Well, as you can see there's a large
blank here for the suggestions are and.
I didn't send an e-mail around
saying I'd suggestions but.
And so we'll have to bust a bit and I
I do have a a matter of different plan
from the previous divisions commercials.
So I think that I hope you have just
write have written anything down.
You have some sort of questions that
we're going to talk through and could
potentially go look through one of the
exercises if there's a very obvious
one that lots of people think is a
really good one to work through but.
A lot of the.
Exercises are sort of turning the handle
rather than massive epiphanies, so.
If there are any. Well,
I think a higher value way of using
this error is if there are questions
that you that that that you can.
Either burning you up,
or which we think are of interest,
or which are puzzling you.
I got the best way you can be
renewed the error I don't really
want to give in and just sort of
trundle through an exercise, but so.
Is it all just easy?
Are there particular?
Exercises that are difficult,
other particular sections that are difficult.
I mean that might be that
might be a good way of of of.
Sort of getting going with with
questions that other other sections
from part one to part 4. That.
You've stumbled over that you're
not sure how how to get from the
beginning of that section to the end
of that section or or that chapter,
or is it all of us have a blur
or are there some of them?
Are there some that are easy?
I mean that's quite usually there's
some sections where you just think,
oh, that's the section that
had made it all clear for me.
Nope. Other sections which?
Where?
Even different parts.
And Part 2 Part 3,
part 4 which ones those?
Where did you find I?
I have a notion of which ones are
probably the harder ones of those 3.
But do you agree with me what which
of those three things hard is?
How would you rank them in
terms of which was your easiest?
Which ones are hardest in
terms of Part 2 Street,
I'm getting the part one,
we're just sort of getting going thing,
but Part 2, three and four and four,
which of those are easy and easy and hard?
Anyone.
That, that's that fan is annoying.
I'm not gonna be. Stop it.
Say again. Part 2. Yeah 4/4.
Right that's that's not what I guessed
right but that that that that's very
useful to know because I suppose the
difficulty there is we can, we can,
we can't sort of just turn the hand
over them with the maths so that
there's that there is back to physics.
Would would others agree with that?
Is that any that.
That's right that should have
a fairly quick noting there.
OK well that's that.
That's interesting because that's
not what I would have guessed,
but I can see why because that
is in a sense where the where
the maths hits the road.
So, right.
Well, that's, that's good then
that gives us our place to start.
If by the way, I'll look back to the page,
if any of you are online and
want to add questions there,
then I don't entirely honestly
then we can do that.
So.
One of these must be there.
Well, just by the way, the.
The videos on the Microsoft Stream
from last from two years ago.
Are any of you looking at those?
Are they useful at all?
But I think it's also
seems rather tentative.
Nor do I'm not gonna ask
you questions about it.
Is it just ask people for information
so so that that that's not.
Sorry. And I think so, yes.
I I think part if part if elected
for part 4 aren't there, then they
certainly should be I and I haven't.
Um, so I might not be able to.
And.
Yeah, right.
One second.
Well, no, but that's what we're looking for.
Oh, come on, I'm Clinton.
Thank you.
Ternals uh.
OK, the lecture 11 from last year isn't.
So I I need to actually,
in fact, why don't I just?
Um.
Because if I don't, then I will forget.
This is 2020. Ohh, there's such
a lot of room, wasn't it? Umm.
Noted. OK.
That wasn't what we do. Oh yes,
that's what we're doing, was this.
So give me that bigger.
So that was done. So the stuff on
the incremental tension and and that
was all about trying to get out.
Explain for the argument potentially
comes from and this notion of
dust so and the point of that the
reason why we're interested in in
in the age of mentor is because
that's what is the source of the.
Changes in in the curvature of space-time.
That's, that's that, that, that that's
the right hand side in 19's equation.
That's the bit which says
Mattel specialty curve.
So the geodesics is space
just matter to move.
Einstein's equation is matched
or specific curve matter?
It's not matter we're
talking about really is it?
It's not matter that just beat
out curve it's energy momentum
that's beta curve are you question?
No, you're discussing it incrementum
that tell specific curve so the so
the step one there was turning.
Our some of our intuitions about matter.
Which is the most compact form of elementum.
How do we turn that into a geometrical form?
