Welcome back to lecture 10.
And this is a second last lecture,
and we are remarkably enough on time,
so we needn't scamper too much.
Now, where we got your last time was a.
They got this far as the I think
I got as far as as as the.
I was revisiting the equivalence principle
and in in this specific form that.
I find the right slide.
All three following non rotating laboratory
laboratories are fully equivalent to the
performance for physical experiments,
and the stronger version of that saying
that any physical law that can be
expressed in tensor notation in special
activity has exactly the same form.
In a current space-time.
And this in a sense,
is why we've been talking about why
we've been missing about geometry.
Because the aim is to is to articulate
physical laws in geometrical form and
what this this tells us is that once
we've done that we can immediately
import that into a curved space-time.
So we know for example well Newton's
laws tell us tells us that excuse me
that the if you if you exert a force
or something then the acceleration
of that object is.
In the same direction of the force,
and proportional to it detail,
the constant proportionality is the mass.
But the geometrical aspect of
that is the important thing.
That's a geometrical law.
That's that.
That is true independently
of the reference you pick.
Independently of the coordinates you
pick is as true in Cartesian coordinates
as it is in polar coordinates and
spherical coordinates and anything
like it's a geometrical law.
We could similarly do things and and and in.
In special relativity we can see the
full momentum is conserved in collisions,
so the total P.
Before and the total P afterwards are equal.
That's a geometrical law.
You know the the P before people
are after are in the same direction
and the same in the same length.
And this is telling us there are
no further complications.
So if we pick our.
An actual local and national
frame of freefall frame.
Then we can do our physics in Sr.
And that still works.
The key thing that this
excludes is coverture coupling.
There are no extra terms which
appear in your geometrical law.
Which are associated
with the local curvature.
It's not that F equals MA
plus a bit depending on R.
So this version of the governance
specifically rules that out.
It explicitly rules out any other
additions to your your physical laws,
and so that is by itself
a physical statement.
So I've,
I've,
I think I've repeatedly
distinguished physical statements
from mathematical statements.
Mathematical statements are things
that follow from other things.
They cannot be false.
Physical statements are
things that might be false.
You could imagine the universe
where that wasn't true.
But in this in this universe,
the guess that that physical
statement is true is a guess.
Which turns out to be
confirmed by experiment.
But it has to be confirmed by experiment
because it could be otherwise.
But it's not so this is a physical statement.
And since you could imagine a
universe where that wasn't true,
but it's it's importantly,
it is true in our case, and that
immediately tells us at least one thing.
Because in.
Special activity.
We know how we we understand.
How how we move if we just stand here?
I just stand here in a reference
frame in which I'm I'm stationary.
That's sort of the simplest motion you
can imagine me not doing anything at all.
What is my movement through space-time?
I'm moving along my own time axis.
OK, I'm just standing here moving
along my time axis, taking taking
out the 2nd until they you know,
so it's not complicated.
And if I were to be,
if someone would be moving past
me and I'm in a reference frame,
a national reference frame,
specialistic reference frame
moving with respect to them,
then my motion, my my world line will
be along my my my local time axis.
In other words,
it would be a time like a time
like straight line or which is
in Minkowski space A geodesic.
So the geodesics of Minkowski
space are time like are various.
But there are some which are time like.
Push and and our our physics,
our understanding of physics
special activity says we move
along the timeline, the timeline,
geodesics in another reference frame,
and we do so in accurate time as well.
Sue. This is telling us
half of the famous slogan.
Ohh.
Free falling particles move on timely duties,
pics of the local space-time.
The slogan being space tells
matter how to move and this was
a famous couplet of I think.
If Wheeler, I think first enunciated,
put it this way,
space tells matter how to move.
This is the point at which we have
to some extent, got rid of gravity.
This is it's saying that once
you have a curved space-time.
The move motion within
it is simple geodesics.
There's no need for gravitational field.
You just followed geodesic in
your field and you're sorted.
Job done.
So that's the second-half of the problem.
Once you've set the problem
up and got and got a a metric,
a solution to answer this equation,
and you've got a solution to.
