.
The straight line is the longest distance.
It's a geodesic, so the.
The person going along the.
The the state home person here.
They were the.
The party who just moving the whole world
line moves along their own time axis.
They're the one following the geodesic.
So the point of this is we
have identified where geodesics
are in Minkowski space.
And this this, this person here the the
the traveller is also following their
duties because they are following the the
the time axis in their own inertial frame.
So time like straight lines you're like
are there UD pics in Minkowski space?
And what the equivalence principle
tells us is that.
OK, we understand how how motion
works in Minkowski space.
Especially every we move along
the our geodesic is moving along
the time axis of inertial frame.
And the Council says the same is true in GR.
So in GR the description of how
you move is you move along the.
Time like geodesics of our
local natural thing,
because the local national frame,
the freefall frame is the one which
is the same as Minkowski space. So.
That's why the that's why it's significant.
This diagram that,
that, that, that, that,
that's the point of the of the of this
argument here because we understand
how physics works in special activity.
We can immediately transplant that
to an understanding of how physics
works in the local national frame.
And because we can always identify
the local natural frame at any point,
we just jump up and down.
We understand how motion works.
We're definitely around a particular
point and thus by integrating that
up we understand how motion works in,
you know,
in a non impossible displacement
in our curved space-time.
So this is the first application,
if you like, of the college principle in GR.
Yeah, importing a bit of
physics from special relativity.
And I think that there's quite next we
think about that because that it is.
No good harks back to me you're familiar
with, I think from special activity.
That I at this point over over complicating
this or talking too much or this.
Have different properties and.
Without spaces.
How can you say that?
The longest distance, yeah,
because of the minus sign in this sphere.
Yes, but the sphere on the
surface of a sphere, so.
The L ^2 is equal to.
The. Theta squared plus.
Sine squared Theta, D, Phi squared.
With the plus sign.
So if you if you think that
the deviations variations,
then any what that is telling you
that what plus is telling you is that
any deviations from a straight line
are going to increase this length.
Whereas in this case a deviation is
straight line. Delta X ^2 being nonzero,
or delta Sigma squared, whatever.
I'm going to decrease that, that,
that, that, that thing. So it's this.
So the answer is the same.
And the signature,
so the the the signature of.
This officer is here.
It's the same with the signature so-called
of Euclidean space which is all, all,
all, all, all positive and the and the
signature of special activity special
activities metric is plus, minus,
minus, minus which is the same as the
signature of all of generativity metrics.
So all of the solutions to instance
equations are plus, minus, minus,
minus or minus depending depending
on your on your convention,
but they they are a ± 2 and
all the solutions of.
Of general activities,
all the solutions of our
sense equations generativity.
All the solutions describe the universe
at the same signature because they are
all mappable to the Minkowski space one.
And so yeah, so other signatures as possible,
mathematicians care about those.
As physicists, we only care about
signatures which are plus or minus.
That's. Question there. Twins.
Move along the geodesics.
Who does?
Both of the twins have, yes,
so they are both moving along
a geodesic in their frame.
So do you mean all the inertial
frame for special relativity?
Moving along the yes so because
sorry I said international so in in
each of these cases and the well so
something like like so that's the.
2 frame access.
That from there to there is also a judic,
but there's a space like geodesic.
So all all of the straight lines
in Minkowski space are geodesics.
And because of the way
this speculative works,
it turns out that there is
always a A-frame in which.
With Travelocity is not
moving in the offering.
So. My parrot.
Heard you could contact. Control tab. OK.
So that was what this this was about.
How do you go from application
application in Windows? Ohh.
Is that that's not very pages. Ohh right.
Did that illuminate? OK. And the?
I think this is a good question to to
look at the what's essentially the.
Um comma goes to semi colon rule in the
case of breach of 420? I think it was.
That's what we're looking at here.
Was and I will, I'll,
I'll just talk through this
rather than writing about it much.
That'll be this bigger search up.
And here.
So.
The way we get to this.
Expression here. Is.
We're doing this calculation
in our local inertial frame,
so we're doing this calculation
in the frame, which is.
Locally flat. And in which the
first derivative of the so.
So the metric is the Minkowski
metric diagonal plus,
minus and the first derivatives of
the metric are all zero and when we.
