Hello everybody.
This is later nine.
We are remarkably enough and I
cannot congratulate myself enough
for this on time, which is very good.
So we have a full 3 lectures to cover
Part 4 which is back to the physics of GR.
So the intense the intellectual payoff
of the. Of the weeks up to now.
Before we started on that, however.
And according to supervisions,
for historical reasons,
the way that the general course had
been covered by provisions has been
different from the other courses.
It's been a separate.
You have supervision sessions rather
than small groups of provisions, but.
There's some flux in the way
that these will be managed,
and it has occurred to me that
it might be useful to have it.
Haven't previously done office hours,
but either an office hour
I could announce or else.
As some others have done for the Honors
courses bookable slots at a certain time.
Does anyone have any feelings about this
or any suggestions on this sort of topic?
Is that a good idea that a bad idea?
Of those two possible models,
which one is is an obviously better one or?
And. Yeah. OK. Business.
And think otherwise.
OK,
the world is I'll I'll identify a
suitable time and I don't really know
your timetables are the afternoons out.
Or other mornings out? Or is there a?
Is there an obviously bad time to
organise to just an officer? Anyone.
I don't know your timetables.
He's normally bad 10 to 1225 bad.
OK, so if I said picked one o'clock one day,
would that be a suitable sort of?
Seems of notes OK I think.
I think with that and sort that out.
I think that is all.
Are there any other questions or thoughts
or anxieties to do with to do this that
you have anxieties not to do this are?
Excluded, OK and let's get going and I will.
Before I I said part 4,
but before I do that I will just
mention a couple of things about
the the the very end of part three.
At the end we're racing for the,
we're racing for the finish line as it were.
And I I I, I I skipped through the bit on
on Judy Deviation but there's a couple
of things I do want to just mention.
I won't work through the.
The derivation of this or the corroboration,
I'll ask you either to look at the
notes or just take it and trust
the point of duty deviation. Is.
You might remember this picture
from part one.
We talked about the idea of of two things
falling toward the center of the Earth.
And because they are falling
to the center of the earth.
The heading toward the same point, and so.
Although they are both in freefall.
Each of the two observers in
at either end of that well see,
spaceship or whatever,
we'll see the distance to the other
one decrease at an increasing rate.
They will be strictly in quote,
accelerating toward each other in
the specific sense that the second
derivative of their separation
will be nonzero.
But they will not be accelerating.
They will not be feeling any push
if they were holding a Plumb Bob,
but that wouldn't work.
If you're holding some sort of
accelerometer then they wouldn't
detect anything as they were
falling even though they were.
There's a second derivative
being non zero happening here,
and that is because these two observers
at either end of the spaceship or
train carriage or they want to call
it are both following geodesics.
But those eugenics,
the separation between those judaics,
is changing as a function,
as a quite complicated function of time.
Because of the curvature of the
space they're falling through.
And we're going to talk about the
relationship between curvature
and gravity in a moment.
But so just like jumping ahead of ourselves,
the gravity that the this earth,
this planet here. Has is for.
Has the consequence that the
space-time around it is curved in
a way we'll learn about shortly.
And so as those particles followed
udic through it,
those geodesics are straight lines in
the space, but nonetheless curved.
And so it's tidal effects like this,
so-called tidal effects that are where and.
The effects of gravity emerge.
That's the picture I used to to
set up the scamper through the
last couple of pages of the note.
The idea was, you recall,
that there were multiple geodesics
and were able to.
This is another version of what we
we what we stepped through fairly
painfully in the last lecture of taking
our vector for a walk around a closed path.
In this case, the end result of that
calculation was what would have been
if we'd gone through step by step,
would have been a calculation of.
So much of this connecting vector at the top,
which links points with equal
parameter on the two geodesics.
How much that changes and the answer.
Jumping to the end.
Was an expression like this,
which I, I, I, I, I, I,
I want to put up here so that I
can refer to it later on when we
when we discover when we can use
that expression in a simple case.