So the goal with that section one is just.
It's just geometric geometries,
the notion of.
Lumps of matter. Next question.
Details based on the curve? Yes.
Does that mean that light curve space?
Yes.
But then why do we think that light
moves along the curved space?
Because the light curves,
because this is the both houses
of that of that.
Slogan that energy momentum
tells space how to curve.
And then test particles within that
space are what explore the space
and are curved by it so as a star.
Curves the space-time around
around it and a planet.
I test particle in that it which is
a a lump of energy momentum does
follow duties and and and and and and curve,
but at the same time that planet is
curving the space-time around it.
So that the, the, the the the the, the, the.
So these are recursiveness of that slogan.
It is part of the the on the thing
to meditate on in a sense.
But it is true that that enough light,
if there was enough energy in a small
enough like energy in a small space,
then it would curve space-time.
And that's sort of The Big Bang.
So the the the The Big Bang is where
there is enough energy and sufficiently
small space that you have the whole
universe being curved around it.
And another thing is that if you
think of gravitational waves.
So these are these oscillatory
solutions in in in space-time.
Those themselves.
Are.
They have energy momentum in them,
so they themselves curve space-time.
And that's why I'm saying equations are
hard to solve because they're nonlinear.
With, um, things like.
Maxwells equations if you take a
solution of Maxwell equations, so.
Like about a library and add another
solution to Maxwell equations,
another library.
Then the sum of the two is a
solution of Maxwell's equations.
In other words,
like light can pass through plate,
racing pass through each other,
so so you add 2 lightweights together,
you also get questions.
That's not true for.
A nonlinear differential equation
like Einstein's equations.
So the solutions to Einstein's
equations are not additive.
You can add two solutions to
identify equations together and get
a solution of intense equations.
That's why it's hard to solve.
So because there's a whole.
Chunk of mathematical methods,
which is all about decomposing differential
equations into ones you can solve,
and that the idea that the
additive is part of that.
So the, the, the, the,
the the recursiveness of of of
that slogan is in a sense talking
to several different things.
It's it's talking to the idea that
it's masters is both the the thing
which explores the space-time and the
thing which creates the coverage.
So the point of that first section is to.
Geometrized the idea of. Mass.
And to remind us perhaps,
that mass is not the only source of energy.
Momentum if the important,
most important source,
instrumentum in our near experience.
But it's not the only source.
And then the second part and we've
got to come back to push back,
we can explore a little more at the moment.
The next part is that the guessing bit.
And just to to to reiterate the the.
It is guesswork.
Einstein was was inspired by the
the formal Poissons equation which
relates the Newtonian gravitational
potential to the distribution of matter.
And to the you know there's a second
derivative equals a mash thing and
and that sort of giving a hint,
but the form of integration is a guess.
And the experimental corroboration of.
Generativity. Is the experiment,
experiment experimental
corroboration of that guess?
And the idea that you might have
the cosmological constant in there.
Was there? In response to an
apparent falsification of that guess,
it appeared that the that the
the expanding solution expanding
cosmological solution that was found
fairly early on was clearly wrong.
Therefore that guess was clearly wrong.
Therefore the cosmological
constant was added.
As in response, but it turned out no,
that's not wrong.
This is actually the case that
the universe is expanding,
therefore there's no need to
add the cosmological constant.
Therefore the original guess
would perfectly good.
And still later it turned out that
with things like gravity or blah blah
blah dark dark energy it might be.
In fact is another case for
adding that term in.
To the to, to, to the equation.
So there's still a certain professionality.
Of a slight professionality,
but certain professionality to the,
UM, the form of Einstein's equations.
And the fact of the equivalence principle
that says there's no coverage coupling,
that says there's no extra things
added to special activity,
that no extra added to physics
because of curvature. There's no.
There's no term in the general
artistic version of of a micro screens,
for example.
Which is involves the curvature,
local curvature.
That's a physical statement which
says no you don't you you won't
have to add anything to intent
equations to deal with coverage.
You won't have any other
other physical with coverage.
So Part 2 there is the guest bit.
And part three is OK, so so.
End of 4.2 is basically
the end of this course.
The G1 is getting up to the point where
we've said we're declared inside equations,
and so 4.3 just a bonus.
No, it's a bonus.