Start that sentence again once
you have an answer to the to the
question of what is the is the
is the is the metric of of of
the of the metric of space-time.
You know what,
you know what what happens next.
So the first half of that question
though is how do you work out what
the metric is of a space-time
in relevant circumstances.
So what is the,
what is the constraint that
we have to put on that?
And that's what we talk about now.
Umm.
So the relevance of this,
but the point being made here,
can I make that I think I can make that
full screen in a second. A little bit.
Get some distractions, are we?
The relevance of this is that.
Although I didn't spend much time on it,
I I in passing earlier mentioned
that it's well when you look
at the energy momentum tensor,
the the team you knew.
There was a continuity
condition which said that the.
That the.
The derivatives.
Of the energy momentum tensor AT20.
And that's just a consequence of you
know what good what goes into a box.
Plus what's in there is what,
what is what you,
what you end up with is a
straightforward continue condition
and that we worked out in a limited
circumstance in a local inertial frame.
But what the?
The equivalence principle is telling us is.
That. We can put a dot on that.
If you like it, you can put a dot on it.
And you're seeing that this law
that was a special artistic law
is no more complicated in GR.
So that also becomes a truth in GR.
That the covariant derivative
of energy mental tensor is 0 is
conserved in invariant terms.
That's the remark about.
Things moving along time right, georgics?
And the summary of that,
so just to recap that.
From college Principal is a variant of
the college principle we talked about.
You know, in, in,
in lecture one of the course basically.
If you think about it, it's.
There is more.
There is more content to that than the,
than the versions of of the coolest
principle we talked about last time,
but only because we have strengthened it
a bit by talking out about specifically.
Because these times and so on.
But it's the same idea.
And the and.
The and and the the the the previous
version that the immediately previous
preceding version of the principle
I mentioned is just equivalent
to what you learned in lecture 1.
And physical laws in flat space take
the same formula local national frame,
and that's also called the comma
go semi colon rule for the failure
of his reason that here we return
this comma straight into semi colon.
OK, so we hear that the common
goal semi colon rule,
that's what we're talking about.
It's just a nice way of a nickname for it,
if you like,
or space tells matter how to move.
Once you get your space then things will
move along duties and problem solved.
So we've solved the second-half
of this problem. First, here we.
And what this means is that we
have to rethink the way that we
think about gravity could.
We are used to standing still,
being the natural order of things,
and falling out of trees being the odd thing.
We're apes, you know that.
You know, we spent millions of years,
you're working out how to hold on
to all the trees and not fall out.
You know, it's a thing.
We have a focus on that,
but we're thinking about the wrong way.
This set should be.
Think of that if you imagine.
I I saw shadow version of us.
Drop dropping through the news as
flushed suddenly disappeared and they
dropped down to the center of the Earth.
We would see that ghost version of us.
Disappearing at an increasing speed,
there would be quotes strictly in scare
quotes, accelerating away from us.
And we think, Oh my God, that's terrible.
They're accelerating.
But the government tells us we're
looking at the wrong way around.
That if you like,
that that version of us plummeting
toward the center of the center of the,
of the gravitational,
local gravitational concentration
is the real, is the natural motion.
That's the real thing.
And we are the ones being accelerated
away from that motion by the
presence of the floor.
To what the floor is doing in this
picture is stopping us joining that
ghost version of us in the in in
free fall and accelerating away.
So as our our feeling,
the pressure on our feet or on other
legs of our chair is not that isn't
just a bit like a force or acceleration.
The force of gravity isn't just
like a force of acceleration,
it is an acceleration.
And the and and so that that this that
this seems slightly equivalent equivocated.
But in a sense if you think about
the right way then it's clear what
that is accelerating away from.
And so this picture of the observer in
freefall and the observer not in freefall.
And you could either view this as being
out in space and this person and the
both out in space but this person is
on a a platform which is accelerating
or else you can rather a standing
on earth and this person is on a.
Platform this person is falling
down and they are equivalent and
in a sense that the, the, the, the,
the physical statement I want to
really get over to you by but it's
just just repetition is is, is that.