All it's saying is that the second
derivative of the separation
between the the the the the points
on 2 neighbouring diuretics.
Separation is a function of.
The tangents to the duties.
The separation itself,
you know we're pointing here to here,
and the Riemann tensor,
so the so that the reason why
this emerges naturally at the
end of Part 3 is this is another
case where the Riemann tensor is
telling us what we want to know.
About the shape of the space
that we're going through,
and this this time it's telling
us stuff about the change
that this connecting vector.
This.
This connecting vector here,
how much that changes because of the
curvature of the space that these
duties are limited are limiting,
so we'll see an example of that.
Do you do you, do you geodesic
deviation equation in not let
me lecture after this one when
you will be illuminated, I hope.
Um. Key points. Any questions about that?
So I I acknowledge I have
effectively missed a bit out there
because because the, the, the,
the details are less important
than the the the conclusion. So.
Onward.
So here we're back to physics. And.
The two things I want to to to,
to cover really in this part are first
of all the equivalence principle.
We touched on the equivalence principle in
part one where I said that all free falling
inertial frames are equivalent for the
performance of all physical experiments.
I think I actually said that in part one.
If I if I didn't then prior to
have done but you, you, you,
you probably have seen that express
somewhere and we find that the
we're able now to put some.
Mathematical flesh on the physical
bones of that of that statement,
that of the code principles.
So we see where the as it were to
use an overused image with with.
This is where the rubber hits the road,
as it were.
And.
My goal,
the overall goal of this of of G1
is to get to the point where we
can write down Einstein's equations
and say that's the question,
now go and answer it.
But it's unfair to do that and not
give our solution to IN equations
and which describes our space-time
of interest to us.
And the one we, we, we,
we,
we look at is how we can recover
nuisance theory of gravity from
Einstein's theory so as the
low energy and low speed.
A bit of of of a against alien
theory of gravity.
Um.
This is challenging, as I've said before,
the point of the the objectives and the
aims are closely linked to assessment,
and this is challenging to assess because all
the calculations are hideously long. So the.
The assessment tends to be explained X.
And, you know, show that you understand X.
Basically, you persuade me that
you have a clue type assessment
rather than other things.
But there are some simple dynamical
calculations that you'll be able to
do in certain contexts, which is.
So there's a there's some simple calculations
that are possible at the end of this.
Um. OK.
And you're in Newton's gravity.
The source of gravity is matter.
Is mass. The the the one of the
terms in in in in the theory of.
Law of universal gravitation or two
of the terms are EMS, they're masses.
So it's not surprise,
no surprise that it's a mass that
is that we have to to to think about
and deal with in this context.
However, let's be relativity.
We can't talk about mass.
The natural thing is talk about energy,
momentum, the so it's not
just the mass of something,
but the energy the intimate
has by virtue of his movement.
So when something is moving rapidly.
There's more energy momentum in
it and so it gravitates more is
what we're going to discover.
We shouldn't be surprised at that,
because if the energy momentum the, the,
the, the, the, the momentum 4 vector,
that is the key,
the thing that's important in
your study of relativity.
And the way we we edge up.
So we'll have to learn how to
discover how to talk about the
energy momentum in extended objects
in a rustic relativistically
satisfactory we we would do that,
we would talk about dust.
Which are beautiful you I want to just
check I'm not missing something out here.
I've I've I've notified of some remark
I might want to make sure I made early.
Ohh yeah.
So just a minor thing.
And I at this point up up up to
this point we've been talking
in about N dimensions.
Of course,
the reason we're talking
interventions is to turn into n = 4,
and so from now on we're talking about four
dimensions with signature of of space-time,
which in this case would be minus,
plus, plus, plus.
And I'm not going to use,
I'm going for indexes.
I'm going to switch to using
Greek indexes for space-time
Dimension 0123 and Latin ones
for it's space only ones.
That's just a slight notational
tweak to fit in with the conventions.
OK, so just. Matter is structured.