That's that's I come up
with the objective say.
Perform some dynamical calculations.
If you look through the past papers,
you'll see that the dynamical calculations
equation are things like can you
draw very simple deductions about.
I can't. There's one that
I've asked in the past about.
The duties equation in our coverage
space-time that if you start off moving
radially you carry on moving really.
Yeah really radially and stuff like that.
So that's that's what I mean by
simple the number calculation if you
look back at past papers that I'm
not asking you to solve as orbits
in in instructional space times,
it's basic simple things.
Because they simply don't have time in
question and that and and that is what
Georgia two is all about in in a sense.
So Patch 4.3 there.
Is really just a. Going, you know.
We're treating one particular solution
just because I can't resist not doing so.
I don't want to leave you with no solutions.
Um, so that's how those three
things sort of slot together.
In terms of the of the sequence
of ideas and and the separation
between the different ideas.
Um. We could dig into,
I mean of those three parts of the other
bits that I should we should be useful
to talk more about in detail which,
which which are the parts of those?
What good to dig into? Could I?
I I I think you're right.
This is actually, from some points of view,
the hardest. The hardest the course.
Because it is actually about trying
to use these mathematical tools.
In practice.
And trying to connect them to the physics
that you've spent the rest of your degree.
Absorbing.
So now tools.
Nice tools, we finally get to use them.
So which of those sections would
be useful to dig into you think?
4.14 point 24.3.
Yeah. Excuses.
Or should I just blather
on about going through?
The second objective section, OK.
Explaining the Congo semi colon rule.
OK, that right that the
comical semi colon rule is.
I think what which section is that?
Specifically.
Comical second rule is come on.
Yeah, is is is this section here on.
The equivalence principle.
So the common goal semi colon
rule is one of the ways of
saying the equivalence principle.
So the current principle is in.
One version of it is the thing that
we sort of covered in part one.
We talked about the we way back in
part one and the idea that you can't
tell if you are if you're free fall,
you can't tell whether you're
way out in space.
Away from all gravitating matter
or you're in lift shaft for around.
You can't. They aren't different.
It's not you can't, it's not you.
It's hard to tell.
The difference is you cannot
tell the difference.
So I as a physical statement,
those two things are the same.
It's what we said in part one
of what we're repeating here.
Another version of that.
Is a stronger version of that.
Is, but because that's consistent with.
And freefall being a sort of special case.
It might be.
There's all sorts of things
that aren't there in freefall
in the local inertial frame,
because being in the local frame,
you're in freefall.
You have the coordinates that are
attached to that are nice and simple,
and you can do calculations
in that frame very easily.
So the local and frame is clearly somewhat.
But it's it's only special because for
calculational reasons or special otherwise.
And the strongest principle is saying no,
it's not special at all.
In a way.
Um.
Any physical law that can be
expressed intention rotation in Sr,
so a geometrical statement of physical law.
The key to the idea of geometry has
exactly the same form in a locally
inertial frame of a curved space-time.
And what that means is,
do I see who was cook?
And with no extra covered your terms
repeating on the right hand side.
In other words,
it's not that there is some
extra that if that's a that's a
conservation law in special activity.
Of other than that is with,
with, with single partial with
partial differentiation.
The tensor form of that is this.
That's. That's a tensor.
The tense of the tensor form the
geometrical form of a special
activity conservation law.
And the strongly equivalent
principle says that's true in GR.
In the story,
it's not that plus some curvature
terms which happened to be zero
in the local inertial frame.
It's there are no extra terms.
So there there are no tidal terms,
there are no. Just do extra energy in.
In the cover.
Which is implicit in the coverture.
It's just there's nothing else
on the right hand side there.
Because the basic equivalence principle
is consistent with there being
other terms there who just simply
zero in their local national frame.
The strong conference says no,
they're not there at all.
And what that is is, is is that for others,
for for that, for example,
for the geometrized, for the, for the,
for, for macro equations, which are
already in basically geometrical form,
there's no extra terms and Maxwells
equations which are to do with geometry.
So late isn't doesn't propagate
differently in a curved space-time.
It propagates in a, A, the, the, the.
There's a very little ways
of saying there's no extra.
There are no, there's no coverage coupling,
there's no curvature terms,
and that is the physical.
The slogan for that is.