Do that we discover Green 419
that the reason that the the
the derivatives of Riemann.
Tanger are that there will third
derivatives of of the metric.
And based on the, the, the, the,
the expression we had for the room
intensive in terms of the metric
in the local natural frame earlier
on a couple of questions back.
And from that equation 419.
We can then play some games late into the
night and discover that there's a this
particular combination of permutations.
Of the remaining of the component of
the Ranger has it is an identity.
The various things cancelled
out so that that that they are.
The Alpha, beta are the same in these cases,
and the new new Lambda cyclically,
cyclically,
cyclically permute in the other three terms.
OK, so that is a deduction.
Fiddly but not you deeply intellectually
challenging deduction from the
expression we had earlier on in the,
you know,
a few things back from for the
Riemann tensor in terms of
derivatives for the metric.
But this is only true in the local natural.
We calculate this in the
local inertial frame.
So it's true in the local inertial frame.
Are we stuck in the local national frame?
No, we are not.
Because this involves just
single derivatives.
And in the local natural frame,
one of the the key property of
the local nature frame is that
all the gammas or the chronicle
deltas or or the chronicler?
All the Christoffel symbols,
all the gammas, are all zero.
Because the local national team is
flat and the the coordinates the
basis vectors in the local frame.
You don't rotate if you like as you
move around the around the space.
So that means that.
If we were to calculate what
the covariant derivative.
Of the remains of these components
of the human tensor.
Was.
Then we would and I got something like this.
We wrote that down and asked, OK,
what is that? How do we evaluate that?
Well, in the local national frame,
all the chronicle deltas
are see chronica deltas.
All christophel symbols are zero. So.
The expression for what this this,
this covenant drive is, you know,
semi colon U would be common
mu plus chronicle delta.
Christoffel symbol.
You know the gamma blah blah blah,
so it would turn into this.
Should be calculated that in the
local natural frame we get this
and we get the answer to 0.
So a question there.
Each term in terms of the partial.
And then you will find out.
So, so in getting to getting to
from 419 to 420. What you do is it
just write that whole thing out and
and and lots of things just cancel
so so you there just by algebra.
A bit. It's just fiddly and you
get all the things wrong.
The question is how it was the,
the, the, the, the, the smart move,
the the the speaker of the move
that gets you from there to there.
And it is that when you evaluate
that in the local natural frame,
it turns into that.
In other words,
you can go in the other direction.
You can see if we found it found
this in the local freedom,
then it would also be the case that.
We'd have this.
Because these two things were evaluate to
the same thing in the local national tree.
But this that isn't.
But this is a potential equation.
And so if it's true in the
in the local natural frame,
if those can put your components
have that relationship in
the local national frame,
then when you switch back to a
non inertial frame this tensor
equation would still be true.
And this is one of the if you like 2
separate comma go semi colon rules.
That you are exposed to this one,
which is the mathematical trick,
which is the.
The IT involves you getting out
of the local inertial frame.
And the other one is the
equivalence principle.
Who says that if you have a law expressed in?
In in geometrical form in special activity.
We are the great derivative is.
It's just with the at the
ordinary derivatives comma.
Then there's nothing more complicated
in in in in going to the.
Equals generativity.
So there's an another
remark about the stuff,
but the the the time logistics.
And our physical law expression geometrical
form is special activity turns into the
corresponding one in general activity,
the comma in social activity,
to assist the semi colon in generativity,
and that's also called the
Komodo semi colon rule.
But the two things happening there,
this one is a mathematical trick.
The other one is a is a is a
physical statement that you're
allowed to do that and nothing else.
There are no extra terms that
appear because of curvature in the.
Does that make sense?
No, no, no, no. I mean that
would take you an afternoon to go
through all the yeah, so, so I.
It's. In some quarters it is
a useful exercise to say,
be able to derive such and such.
It's sort of useful if it obliges people
to think through the notes and so on.
But most of the things in general
relativity and everything that's
terribly useful as an exercise anyway,
has or as an assessment thing.
But in most cases in general activity
is just infeasible in an exam because
it just takes too long writing exam.
Writing exam questions for GR is murder
because anything is either trivial or an
afternoon's work to to to to calculate.