Any questions about that that that
I I think I've done that today.
It's just so nice to be able to make a
physical statement in this course rather
than just here's here's more maths,
I think rather indulge,
indulge you OK?
So the question then becomes,
how do we work out? How?
What do we, how, how do we?
Constrain what that?
Metric or space-time should be.
And again this will involve
actually a physical statement,
a statement about the universe
which could be false.
So we have to to guess or we don't
have to guess because Einstein
did the guessing on our behalf.
So that that guessing I think happened
more or less over the summer of 1915.
So you've done a lot of the
work to to to sort out the maths
of this to learn the maths of
different geometry over 10 years.
You think you had a tough over 10 lectures.
He took 10 years from 1905 to 1915 to to
to sort that that math out in his head.
He claimed he never understood it really.
But it's only at the end,
in a sense,
that he did the guesswork to to to
work out what the the constraints
were on the the the space-time.
And when we start is by going back to
Newton and this is Poisson's equations.
This is the.
It's actually the the.
The Laplacian of the
gravitational field is that.
The coverage of that gravitational field.
Essentially this is just the Newtonian
gravitational field it's governed by.
The density of mass.
The local density of mass. And.
Big new gravity should constant.
So that's a statement of Newton's
law of gravity, if you like.
Not in a way that Newton would recognize,
but it's due.
That's Newton's law of gravity.
From that you can deduce F
equals GM or are squared.
And we can take and and the the the
vacuum version of that is similar
except obviously with with no mass,
so, so the, the, the.
Meet with the graphical field
when there's no mass is well,
it's not complicated.
It's it's just flat.
There's no gravitational field.
And we could take that as inspiration.
For what to do next.
Because. And.
OK, just a quick question and just to at
least give us we probably brief pause,
we thought what does this Phi
comma I comma I represent?
The diagonal of a tensor there.
To this contraction or this? A
construction here. Who is it?
Was the first one?
Who is it with the second one?
Who was it? Was the third one?
Well, I thought. Who was there?
About her brief chat just to.
This isn't saying you're wrong.
You have reached.
Thank you. OK,
let's you know chatting is always brief,
but so so the diagonal of a tensor.
This contraction here.
The third ring. I see a lot,
a lot, a lot of indecisions.
They're there, they're still,
but it's not a big deal.
But it's the.
Is it just this contraction,
it has to be the contraction
because in the case of.
The third one.
We'd end up with.
Two eyes in at the bottom if you're like,
so it has to do in order for
the there to be a this be a
correct term with a sum over it.
It has to be this contraction here.
So the contraction over those hard
to hard to pick out G's there.
So I I mentioned that just because
it's new it's about time for quick
question but also because it's
it looks a bit strange that you
haven't previously seen commas.
Upstairs if you like.
So that's.
That notation looks slightly strange.
OK, so come back to that mode. And.
So we have. Possible equation?
I comma I comma I = 4 Pi G. Rule.
And. Fine comma I comma I = 0.
In the vacuum. Now we can look back.
A bit. And again,
lecture one who knew lecture one
was going to be it was significant.
We can find a thing in lecture
one which described the you know,
you remember this.
Idea of the of of the
objects falling toward um.
Earth.
And getting closer without experiencing
any acceleration and we found an
expression for that. Which were the.
I think I I think I I think I I I
mentioned this and so showed the result
without so walking through through it
step by step because because the the
details are sort of fairly obvious,
you walk through but it wasn't
worth delaying which was GM over R.
And. Uh, sorry.
Which? Also jumping a few
steps here is also. And.
Not very unique side. That's better.
OK. And that is the that was
the 2nd derivative of the.
This looks like I'm sort of missing a
couple of steps in the middle here,
but the point is,
the point is the details of that
then that we can get an expression
for something that that makes
sense in our identitarian picture.
Which reminds us of the.
The expression we got last at the
end of the last chapter. Which had.
You know, what was it?
Which was the expression for geodesic
deviation which involved the.
Rementer. And the point of this?
One can go through this
argument in in in more steps,
but the point of this is that it's
that this is suggesting hinting to us.