There are all sorts of things that.
That their whole disciplines
devoted the structure of matter.
That's complication.
You can simplify that and talk about
a perfect fluid which something with
no viscosity but which has mass,
mass, density and has pressure.
You can simplify that so further.
By talking of our of of of dust,
which in this context is something which has.
No internal structure,
so there's no viscosity and it's all,
there's a there's a frame
in which it is at rest,
so there's no pressure.
So and the idea is a collection
of dust particles.
So this time we push has mass density.
And nothing else.
So let's imagine our box of this.
There's a box of what?
Plenty dot X dot Y dot Z. And.
You can imagine it being moving at some
constant speed with respect to it,
or or we moving at some
constant with respect to it.
Now in that box will therefore
length contract as your
recollection or special activity
will will immediately remind you.
So I want to go through
this in careful order.
So the. And. The the volume.
Ohh of in this other in other frame
we just X prime delta Y prime delta
Z primed with an extra factor of.
The. Look at this update 10 no.
With an X Factor of gamma.
But that's the usual range factor.
So the thing will length contract to the
volume will go down by a factor of gamma,
which means the density goes up by
a factor that the number density
goes up by a factor of gamma. Um.
So if there is a, if we look at this
and ask what is the flux of particles
through the end wall of this with
this thing is moving with respect to.
I think the perspective is wrong in that.
I think it's actually I need to
redraw that I cannot tolerate
perspective being wrong and then
the the flux of particles through
the the the end wall will be the.
Volume of this X to the Y.
Doctor Z.
Divided by divided by.
The flux is the number per
unit area per unit time.
Which they're going to end up with a flux.
In the X direction. Being.
And.
Yeah, but I'm gonna let you
jump to the end being gamma.
In the X where N is the.
The the number density in the rest frame.
Turn off auto focus. OK.
And. One can go through that in
more step by step, I hope you you,
you you agree that's that,
that that that's reasonable.
The point is that that that the
number density is proportional to
the velocity but with an extra
factor of gamma because of the
of the increase in the number
of density because of the length
contraction of the side of the box.
And. That looks sort of familiar
that that that looks like.
N times.
Gamma X and so we can jump to the
conclusion that the there is a.
Our flux vector here.
Which is the number density times the.
For velocity of the.
Or the motion. So we're not just,
it's not just general,
not just specific.
This is to the X direction.
We'll jump to that conclusion.
I mean, it will turn out to
that as a sensible thing to do,
as what I'm doing here is
motivating this as an idea.
So that's the.
Flux factor.
And the components of this.
Are going, they're going to be MN.
Yep, gamma, NVX, gamma,
NVY, gamma, NV. Is it?
How do we extract those components?
Of the of the of the of the flux
vector in exactly the the usual
way by plugging a what one of
the basis one forms into the.
Into the. It's ****** factor.
That's ****** vector. And.
With the alpha basis one form
plugged into it. And here as I say,
I'm switching to using the the
traditional thing of using Greek
letters for the the 00 to 3 indexes,
and I'll later use Latin letters for
the spatial components, so that's.
I'm just recapitulating stuff that
you've you've seen before about
the way we extract components or
vector from the vector itself and
and one of the basis one forms.
Umm.
So.
We've here talked about the change in the.
Numbered entity of the particles in the
box and that number density has gone up
because the size of the box has gone down.
Usually retract. However,
there's also a because these
these these these dust particles
in the box are moving at speed.
The energy that they have.
Is acquires another factor of gamma.
So the energy of the particles
in the box. Uh.
Let's write down the, the, the, the.
Energy ends up being gamma
N the increase to the. Yeah.
The increase in the number density.
And each of these particles has a mass M.
We inquired another gamma the energy in the.
Of the particles of the box has acquired
a gamma squared and that is a hint
that we are not going to be able to
describe this the energy of the particles
in this box by a vector because.