Things in the middle which are,
you know, not just insulting to you,
but which are still doable in a
readable amount of time without
panic is is is tricky.
Share my pain.
I hate it. I actually mind marking less.
OK. That was. Um. So that's sort of.
This put another thing it's useful to.
Right, I'll come back to that and
useful to look at this at 3:29.
If you do anything.
After with you. And which was?
I think this would also be a
useful thing to talk through.
Yeah. And so it it. We have bigger.
So. It appears like say you can't
do the christophel symbols for the
surface of unit sphere and just asking
you to at that point turn the handle
and calculate the component of the
cover Centre in these coordinates.
So extra 3.4 we got.
I'll even write them down on this
bit of pig paper. These would be 4.
Just for reference in a moment.
And.
We we had a G Theta, Theta equals 1G Phi,
Phi equals. Sine squared Theta.
And other zero and the because that's
diagonal the components of the. And.
And.
Inverse matrix. The matrix with
indices raised is just one over those,
because the, the, the, the, the,
the those two matrices are diagonal.
And the Castro symbols we worked out. Where?
And.
In one may have worked out in
that where if they're nice neat
table of them, yes there is.
That. Gamma. A Theta Phi Phi
is equal to minus sine Theta.
Cos Theta, gamma Phi.
Theta, Phi which equal to.
High Phi Theta equals.
Same teacher and was there another one? Yeah.
I think the other ones were all zero. Yeah.
You know. Thank you.
So that's that that I'm just copying
and actually 324 at that point.
So calculate the components of the
coverage sensor for these coordinates.
So what's R? In this case,
and what we we know is that our.
The answer is. I think it's.
349.
Yes, I've got that place, 340.
It is this.
So it's a fairly messy thing
which involves the Christoffel
symbols and derivatives of them.
So if we ask. If we.
Pick out one of the components of
the human tensor. I think what I.
What I'm picking here is R. Utah Phi.
Peterffy. Then that's going to be.
G Theta Alpha, R alpha. Phi.
Teacher's fine. So that.
So for instance this,
and we obtain that by doing that contraction.
And looking back at it agreed to 49.
That's going to be G. These are alpha.
Ah, right, that's great.
Tedious gamma alpha.
Chucky and five. Fiji.
Comma, Theta minus.
Comma I was alpha. And GQ.
Phi, Theta comma. Phi plus. Alpha.
How can we you? Why don't I
just copy that down rather than
trying to to to to do it? Sigma.
It. Sigma. 5-5 minutes. And.
Alpha. Sigma Phi. Sigma Theta.
Bye teacher. And. So all I'm
doing here is looking back at.
This expression here and so
filling in the filling, filling,
filling, filling in the slots.
And he got a bit of a mess,
but I can do the alpha summation immediately.
Very quickly, because we know.
Yeah. That. G Theta Phi.
Is 0. And GT to Theta is 1,
so the alpha will be the the some
some there where alpha is equal to
Phi is 0 and what we're left with is.
Just that expression there with alpha
turned into Theta. And so we have.
Theta, Phi, Phi comma, Theta minus.
The Phi, Theta, comma Phi plus.
Sigma Theta. Sigma Phi Phi minus Theta,
Sigma Phi, Sigma Phi Theta.
And then we look back at.
This expression here.
And we discover.
That.
And.
That's. With this.
Did I see that was going to
be for that with that teacher?
That was theater 55. And.
2424.
And. If you don't, yeah.
Three to five Phi. At the.
Teacher is going to be.
Minus Cos squared Theta. And.
Plus sign. Squared Theta.
Um Theta 5 Theta. Is is 0.
Because it's one of the others 30 here.
Theta. Something Theta
is a is is going to be 0.
For any for any any Sigma because
it's one of the others are zero.
Phi, Theta something.
Phi is going to be nonzero only if
when when Sigma is equal to. Phi. So.
That the only time that was five from
the sum and that would be minus gamma.
Peter, Phi, Phi. Phi.
Five future. Which is.
Equals to future Phi.
Phi is minus sine Theta Cos Theta times.
Costita over. Sine Theta.
That gives us our. Cost square Theta.
My mosquito it so that cancels with
that we end up with sine squared Theta.
I'm not impressed about that.
I didn't make a assign error there.
Immediately impressed, actually.