That the thing that corresponds to this.
Derivative here is something to do. With the.
With contractions of the Riemann tensor.
And in particular. It's possible
to do with the Richie tensor.
Which is that that particular contraction
of the Riemann tensor, and so one can guess.
Is the analogue of this vacuum equation.
Something nice and straight
forward like our. Mute, mute.
Alpha beta equals zero.
Does that count as our?
An analogue in GR of
this vacuum equation. At.
And the answer is yes, it does
sort of so that that that is.
Only the halfway at each year,
but that that is that is true in the sense
that that does boil down to the the.
Those are the vacuum field equations
for for GR so that the shape of the.
If in a universe with no
energy momentum in it,
the possible shapes of the metric are
ones where the curvature they're tensor.
Sort of curvature is 0,
which sort of makes sense.
No, no mass, no curvature,
because that's that's not
surprising therefore.
But we're still not there.
And one thing we can do is we can
add to it since it's also true.
That if our if our alpha beta is zero
that would imply also that G. Alpha UR.
You. Beta would be equal to 0 trivially,
so that that that implies also that G.
MU equals to R MU nu plus minus 1/2.
Or, Gee, you knew equals 0,
so I haven't added this is equivalent
to that, because if that's true,
then this must be true.
But the point is, I I'm,
I'm you know, knowing what,
I'm knowing what's coming here.
And I I've I've just switched
from our to the I sentence
G for reasons which are,
at this precise point, obscure.
So that's the vacuum field equations.
What are the you're not interested in that
we're interested in is the field equations
in the presence of energy momentum.
So you've got a star, you know what's the,
what's the field around that question?
Or equal to zero. Yeah, we can.
Can we immediately assume that
the G will be the Euclidean G?
No, because I think that.
I think we cannot know because
I'm not 100% opposed to this,
but I think there would be
non Euclidean G's which would
have that have that property.
I'm not sure what they would
look like offhand, but I.
I think the answer is probably yes,
but I I I don't think that completely
constrains that because the thing that.
Yes, yes this is.
If you start with two lines,
they will never meet the basic.
Yes, I think actually it might I mean I,
I I'm, I'm hedging here because I'm
not 100% sure but I'm fairly I think,
I think I'm fairly confident you're
right that that, that, that,
that that it's probably the only
solution of that is the Euclidean one.
So I think I think that that would
be the case and point for that for
that extra auxiliary reason. Yeah.
So.
I guess for this for Winston in the vacuum,
in the field equations in
the presence of matter.
So I guess if this is sort of right
is to say, well how about seeing
you you equals TU where T is you
remember the energy momentum tensor.
OK, that's plausible.
That as a first step, that's so plausible.
Or some. Proportional to that Kappa.
And. No, but the conservation law.
That we saw here.
Well then tell us that so T.
MU new semi colon.
Neu equals zero would then imply that
R MU nu semi colon U was equal to 0.
Um, OK, that might be the case,
but that would also imply
that when we contracted it?
That the curvature scalar.
Would also had a 0 derivative.
And that tells us that there's no
coverage at all, basically. And why?
The third one? The third one or
the the second one? The answer
is the second one second, yeah.
Yeah. Yeah, which is? So um. If the.
And. So if the coverage scaler is.
Has zero. Derivative then that is
essentially saying that the that, that,
that that the universe is flat and since the.
The. That would in turn mean that the.
Contraction of.
Gu Alpha T alpha. You.
And going you would go to zero.
That would in turn tell us that
the universe had constant density.
So this, this. This.
Possibility can't be right.
Because the the conservation law of.
The argumentum tensor ends up leading
us to conclude that the universe
has constant density, which is.
Not the case. Universe is lumpy,
so this can't be right.
It's clearly not far wrong.
We're on the right track,
but this can't be right.
But we have been hinting
at the right answer here.
Because I picked this as the. As.
Apparently trivial deduction from the.
Richard James from the plausible
vacuum through the equations.
So what we can the next guess,
the next the next step up is to guess that.