The. But by a rank one object,
because the transformation of a rank
one object from frame to frame would
only pick up one factor of gamma,
whereas it's clear we need two,
so we're going to need a higher rank object.
What do we have to to build
that higher rank object?
The things we have available are the
momentum 4 vectors of the dust of the dust
particles and the overall flux vector.
So the the the metal dust particles is.
It is telling us that in the in
another frame the momentum of those
dust particles would would change and
we also are told that there is a an
overall number density corresponding
to the flux vector which will
also change and so those are the
two vectors we have to play with.
And. So what we can do is.
Guess and here are motivating
again rather than deducing
that there is our tensor.
Which is the. The.
Momentum for vector of the dust particles.
Outer product with the. Sorry.
The. Flux 4 vector of the particles
as a whole. Which is going to be.
M. You. Cross. And. You from above.
Which is just. Role you.
Because you were rules. You could.
Amen. And I guess it's obvious M here
the rest mass of the particles and N
is the rest density of the particles.
In, in, in the dust.
And we can extract components of this.
Kendra in the usual way by applying.
Thesis.
One forms to it. Which extracted which
ended with being P. The. Xmu. Times.
In.
The. X. You.
In again illustrating the way that the.
The. We evaluate a natural product.
By plugging the. In this case,
one forms in turn into the two
components of the outer product,
and simply real number
multiplying the results together.
So that's that's that's an equivalent
saying rather than an equal sign really.
Did that. I don't see any looks
of shock and horror. That's good.
So the so the thing don't need to do again.
Reassuring you that the sensible
thing to do is to start looking at
individual components of this object,
which is the energy momentum tensor or
the stress energy tensor of the dust.
So this is the tensor which I assert
is characterizing the energy and
momentum of the dust as an assembly,
which we are going to discover is the
source of the gravitational field,
gravitational field that the the
the change to space-time around it.
So let's look, for example, T00.
There's usually a component. That um.
P.
DT times N. DT. And the.
The same component of the Momentum
4 vector is gamma M If you
recall your special relativity.
And the time component of
the flux vector is gamma N.
Which gives the.
The. Term that, that,
that that we sort of in a slightly
hand WAVY way deduce earlier on.
So that's reassuring.
Um, the.
T0Y and again this is me using
Latin letters to indicate the
space components of the tensor.
That's so is 1 two or three?
It's going to be P.
ETN.
The XI. Which will be gamma M.
And gamma. And VI.
And if we, you know that's different from.
Here. And we look at that
that has the dimensions of.
Mass. Per unit area per unit time.
So that is the. And match being in
this context the same as energy.
That's the energy flow per
unit area per unit time.
So that's the energy flow across.
A surface of constant X per unit time.
So that's the essentially the mass flow.
That's been. The represented by the.
The mass flow of the dust. And.
That we can have a similar sort
of thing argument for what TI0 is.
That's the the spatial component of the
momentum and the time component of the dust.
And you can. It's supposed to reassure
yourself that is the same as.
At T0I you know there was this
tensor is symmetric, turns out.
And we can look at T IJ so the space
space components of the tensor.
Which will be equal to P. The XI.
And. The X. G. Uh, which will be um?
M. Gamma M Phi I times.
Gamma
NVJ which basically from the
dimensions ends up being the.
The. Flux the flux of momentum
per unit area per unit time,
which ends up being the force per unit area.
And. There's typically argument argument
here but but that that is picking up the,
the, the, the, the,
the the force that there would be on a
wall of if if this body of flux but the
this body of dust particles hit a wall.
And that's that's. Force.
Per unit. Area. Umm.
So more more generally,
what we're doing here is we're
interpreting the various components of
this tensor as the flux the way I put the.
And the flux of the the the T Alpha beta is
the flux of the alpha component of momentum,
so energy or momentum across the beta.
Our surface of constant beta the constant
constant time I into the future or surface
of constant of constant position, IE.
The flow you know through the space.
OK. I'm aware this is all
slightly hand waving,
but it's I as I say again,
it's intended to motivate
the the identification these
things rather than reduce them.