But the point even if I had made
a sign error, then the point of
this looks rather more forbidding.
The thing that needs practicing here,
the thing that something like that size is
the sort of thing that I think would be,
is the sort of thing that would
fit in an example question.
That's not a promise,
but that's the sort of side of thing that
that that fits in exactly that question.
And it's easy if you've practiced
them and you've you've failed to.
So you've managed to avoid just
losing track and getting exposure
of of signs all over the place.
The practice comes in in in saying I,
you know, I spot that zero,
so I don't have to do anything else.
That's zero for all the all the
possibilities of stigma, this one is.
Is nonzero only when taking equal to five.
So which to identifying which
terms survive the sum and and and
just keep track of the algebra.
So an easy question is if you mark
if you have practiced it because
it's easy to understand,
it's easy to understand than to do
on the on the hoof, as it were.
And that's really one of the
components of the Riemann tensor.
There are what? Two by two? By two by two?
There are 8 components of the of of the.
Or even tension. So what are the other ones?
Well, that was our Theta Phi.
Peter fight.
But we know from the symmetries of the.
Remind tensor that were noted down
that is equal to minus R, Theta,
Phi, Phi Theta.
We were just swapped over the last two.
Terms is equal to minus.
Are Phi, Theta, Theta,
Phi whose walked over the 1st 2?
And it is equal to the swapping over
the first pair and the second pair.
That's just gets back gets back
what we what we started off with.
And what I'm looking at there are.
And.
And the two.
These things here so the the.
The symmetries of the human
tensor which are obtainable
from this expression in the.
Local national frame.
Which is and and and that
expression there is not.
We can't turn the this comment
into semicolons in that case.
Can anyone see why not?
In in this case here, why can we not
just turn those comments into semicolons?
It's Robert geodesics.
It's because the second derivatives.
Because the it's that that
is not an obvious remark,
but these are second
derivatives of the metric,
so if you think of what the the.
colon MK would be it would involve
terms which were, you know,
comma M comma K plus derivatives
of the Christoffel symbols as
well and they don't necessary.
And the derivatives of the christophel
symbols don't necessarily disappear.
So that by itself can't turn doesn't is is
not the local national frame version of our.
Covariant derivative but this one here.
We can use that to calculate
these symmetries here.
Because these symmetries don't
involve any current drives at all,
their attention equations again.
Even though we covered them in
the local national frame and so.
Be intentional equations.
We can say they they are they
are true in an arbitrary frame,
but that's a long way of pointing
out that that that these particular
index swaps give us information about
what the other things are here.
If we look at something like our Theta.
Theta, Phi, Phi.
We can.
Know that that change is sign if we swap the
last two indexes so that is equal to minus R,
Theta, Theta, Phi, Phi.
And which means there's of course
there has to be a has to be 0.
So although there are two terms,
2 instructions,
2 components of the Riemann tensor in.
On the surface of the sphere
in two dimensions. And.
Most of them are equal.
Some of them are zero.
The long way I have to actually calculate.
Is a sample one a sample code 01
which we do by doing this relatively
simple once you get used to it some.
Did that make sense? So anyway,
in, in, in, let's go back to.
So with that, yeah, because this is
the excitement of mine in a way.
There's slightly less to that
exercise than meets the eye.
It looks so forbidding, but it is,
it's actually quite useful exercise
to go through step by step.
And you've seen me do this here,
go off and do that again.
You've in a sense we,
we from a standing start and and and and
and and and rehearse that sort of thing.
It's just a matter inject gymnastics.
Put the hand together as I thought.
OK, question there.
The matrix. The matrix, sorry.
And. Well, we have, we have,
we, we, we, we did there.
But we we sum that over alpha that was that.
That was that with alpha equals Theta and
that equals Phi but the G Theta Theta.
Is 1 and G Theta Phi is 0 so the
second term with alpha equals Phi.
Is you know this is something
they multiplied by zero,
so it just disappeared.
So that's how we got from there
to to to to to there that that
that is the answer to that sum.
Again, something that's really
obvious once you've done once,
you've got through it by hand once.
With the water of that exercise,
I think was a was a sort of
codicil to that exercise. Umm.
Why would you not use equation 355? Um.