But that is upload applicable
applicable version. So note the
Richie tensor being proportional
to the argumentum tensor but this.
Construction. Thanks. Attention
this constructed from the Richie
tensor being proportional to the.
The incremental tensor,
and that is plausible because we know.
From when one of the correct
contracted Bianchi identities that.
Gu.
The new new semi colon new is equal
to 0 as a mathematical identity.
Based on the properties of the tensor,
so the conservation,
this conservation law is satisfied.
Mathematically, by this.
So that's plausible, and it's right.
It turns out turns out to
be matched by reality,
and this equation here is I sense equation.
And. What I've given you is not a
mathematical deduction, deduction of it.
I'm seeing what I've said I said earlier on
it's not a mathematical reduction of it,
it's I guess it's a plausible guess that
we have motivated by analogy with patterns.
Equation is against the Einstein made,
eventually offered several
goals in the summer of 1915.
And published in the famous paper often 15.
And it's that that is the physical
statement on which all of the.
Solutions that follow have been
based and all and which all the
tests of GR have been based.
So the the the tests of
GR GPS has satellites.
Black, you know, refugees from,
from, from,
from slowing down neutron stars and so on.
Our tests of that guess.
And it's passed every single one so far.
So that. Through there we can just look
at that from for that's the sort of thing,
and that is the thing we can all look
at and admire and put on T-shirts.
Now that is. Attention equation.
So it's a geometrical thing.
But we've quoted it with.
It in component form.
T is our symmetric tensor.
And we as we discovered as we as
we reassured ourself last week,
which means G is also symmetric tensor.
Which means it has 4 + 3 + 2
+ 1 independent components.
That's 10 because if it's symmetric tensor,
then as a matrix it's a symmetric matrix.
I am not antisymmetric so that there
are 10 independent components in that.
The. And we try to Bianchi identities.
Add 4 code. The MU equals 1230123.
Add 4 constraints to that.
Bring this down to six
independent constraints. And.
And the remaining degrees of freedom?
Are based on the fact that we can rescale
the coordinate functions as we wish.
We could measure in feet rather than
rather metres, or rotate or our axes.
So this ends up constraining our
the solutions as much as we need.
Because remember, think back, unpack this.
That GI Sentencer is composed
of the Ritchie Center.
The Richard Center.
It depends on the Riemann tensor.
The Riemann tensor depends at least
the local natural frame on derivatives
of the metric and particularly
second derivatives of the metric.
So this is a collection of of 2nd
order of 10. Plus constraints 10.
2nd order.
Differential equations in
the components of the metric.
The metric has it also auto metric tensor,
so it also has ten.
Yeah, sorry this is a said that
I said that in the wrong order.
This has 10 has 10 constraints,
four of which are taking up going
forward these if we want to.
So it's a differential equation
for the components of the metric,
a second order differential equation
for the component of the metric.
It adds 6 constraints.
So there are four degrees of
freedom left unconstrained,
they are taken up by the freedom we
have to change our our our coordinates
in length and and orientation.
So this is what this does is it constrains.
The metric.
So the metrics that are allowed around
our source of energy momentum are
those which satisfy that equation.
And those are the codes of all
solutions to generate to generativity.
Doing so is not trivial because you
have 10 simultaneous differential
equations to solve.
But if you symmetrize the problem enough,
then you can solve that problem.
And the first. Predictions of um.
Which come up come ohgr, where?
I Einsteins in sort of 19 fourteen
913 I I believe but they were based
on the gravitational redshift that
we talked briefly last in lecture
one again the idea that a photon
claims through gravitational field
it changes frequency and to the
extent that frequency is the type
of clock that means that that that
time moves differently at higher
and the shield photon argument said
that time moved differently at a
higher gravitational potential.
From at a lower one.
And you can deduce from that
version of the metric of.
Of a weak field space-time.
Which is what you and from that you
can deduce things like that there
will be some deflection of Starlight
as it goes past a large mass.
But I distract myself because
that is not that.
That although it's significant
and certainly significant,
and was one of the things that the famous
Eddington Dyson measurement of of of
the Solar Eclipse aimed to rule out,
it's not by itself a solution
of intense equations.