That.
Good, because there that the the the
the the the point being made there
is that you can go from the special
artistic version team you knew comma
new equals zero to the generalistic
version tbu semi colon U = 0.
And that sort of physical law so that
is so this version this this quote,
comma go semi colon rule is an A way
of thinking about that that that that,
that physical statement.
That there's no extra coverage
coupling now and and and the point
of the of the of the Congo Senegal
rule is that there's sort of two.
Only one of which is being referred to here,
because if you if you remember
all the stuff about going to The
Walking in the local natural frame.
The reason why we work in the
local national frame is because
the calculations are easy.
But if we have our if in that easy frame,
we have a an equation. Which involves.
Angled involved just components or
single derivatives of components.
Not set double second derivatives
but single Dr Components then.
That is the.
And although we calculated that in
the local natural frame these special
coordinates those that that all
the bits that would survive of are.
More complicated expression which
would be which are transformed
into these coordinates.
In other words that is equal to the.
Covariant version of that.
Expression of trying to for example,
for example, for example.
Let's go back to. Here.
And I was looking at this.
Earlier I think we may have.
Talked about this briefly last time. Umm.
For example. I'll be better, bigger.
That's one of the exercises later on,
little later on in in part three,
and it's about. Completing.
You're calculating this expression here,
so that going slightly beyond that.
OK. It's about it. It's it sure that.
And the point is that there the.
We do the calculation in
the local inertial frame.
The point being that the
rest of this expression,
this complicated expression here,
is all zero because the
Christoffel symbols are zero
in that in those coordinates.
So in this frame.
This potentially very complicated
expression turns, you know,
collapses to just that.
And.
We can. We can. In these coordinates we
can compare that with the expression
like 3349 for the Riemann tensor.
And discover that that question there.
Is all the bits of the of that
expression in 349 that survive.
In other words, this. In other words,
this is equal to this in these coordinates.
But. This equation here. Is an equation
between components of a tensor.
Which is true in in this. Frame.
Yeah, in this frame, in these coordinates.
But if it's if those components
are equal in that frame.
Then they will be equal in any frame.
If you transform the left hand side from
the local frame into something else,
and transform the right hand side from
the local industry into something else,
then you'll still have an equality there.
You have a massive more terms right hand
side, but you still have inequality.
In other words, this is a tensor.
Equation.
It works because the sub the sub
scripts of the knob line different and.
He's well, yes. Because because this.
Hear that? That's natural.
I apology VK minus nabla G
nable I VK But if they were?
OK, that would work because
then we would have a second.
Ohh alright but.
But that we wouldn't see,
but that that's true in principle,
but we would what we see here would be.
A zero here because I think if you
swap I and G then you choose sign,
so you end up with you know.
One more thing.
We have a Ji comma hi.
Yeah, as well. So then we have 3.
Over M I'm not sure.
I'm not sure what you mean.
And we also have it.
J5 comma I.
I'd have to go back and look at the,
the, the, the, the, the, the,
the intermediate steps there,
but I think that we would end up with
this if that I would end up with zero
on the right hand side. Because again,
I have to go back through the steps,
but because because that's, that's a so, so.
I V minus Napoli Napoli V will be equal to 0.
But the point here is that here
you have a changer. Here you have.
Yes, so it's a tensor.
If the component of the tensor are equal,
then the tensor are equal
as geometrical objects.
And that's sort of a common
goal semi colon rule.
In the sense that you can go from.
And that's not perhaps a terrifically
good expression, but but.
Is there another version of that
which illustrates that better?
Which I?
Umm.
Let's see.
And I'm not going to find
one immediately, but.
Four and.
I'm not gonna be mediately but but the.
I mean, I, I think I, I,
I'm risking over complicating
this by speaking too much.
So, so the, the, the, the, the,
the key point I want to to stress
is that there's two things.
There's two cases where you go
from a a commentary semi colon.
One is in the general context of
this thing about doing calculations
of the local national frame
and then deducing that you've
actually got a tensor equation.
Therefore you can turn the
comment into semi colon.
And that a mathematical trick.
And the other is this version is
rephrasing of the equivalence principle.
To see that there's no coverage coupling.
There's no that that, that.
A. A differential equation
such as that conservation
equation in special activity,
which will just involve our.