Now I said this was a perfect,
this was just it had no pressure.
If we relax that slightly
and talk of a perfect fluid,
a perfect fluid has no preferred
direction so that the spatial part.
So for the for the incrementum
tensor describing that perfect fluid
therefore cannot have any asymmetry,
must be symmetrical.
All directions in the all spatial directions.
So that for a perfect fluid.
And the.
IG components the spatial.
Spatial sector of the energy momentum
tensor has to be proportional.
To the to the identity
matrix and by comparing what?
What we've just seen is that
it's plausible to identify the.
The the spatial part of the
energy momentum tensor as the.
And at having the dimensions
of pressure per unit. Time.
Of of force per unit time,
which is pressure.
Then the spatial sector.
Is it good to pee?
Delta IG. And for a for dust put,
the pressure is 0, therefore the spatial
sector is going to be 0, so it's
detrimental density is is very simple.
Umm.
Key now.
In the.
And.
I. OK.
Through for further. So yeah,
like hand waving, pushback.
I don't like this hand waving,
but it's, it's, it's, it's sort of
necessary to get to the the, the,
the end point we can end up concluding.
That.
And.
For a perfect fluid.
That the form for the energy momentum
tensor of our perfect fluid.
So that's something which has no
viscosity but does have pressure.
Is ends up being a very simple form,
which is just that you cross you.
With terms involving the mass
density in the restream. The.
The pressure, so P here is pressure
rather than momentum slightly
unfortunately with an extra term
which is the just the metric.
And in the case of dust
where the pressure is 0.
That ends up being simply rule. And.
000 for justice.
In specifically the momentarily comoving.
Reference frame MRF.
So that's a long way of of saying
that this energy maintains a
clearly can be quite complicated.
There's also things that could be in there.
But the key things are the.
Masters there.
The actual objects there,
which are expressed in things like
the density, the mass density.
And other motion,
other sources of energy such as pressure.
Which appear when in the case of a
fluid but not in the case of dust and
in the case of of the of the of dust,
that energy metric tensor becomes,
you know, reassuringly simple.
It's it ends up with only one known 0
component in the the framework which
is not moving and Lawrence transform
Lorentz transform into other forms.
Similarly that's.
So this is a diagonal matrix there.
OK,
going on to slightly more substantive things.
No.
Section 412 is another way of arriving
at that sort of sort of inclusion,
which is is more satisfying in some
respects and less satisfying others.
This way of of getting
to that conclusion does.
It celebrates our return to physics
here by by returning to slightly
heuristic physical arguments.
The other way of approaching this is to.
Talk about the.
The way in which you can describe
volumes and describe volumes in
a geometrical satisfactory way
within relativity using one forms
and two forms and similar things,
because one forms as the the clues in
the name at the bottom of a tree of
more complicated geometrical objects.
There's nothing else I want to say about
that other alternative approach to this,
but I but it doesn't have a
dangerous because you might be
interested and it's sort of relevant,
but I'm not gonna say
anything about it just now.
And.
OK.
Would you were doing well?
And now what we're going to do is
to actually talk about the laws
of physics in curved space-time.
So this is really where
we start to discover how.
How all the maths we've done
so far links to to physics,
what we want to understand.
Before we do that,
we ought to make a detour back into
into math by defining defining
a couple of extra objects. Umm.
But the they're fairly routine things,
so they are just contractions
of Riemann tensor,
which turned to be of more ready
significance. So #1 is. And.
What? Is it the remote sensor?
Alpha Beta you knew and contract
it over the 1st. And 3rd indexes.
Then we get an object with a.
We've got a rank two object.
With with the two unmatched indexes,
which is called the Richie Tensor.
So just a contraction of the of
the Riemann tensor you plug in,
plug plug two of the
components into each other.
If we further contract that
and take the the Richie tensor.
And contract over those.
Remaining 2 indexes, then we get a
scaler called the Richie Scaler.