So what are the solutions of
intense equations? Can I think?
And.
Yeah,
I think we actually we're actually
going ahead of time here.
So I think we can we we'll we'll move on.
There may even finish early next next week,
but.
Before you do I do move on to the
first simple solution of this.
Other questions on what we've done so far?
Are you happy or perplexed or a question?
There were two questions here so.
Right, so here this implies
that the density is constant.
But why does it notify?
And so the question was.
How does this imply that density is constant?
But this doesn't?
And I think the answer to
that is that if you if you.
Stick to just that.
Then that's.
Our function of second
derivatives of the metric.
So that's a constraint on 2nd
derivative of the metric. But the.
The the mathematical remark
that the argumentum tensor that
the Andrew Mentum is conserved.
If this is true,
forces the Ritchie Dancer.
To be have have A to be
conserved as well if you like.
Which in turn implies that the
curvature scalar is conserved and.
The other but if if that's true,
then the this contraction if
that's true and that's true.
Then this.
This could be written as
this contraction of the.
Andrew Mentum,
tensor having with you interactive
since G is.
And.
I think I'm saying missing a step here.
Umm. If you could see me going
on struggles at this point.
OK. I I think I may have.
I have a comment misplaced the
careful description of this of this
step in the notes, but the the the
to the question of why does this up?
It's gonna be like, why? Why does this?
Make.
Give the got the argument rate,
why does this make predict that the universe
is constant density and this one doesn't?
I think it's basically because there's
this extra freedom in here. So the. Umm.
If if that's zero in a in a vacuum.
Then you have the extra freedom
of of changing G if you're like.
Without changing the Richie changer this is,
this is rather handwaving argument, but
presumably that's a more complicated object.
I think is is a slightly
more complicated object.
This, this argument that that, that,
that, that that the potential ends up
being being constant doesn't work out.
OK, I'm going I'm annoyed that I'm
going to have to go to reassure myself
that's rather hand waving argument,
but I think I think the key,
the key point there is this is
simply a slightly richer object
so it has slightly more freedom.
You ask questions.
Yeah. You write. On the north you're
right that is divided over RI was
wondering if because I I was looking
back at the nodes like set first
set of notes and it was divided by
zero Q and I was wondering it. Ohh.
And that maybe just my mistake.
Yeah. Yeah. And in fact, yeah, yes.
When I was writing that up,
I thought, is this right?
But I think our cubed bread make a lot
more sense because I think that's right.
I think that should be.
That, that, that, that probably
should be an an arc cubed there.
All right to him.
But. OK.
Thank you for that because that's that
could have been a distraction. And the.
In the beginning of this section,
we can approximate the perfect
fluid with a delta function, yes.
Chronic delta, yeah.
Is that what gives us?
Is that the only case where we
can have one solution to those
10 differential equations?
Perfect fluids or can you
parameterize it first?
I think in that case we're jumping
back a bit and alright so,
so I I think the question
if we go back to. And.
Basically here.
Yeah. And so. And this is the energy metric
tensor or the components of the metric
tensor for the perfect fluid which we have,
and we persuaded itself of that because.
The answer we we, we,
we guessed there has to be independent
of what direction we're facing,
what what, what, what we're picking on.
So that isn't a solution to anything.
That's. That's our input to the to to
to to the to to to to to Einstein's
equation we're seeing this is how.
This is what the arguments must be.
Must look like this.
So the only parameter there is
is the pressure so that the.
I think this will be.
Well, that, that,
that I I think in in local natural
from that would also be a diagonal.
That, yeah, that would be a diagonal
matrix of coordinates in the local frame.
Is that what I'm saying?
I said what you're asking.
Be part of the left hand side
of the equation of this yes then
financial equation yes means.
It doesn't mean that they
are have one solution, yes,
but at least in a perfect fluid.
Since we have that,
we know that there has to be
only one solution to those 10.
And yes. Yes, and and and and
and and we won't cover this,
but in the case of a,
because they're recovered next in in G2,
so G2 is picking up where this leaves off.