A comma, I think.
I think, I think partial derivative
can be turned into a physical law,
that physical law and specialty can
be into physical law generativity by
turning the common into semi colon.
And that's a physical statement,
not a mathematical trick.
So that the the distinction
between those two things is the
important thing. And so, so.
So where are we? So that. Objective.
Is basically see what I just said?
It it it's, you know.
In your room or to explain.
That. In a way which indicates
that you do understand it.
Is it cool there?
So I think the objectives in part
four are not actually terribly hard.
I think the idea is that the
aims in that's one of the ones
with the aims are quite hard,
but the objectives aren't massively
hard for what I want you to be
able to to be testable on. But.
I think these two things
are both explained things.
The not do a calculation,
but you show me you understand.
Good questions which are?
A pinch to mark,
but I think quite good as ways of I
think the quite useful objectives in
terms of if you can do this then I
sort of believe you understand it.
Which is the getting back to
the point of exams and the fate
of rubbish we're going.
But obsessing things would be so much
simpler if we just say they are the aims.
Do you have you tried the aims?
Yeah, yeah, yeah, I would agree.
That would be simpler.
4.2 OK, don't look at that. Umm.
So 4.2. Ecomo.
Alright, did you have all the answers? Right.
OK. I think there actually was one of the.
Yeah, the things mentioned in the thing
last week where I think I say I can't,
why I didn't talk about that last time,
I think whatever, right?
And. I remember this do the whole sensors.
OK. And one graduate.
I'm good, I think rather than right.
And we'll talk through the,
the, the, the, the, the,
the, the, the, the note.
4.2.
OK. And? No, it's quite brief,
isn't it? Yeah, as as a note,
perhaps should expand that.
Well, OK, let's let my body should break,
break, break this down to some extent.
Just what's in place to?
And.
OK.
So the.
I think I think that this
ansatz is suggested in.
In the question so.
A row plus B. P. You cross you.
Plus CRO. Plus DP.
G and equation 4.4.
Let's go back to. This.
Oh, oh duh.
Um, so for dust. Um.
So.
What was said there? Is.
If we are to be consistent with.
Equation 4.4 as equation will be 4. And.
Why is it? Why is it obvious that
we must have equals one is equals 0?
Right.
If we are in the movie reference
frame, then G is. Diagonal.
What are we choosing?
What's minus plus here?
Just one. OK. So. If.
In the case of dust.
This is a question.
Here is if we going with
this and that's here. Then.
If that's consistent with this in the
case of dust, dust is the remember
the case where the pressure is 0,
so it is a an ideal fluid,
which being made of dust,
you know, which is stationary,
is not banging against the dust particles,
not bang against each other or banging
against the walls of notional container.
There's no pressure. So, so, so.
P in this and that is 0.
That turns into a row U cross U + C
row G and if this is to be the case.
If if she wasn't zero in
this general expression,
then would we would see an AAG in this term,
which we don't.
So this is so this general term.
The general expression here has to.
If it's to be consistent with that,
then all the way that can be is
if A is 1 and C is 0, so we can't.
So we deduce that that term can't be
present in the Azure momentum transfer.
So. And.
Dust implies see a equals one,
C = 0.
Um and.
OK I I can pop you expand on on
on the note for that a little bit
to to make it less need a little
less unpacking now in the moment
I'll move reference frame the the.
There you is equal to 1000.
And G is. You could just that.
So there T00 is a row. Plus BP.
You 0U0? Plus. And. DP.
G00 which is minus one which is.
Row plus BP.
You one thing one is 1. Minus.
Probably DP. Yep. So. OK.
Does that make sense? So so. So. So.
So again we're we're we're picking
A-frame that with with this simple
so this is the frame in which the
the dust is not moving so all these
motions of dust are just sitting there.
Not moving with respect to us and not
moving with respect to each other.
OK. So all for so for each of
them or the moats of dust. The.
Velocity 4 vector is nice and simple.
Just one the the the time component
is 1 and the IT is that familiar from
your recollection of special activity.
I'm seeing some some nodes and
some sort of slightly,
I'll look that up later take notes
and G is a simple in this frame.
So just plugging these numbers
in with T00 is that. And tig.
It will be raw plus BP.