Hi yes so this expression here
looks like it's a a a shortcut
to what we calculated there,
but that applies only in
the local national frame.
So the the the which is
office of the tangent to the.
But in this context that applies
only to the tangent to the
the the surface of the sphere.
But what what we what we want is the
agreement tensors in the in the on
the on the surface of the sphere.
So you had waiting yesterday, but.
Uh. So more than with the pilot
as I speak, speak faster.
Ohh, thank you. Right, good.
That was a an e-mail question.
So this I think that that that's
a natural move on to what the
richest tension scaler and.
Mean and I think that this naturally
follows on because ohh yes.
So the rest of that of that question 329 and.
Had eyes calculate what the?
Richie Tensor was in these these on
on the in the in the space and these
coordinates are are Theta Theta equals one.
Are Phi, Phi equals sine squared Theta and?
Now I'm afraid I'm going to
give you a slightly hand
waving answer to this question.
What does what do these things mean?
Because physically.
I I think I I should be able to find
a deeper way of of saying this,
but if you look at those.
Those expressions then what you. Can see is.
And.
Yes, that's not a very pretty.
That's going to be. Five.
And so that is going to be.
Teacher. What is?
And that's going to be. So if you
look at stephco pullers. Then. The.
The fact that as you. A change.
Fi are you change change Theta
I as you change the the,
the sort of longitudinal coordinate, the?
The the volume element,
the area element at that point is,
is is is proportional to the changes
in proportion to the site to the the.
Change you make in the. Theta.
But as you changed Phi as
you open up that that angle,
the area element in that little block.
There.
Changes in proportion to the sign
of the of the of this angle here.
So it what the the the Richard
Chance is picking up is the the
sort of sensitivity if you like.
I'll just be that I've just was just
popping about the the the sensitivity of
the this area element to the changes in
depending on where in the coordinates
in the coordinate surface you are.
And. So it's.
Another slightly hand waving.
We think the Richie tensor is giving you
information about the way the coordinates
change as you move around the space.
In a way that's useful for the sort of
sort of calculation the Riemann tensor is.
Is giving you more information about exactly
how you how how a vector will change.
All its components will change
as you go in in a,
in a circuit or in a space,
but the remains the richest center
is of summarizing that to the
sort of need to know bits.
For doing calculations on the
surface of the sphere in a way.
So there's more in the in the Riemann tensor
than you sort of need for calculations.
And the richer tensor is the
encoding the the the behaviour of the
coordinates I think as you move around.
And the and in this case so, so.
So the way we we we we calculate that is.
Well,
the way we we we we calculated by
seeing our Theta Theta is equal to and.
She's.
Um. Could you be G?
Alpha. How do I write this? And.
Yeah, so
RGL is equal to G. Key R.
IG KL so R, Theta Theta is going to be G.
And. Let's see Alpha, beta R.
And. Of. Theta, beta. Teacher.
And that will only be.
Which would be equal to G
Theta Theta, R Theta, Theta,
Theta Theta plus G Phi Phi. Are.
Phi, Theta, Phi Theta.
And since that is equal to 0,
that is equal to. Umm.
1 / 1 over sine squared Theta times.
Sine squared Theta equals one.
That's how we calculated these.
And the curvature tensor,
I'm not gonna go through it.
Similarly, ends up covering scaler.
Similarly, ends up being a number.
Which is constant over the whole
surface of the of the space.
So the, the the the the sphere
has a constant curvature,
the same curvature at at at every point.
The, the, the, the behaviour,
the sensitivity of ordinary element
to the the position on the sphere
varies as you move over the over the
sphere in ways that this picks up,
but the but overall curviness of the
sphere is is the same everywhere.
The, the, The you know the the
units of of of that culture are
not particularly interesting but.
I would like if you have a
space that changes curvature
then I would not be concerned.
Yes so I speak so sorry.
For example so see this with with
the the surface of an ellipsoid.
Then there are would have a probably a
variation on depending on the on the on the,
on the feature, the feature parameter.
Yeah so there would be it would
be a coordinate dependence to
the to the curvature.
OK, so talk faster, talk faster. And.
Right, I think.
I'm not going to mention that
just now because that section 413
is the is the alternative way of
getting the engagement retention,
which I I isn't that dangerous band thing,
but which I sort of wanted to give up,
give bodies worth to.