Um.
And this from the from the
symmetries of the Riemann tensor
that we saw briefly last time,
we can deduce that that.
Richie Tensor is symmetric,
so R Beta nu is equal to RU beta.
By we also last time discovered that
in the local and national frame.
That is an explicit expression for the
Richie tensor, which is in terms of
derivatives of the metric. So if we.
And this is not obvious if we ask what about?
Peter. You knew? Comma Umbra and
and differentiate the Richie tensor
with respect to one of the other.
Yeah, according to functions.
And. Do so. Alpha beta.
New Lambda or new? Ohh. Of a beta.
Under you, you we discover.
Or rather, Yankee discovered.
That's zero, that there is a.
Asymmetry of the Riemann tensor,
of of the of the of the of
the differential ruin tensor.
So so these this is the each
of these expressions is a mess
of derivatives of the metric,
so I think it's not too hard to
work that out for yourself to to
to to do the calculation yourself
and discover that it is all cancel,
but it's not terribly exciting.
But this is being done in the.
Local, natural, free.
In the local inertial frame
because the expression for the
Riemann tensor in terms of the
metric only applied in the in
the in the local metal frame.
In the local natural frame,
however, these are.
In the local natural frame
covariant differentiation.
Is the same as ordinary differentiation.
So in that frame it turns out that we can
write turn these commas in semi colon.
This is a trick we did also last time.
Because the in the local inertial
frame the the the gammas that appear
when we go the other way are all zero.
But that is a tensor equation.
So that although we worked
out in special frame,
we've worked to attentional
equation in that special frame.
Which means that it's not just true,
it's it's it's true not just
in that special frame.
And and this. Expiration here is
called the the Bianchi identity,
and it is used for at various times
in the algebra of doing the the
the calculations here to to to get
rid of terms, rearrange things,
and make things disappear.
And if we do this contraction
here on the UM bank identities,
we get similar one which
I'll just write down South.
It's on the sheet due to the new.
Semi colon Lambda +3. Minus R.
Beta Lambda semi colon nu plus R.
Which new lambdas? And accordingly
you contracted Bianchi identity.
And if we can track that again.
We get an expression G alpha beta
semi colon beta. Equals zero.
And you know I I'm missing out a
lot of very boring algebra here
where this could this tensor G.
Is. Formed from the Richie tensor.
And the metric and the Richie scaler.
And this tensor here.
Has this property that is that that
that as a deduction of from this
that is convenient derivative is 0.
And that change is called
the Einstein tensor.
And as you might not be astonished
to discover, that tends to plays
quite significant part part in our
in what we are about to discover.
But the what?
The what the the Riemann tensor? Uh.
There are a lot of tensions happening here.
The Riemann tensor.
Is the one that we discovered contained
all the information about how a vector
changes as it goes around a path.
So you you you plug your your question
in this path that this vector go
around this path you plug that that
that data into the rementer and
what comes out is an is information
about how much that that that
vector vector changes and or else
you talk about the tangent to two
geodesics and the link joining them.
You plug those in and what comes
out is information from the Riemann
tensor about how much that.
That, that, that,
that connecting vector between these
two things, how much that changes.
So there's a lot of information in
the Riemann tensor, the Ricci tensor.
Has fewer degrees of freedom.
A lot of the of stuff had been contracted
away so that that there are animatrix.
There are 4 by 4 components
to the Richie tensor,
where there are 4 by 4 by 4 by 4
components to the Riemann tensor.
But so, but the the virtue change is,
in a sense, all the bits of of information
in the human density that you care about.
It turns out that that that's the bit
that matches most closely the stuff that
is informed of to us about the curvature
of the space-time going through it.
And this Einstein tensor is just a adaptation
of tweaked version of that Richie Tensor,
which is the one with this
interesting property. Which?
Is of significance to us.
And and having. Carefully got to the.
One reason why it's important to get through
this step through that section 1441.
To an incremental tensor.