Well. You I the the the UI are
are are are 0 * 0. And plus DP
and and and the. Metric here.
Is 0 operational. And all of the terms on
the diagonal are ones, so that will be.
One if I and J are equal and 0 otherwise.
So you see again.
Because in the end if you think
of the special artistic 4 vector
for velocity. And that is.
So. And. If you could X well.
New by TD tour. Which is.
Well. If, if, if our particle.
Is is is seeing in one place then the.
So the the the four velocity.
Is the rustic velocity.
It's the the the special
components of it are well.
Be X0 by D Tau, DX1 by D Tau. And so on.
But X0 the, the for for a displacement,
the the, the sort of.
X cosky diagram. I I displacement.
We'll have a.
The X and A.
The T so if a displacement has no
displacement in the spatial direction
then that will be purely timelike
but purely parallel to the time axis.
So X0 will be DT by D.
Tall.
000, which is equal to 1000
because the proper time is the
same as the coordinate time.
In the case where the displacement
is purely in the time direction with
displacement isn't moving, so some.
That's a very potted recollection
of what the special University
for for velocity is. But.
Team work for at the moment that
that that velocity is. Simple.
In the moment helical moving reference frame,
because a particle is
then moving only in time.
That's another way of thinking about it.
So the the the speed of this particle,
it's just sitting there.
The speed of me in this in this
room is I'm moving through time,
but not through space.
So the rate of the rate of
change of my time coordinate is
the same as the proper time.
And I'm not moving through
space at all in this frame.
So plugging those in to this
I recover this and this one.
And equation 4.7.
And is the one which I know we've
talked about a bit before about the.
The argument of a perfect
fluid being diagonal because
there's no preferred direction.
Because there's no shear.
So if that's to be if that this
question here is to match that,
then this D will have to be 1.
And. Uh. Throughout 4.7.
And. Put with other thing
is sage. 4.4. About 4.4.
Um.
Yes, which is that that
implies that T00 is equal to.
Rule plus. B P -, P.
But 4.4. In the stream.
Is that T is is equal to rho.
And U0U0 but equal to row so
this has to be equal to row.
Which implies B is equal to 1.
So the. The, the,
the the logic there is that.
When we were talking about but this,
but here we were asking what
essentially we're asking.
So this is modelling,
this is mathematical modelling.
You're asking what mathematical
structures can I use to?
Pick up the important features
of the these physical.
Objects, and so the mathematical
modeling of dust. What do we
have available? We have the.
Momentum of the particles.
Which is, you know each each
individual particle has a momentum.
It's a momentum 4 vector.
It's a special artistic you know
the mass times the the you you get.
M * U the If the special artistic
form momentum we have the flux that we
worked out asking how how these this
assembly of particles so so that's the
momentum of each individual particle.
We also worked with the flux of vector,
which describes how the assembly
of particles is moving.
How many particles are there going
through a unit area in the speaker
directions and the time directions,
and and and so on and.
So those two things we have to play with.
Richika punch. What?
What happens if we multiply if we take
the tensor product of those two vectors?
Call it the this this this tensor tea.
Because we can compose,
we can construct our second rank
tensor from first rank tensors by
somebody taking the other product.
And we know we have to have a
second rank tensor here somewhere,
because if you remember if we look at this.
And. Box of dust.
Then as the dust is moving and
then it's got dust in it.
And add the box is moving if we
are instead in a different frame.
In which the box is moving.
Then it will be length contracted.
By a factor of gamma? By of of gamma, yes.
So the density of the number density
of of the things of the box will
go up in our frame similar to
the boxes got smaller.
But when those particles are moving past us?
All the same speed because they're just,
they're not moving relative to
each other at the moving past us.
There's another factor of gamma which is
the you know the gamma in gamma v ^2.
So the.
Number density goes up by a factor
of gamma because of the contraction.
And the energy density goes up
by a factor of gamma squared.
Because of that times the the
change in the rush momentum of each
individual particles so that gamma
squared so when you change frame.
From the moment alcovy frame,
moment alcove with the dust
to something else,
you're getting a gamma squared.
That's telling you that is not
lowering transformation of a vector.
It changes that that that gamma
squared is telling there's
rank 2 tensor somewhere here.
That's hinting.