I can put pick that up in an office hour
if anyone wants to grab me next Thursday.
Like I said, this point here
is picked up by the stuff in.
Part 4. Which? Is. Um. 4.
Yeah, the stuff in 413 is quite nice,
but I I, I it's, I want to think
of it and it's quite nice.
I don't want to put a dangerous
bend thing there to discourage
you from have a look at it,
but I sort of don't want you
to spend too much time on it.
What I said back here was.
There's this thing here.
So the question. Was. With the.
Right the the the the audio component.
But when I'm saying so,
I think this is essentially it'd
be useful to to. Pretty good.
To. Go go to these arguments.
I again. And no.
And. Trying to recall the steps
just before this, so the.
The the. It seems comes
from from rearranging this.
This version of the components in this
frame here so the the rule comes about
from the consideration of the the dust.
So the energy momentum in in in in
this box just sitting there moving
forwards into time is dependent on just
how much mass there is just so much
material there is in the case of dust.
Dust remember is this idealized thing
which is isn't moving it there's
a frame in which it isn't moving.
So there's no.
Contribution to the argumentum
from from that.
There's no, there's no internal,
there's no internal shears.
It's it's a perfect fluid.
There's no internal stresses and strains
and because it's not moving inside the box,
it's not banging off the edge
off the side of the box.
So it's not creating any
pressure in the in in the box.
So the only source of energy momentum in
that notional box is due to the the mass,
the mass density move forward
into the future.
And that's so that's where the the the.
The the, the,
the the T00 component comes from.
I have to just take the the the energy
density in the in the box which is is in
this country the same as mass density.
And. This.
This question of of why the?
Engagement engagement tensor is
proportional to the unit unit matrix.
Is um.
And I have to find a better
way of explaining this.
Because one of these things
that when you so feels right,
but I'm not sure how I would
go through the steps to to
to to to to expand on it.
If it isn't a preferred
direction, then the that.
The the that tensor had to
be rotationally symmetric.
And if there's no shear,
then there can't be any.
Contributions to environmental
push depend on as something moving
in this direction across a plane
in a different physical plane.
Um, and so you can so I
see this the the expansion,
I should tie down a little more.
Fundamentally, it's the rotational symmetry.
Rotational symmetry of of that situation
means it has to end up being diagonal.
And and and so if you've got that far,
then the rest is just sort of algebra
in the sense that you can expand
that or you're adding subtract this.
Matrix here.
Turn that into something which is.
Purely temp time like I think at
least one question, 22 questions.
Also important because you have the matrix.
They don't understand.
Yeah so so this we've just because
we've we've we've taken that out
into two matrices added AP here
into practical corresponding P
there purely so that this is then
proportional to the the the metric.
So that the so that ends up
being paintings the metric.
And in the local national freedom,
that means that being proportional to T
times the velocity 4 vector in that frame.
Therefore we can write this as
an in that frame.
In the local inertial frame,
the we just got we can write this.
So we can write down this in the local frame.
And because the at that point
at the local dash frame,
the still instantaneously
call moving reference frame.
And because of tension in that frame,
it's a tension equation everywhere.
So that's our. That, that,
that, that is quite a lot
happening in that paragraph.
There's a couple of paragraphs,
there's certain amount of of of
of of physics motivated see to the
pants hand waving combined with
a certain amount of, you know,
like cunning in prisons up right away.
Combined with.
Identifying the fact that in in this
moment helical movement reference frame
these vectors take a nice simple form.
Combined with turning this into a matrix,
therefore turning it into a
matrix tensor equation in that
local moving reference frame.
At which point you realize that
once you've written equation,
you've written it down quite quite generally.
So there are a number of different steps
using a using everything from physics to,
you know, dumb algebra.
In that passage.
Which because.
With the general through
between everything, yes.
Pretty specific, no?
If a tensor equation, geometry,
geometry because it can't be
from specific the the components
might be worked out in A-frame,
but the tension equation,
because it's purely topic geometry,
it's really dependent has to be reminded.
OK, that was more talking
than I braced myself for,
but I'll see you.
The last lecture is on Wednesday,
so I shall look forward with
the contribution to your
right shining faces there.