I will mention a solution of incense
equations just because I show you one,
but G2 is essentially solutions,
plural, of Einstein's equation.
And solution number one that you'll teach,
you'll learn.
Is this the Schwarzschild metric?
And the structural metric is the solution
to the case where you have a central,
a single isolated source of mass,
so just a lump of mass in the alone in
the universe and the and and and that.
So that's a very specific distribution
or argumentum is a lump there.
And the the solution to that
is a particular metric which is
called the structural solution,
which has properties which
are are of interest.
But that is the solution to the case where
there's only one dot in your universe.
So the and and and jumping ahead
of myself because we have a whole
5 minutes of time jumping myself.
Or do you have Twitch? The. That was.
I seen the the 1915 people
was in November 1915.
And Schwarzschild, who was,
I think in the German army at that point
in the middle of the First World War,
you know, piles Louise time
under shell fire, you know,
solving Einstein's equations as you do.
And he did it remarkably quickly.
And I think in early 1916 he wrote Einstein
saying this appears to be a solution.
And I said went ohh, I didn't think
I thought take longer than that.
And said, well,
it's quite clearly classically right.
So um.
That so that's that's an example
of the case where although you
have ten couple differential
equations if you make the problem
symmetric symmetrical enough.
And you're you're move.
Complications by saying we
only have one bit of mass,
then you can then half of these equations.
You know the large chunks of the different
equations of instructions fall away,
and you're left with much less to solve
than you would start off with more
complicated things like other solutions of.
And.
And wrong what I said before. Because the.
The the free space.
The the the the vacuum
form of instantiations,
which is what that really is.
Or or that has another solution which
isn't a constant flat universe.
That solution is gravitational waves.
The gravitational waves are
a solution to that equation.
In the presence of no matter at all.
So graph you could have graphical waves in
a universe with no matter at all, because
what graphical waves are is a dynamic.
Perturbation in the metric.
This metric is perturbed,
perturbed the next bit and and that
and that propagates. So that's all.
Gravitational waves are one
another solution to up to I said
equations in this particular case
of a 0 right hand side. Question.
So again, how would there be a
perturbation if there isn't any master?
And well, the question of how you how
that is started is a different question.
But so, so that solution says that
if given that it was perturbed,
then the perturbation, it's like Maxwell
equations support a perturbation of
the electric and magnetic fields.
They don't sort of care how
that perturbation was started,
that's what. That that's.
Another another question if you like,
but they support a perturbation, yeah.
Thank you.
Good. So that's a. That's a fine thing.
So and and that gives the
other half of the slogan.
So space tells matter,
how to curve, how to move.
Think we're lunatics, Mattel speed,
how to curve instance field equations.
So you have doubtless seen that slogan
in various popular accounts of of general
activity or possibly or or whatever.
Or you may have heard of this,
but that's that's where those two,
those 222 parts of that slogan
are both deep physical statements.
Expression or jocular form,
but they are both physical statements.
Push, you know no the underlying.
Underlying and that's just
a nice picture really.
The point being that once
you have a curved space-time,
then doing going in a straight line
in that course space-time can end up.
Producing port. As it were of youth,
most eight would be a a closed loop.
Key points. Now I I, I, I, I.
OK. Have you been ahead of time?
I have. Look up the last of
the stuff before section 4.3.
But second 4.3 is fairly self-contained
and so next time what we'll do is
we'll go through section 4.3 which
is about doing most of the solution
of Einsteins equations minus chunk
of boring algebra in the very
simple case of attaining mass,
attaining central mass and and we
discover a nice thing at that point.
And time is the last.
Let the last lecture,
I think it's the Friday of next week
that we have the next supervision.
What? Sorry, this week is OK, right?
In that case I will.
And I haven't yet,
but I will put up a a a little
heading on on on the on the pilot for
Supervision 2 and I exhort you to add
questions or puzzles or things there.
Comment on what someone puts up
something which is almost what you want.
Then add comments like things you know.
I will use that as the as the
skeleton of the supervision this week.
I'll see you then.