You've gotta find a Rank 2 tensor
to play with this here.
What you said is that number density
goes up when two objects several
have speed relative to each other.
Meaning that if I were able
to move at the speed of light,
would they see everything be a black hole?
Can't really.
And so that's just not a question.
I mean, I think that.
If you were. If something's
moving past you very rapidly.
Then yes, it becomes smaller and and
and more and more compact and so and
so the space-time around it would be.
The measure would be different
because you're moving.
In the same way that that, that,
that the length contraction,
time dilation, the measurements of
things happened differently because
you're because you're moving.
So that's the so the the the metric
tensor will be the same but the
coordinates in a different frame.
Would be differ.
Because everyone's frame is is
equally valid that that's that's
the relativity in the relativity,
but in in in the generativity.
It doesn't matter who where you are,
who you are, how fast you're moving,
your coordinates are as good
as anyone else's.
And GR is all about the mathematical
consequence of that statement.
You call your quarter perfectly
good at everyone else's,
so when you move at some very
high speed and extra mass.
Your coordinates would be
different my coordinates.
The metric will be the same the metric.
The distance element of the
space station moving through will
be the same as geometrically,
but the numbers you get out
will be different.
And what we learned from special
activity is that if you have a.
Collision of.
This is this is it.
One of the things that that provokes
system of alarm when I'm teaching
this in second year is if you have a.
A collision between two relativistic things.
Yeah,
you have a a lump of a putty
and a A some sort of barbarian
commanded relativistic speed and
hit the party and the and the two
inelastic collision and and and
and the thing goes off afterwards.
And you,
you balance Ross 54 momentum
and you work out how much the
different components are and and
and the mass and blah blah blah.
And there's more mass afterwards
than there was before.
And folk go what?
How can we be more mass afterwards
than before?
Does that mean there's more?
No, it doesn't mean. Yes, that's it.
That's it. Once you have that
in in that box if you like.
There's no because this box
has now had this other.
Rustic party will come in and smash
into the party and and go off.
There's an awful lot of kinetic energy there.
And that adds to the to the
incrementum in that box.
So there's no more gravitating
stuff in that box.
Not there's more mass in the box,
but because or there's more mass
in one frame and another frame,
there's not more mass.
They're just in the environment.
Yeah, etcetera.
I see running again,
but but the the the point is
that if the argumentum that is
different is the source of the mass.
Is why we have a tensor here,
a geometrical object,
and how we were able to, you know, see.
These things to play with her,
to make changes out of that.
And so that question,
that exercise is seeing, OK,
stepping back a little bit,
if you really were starting this
from scratch by saying what
do we have to play with,
how could make changes.
Then all you have to play with are you,
you cross U&amp;G,
so how do you add those together in a
way that's consistent with what you
have sort of seen in in this case?
So.
So yes, that's a. Well,
people are putting on their coach now,
so I'll be quick question,
but why do we need to,
why do we need to add on top of the
velocity vector here the the metric?
Because physical condition can tell
us why we necessarily need attention.
Because we have this, yeah.
We we don't have to I mean this
isn't telling us we have to but
but the the the point of the of
the exercise is saying if you.
If you start from nothing and see
how would we go about about making
attention from what we've got.
Then what do we what do we have?
We have, we have you and we have G.
And and we can put them together in
what seems to be the most generic form.
Given some constraints.
What we otherwise know the expression.
And here this has to be consistent with this.
What does that do?
What does that tell us about the,
the, the, the, the, the, the,
the in principle things,
the, the, the, the,
the values of EBC&amp;D in in that express?
So that's that's using your imagination.
What does the other part of the argument
constrain that imagination to to to be?
And I'd better stop there. And.
No. Could you get somebody
else to go but because, well,
you do have any lectures to go to,
but you probably have somewhere
else to go this evening.
And that I think that is I think
and 5:00 o'clock this Friday is the
end of teaching for this semester.
OK, one moment too soon.
And I hope you'll have a good time
over the Christmas break and New Year.
And I look forward.
Well, I won't see you next semester,
but I will I think keep the officers
going every, every, every couple of weeks.
I'll try to make an announcement
of that in the next semester.
But unless you are the officers.
And have a good time. I won't.
I'm sure we'll bump into you.
Enjoy and.
Or enjoy?