The previous thing is that the the sort of.
The the punchline of the.
Of the discussion of the elementum tensor,
is that the argumentum tensor?
It's such that. For.
So on physical grounds,
based on the idea that if you have a box,
the total amount of going
into it must be this.
Arrangement going into it must be
the same as the total amount in the
box minus A in the coming out of it.
And a mathematical statement of
that is that the? Convenient
derivative of energy momentum tensor.
Is 0. And the fact that the. The.
This property of the.
There's energy intensive
describing the amount of energy,
the amount of stuff in an area
of space that it has this.
This property here,
and the fact that this potentially
there's the ancient tensor which
is based on the curvature of
the space-time has a similar
property is heavily hinting
that these two things are very
closely related to each other,
and we'll discover that that's true.
Um. OK.
We'll hold on to that thought because
we want to come back to that.
And. I keep forgetting I've got
these quick questions and then
so this is scrolling past them,
it's you have these slides.
It's very useful to have a
have a think about those quick
questions in the way I suggested.
In the past and doing the mental.
What's my first reaction to this?
Then thinking about it and going up?
I I. These questions are useful not just
because they remind you what's going on,
because they give you a bit of
a break and stop listening to
me talking for an entire hour.
Anyway. Um.
Um.
In the last couple of minutes.
I'm going to just edge into.
I'll start again from 2nd 4/4 to two but the.
What we have now.
Now we have all the materials that
we can start doing physics again.
Because we have some mathematical objects
which have properties that are that are.
And that we can attach physical meaning to.
And that have a continuity conditions
such as literal amount of energy,
mental which conserve.
That's conservation law.
Which is implausible.
We have something which describes curvature
in a way which we think is plausible.
Now we have to work out what's the
relationship between those and the
way we do that is by going back to
the ideas that we talked about in
the in part one that they brief,
part one we talked about the.
Things falling in lift shafts
and things being freefall,
and the idea of the local national
frame being the one you're in
when you're either away from or
gravity or in free fall or lift
shaft or or or something similar.
And what we what we've discovered
there is that you can make sense of
all these things happening by talking
about the equivalence principle.
And I think we saw a couple one or
two versions of the equivalence
principle in the in part one.
Which would be to do with seeing that
the experience of being free fall.
And the experience of being.
Of not being in a gravitational field at all.
Are equivalent. Not just hard to separate.
Not just impossible to separate,
but they are actually the same thing.
And So what we're going to do next
is see two further versions of the
code response 2 for the statements
of the equivalence principle, which.
Restate that in terms that we've in
slightly more sophisticated terms
that we have available to us now.
And also we tell us what to do next,
and spoiler what it tells us is that
the physics that we understand.
In special relativity.
Which is basically Newton's Newton physics,
except with with some tweaks.
So the physics we all of our our experience,
the physics we understand when
we jump up and down.
It turns out that because of this,
we can import that directly into the local
inertial frame in a in a curved space-time.
And because we can transform from that
local natural frame to anywhere else,
we at that point can use the fakes
we understand in free fall and
apply it in a curved space-time.
And what that leased with is
the problem of how do we,
how precisely do we discover what
the constraints are that matter
applies to that space-time.
In other words,
how does matter change the
curvature of that space-time?
At which point,
if we understand what the space-time is,
what the what the shape of the space-time is,
and kind of work out,
and we haven't done this yet,
you can work out how things will
move through that space-time.
Spoiler is logistics.
Then we have walked out how things
move in a curved space-time and
we have reached our goal.
So there's only a couple more steps to do,
and each of those steps is in a
way quite short, quite, quite,
quite, quite small step at this
point because of all the work we've
done about understanding duties,
understanding curvature and
understanding the metric.
So this is where it all comes
together to get Einstein's equations,
which are the instance,
the end point of who we are.
So the next time I aim to get to the
point where at least we're at two
and beyond the point where we are
talking about some equations and we
got onto one simple solution and all of G2.
Solutions to integrate.