Hello everybody. Welcome to Lecture 11,
which is our last lecture of GRG one
preparing you for the likes of G2,
so our next semester.
So we are astonishing for
me in very good time.
So we've made excellent progress
and so I think we might well.
We'll have we're not under
time pressure this week.
It might even finish early.
Where I got to last time was I was
talking about and I hope justifying
the idea of the the the the physical
statement of Einstein's equations
relating the Einstein tensor which
is composed of the richer tensor
and the cover change and the metric
with and equating that for making
that proportional to the energy
momentum tensor which characterizes
the distribution of energy momentum.
In our particular volume,
that sounds rather abstract.
What it means is if you have a single
mass in the in the center of the universe,
that is a characterized by an
appropriate Environmental Center.
If you have an extended object
or you have a universe full of
certain distribution of argumentum,
that's characterized by the energy
momentum tensor and intuition
tells you what the consequent.
Distribution shape or space-time is.
And that is.
Basically it as far as the
the equations it goes.
However,
one of the first one of
the solutions to this was.
Which you will learn about N G2
I think all you which you've also
learned about in in in the cosmology
lectures you've had I'm sure is the
notion of the expanding universe.
There's a solution to Einstein's
equations which consists of a
universe which is expanding,
and this was felt.
And at that time in the 30s,
I think to be obviously unphysical,
obviously unreasonable.
That can't be, that can't be an
answer so thought to be illegitimate.
Solution, and that was therefore fair enough.
That's fine. That's that.
If physical solutions.
So I'm saying to you that he his
work was not done at this point
and he added another term to this.
And
ah.
Sorry, which I will write down.
In that case which was G?
You knew plus Lambda. Lambda G.
You you equals Kappa T mu where Lambda is.
Auto focus off where Lambda is just a scalar.
Multiplying the.
The metric and he said OK that's
the that that's clearly what I
should have said first time and
and when you add that term one of
the the the the the whole universe
solution you can get a a static
whole universe solution out of it.
And it was only later,
when the Hubble Metro expansion
was discovered that that you know,
and it became clear as a matter
of observation that the universe
was in fact expanding,
that he referred to this
as his greatest blunder.
That he had wimped out effectively,
effectively by believing experiment
to by believing experiment too much
and not going with the original plan
of just sticking with the original
version with Lambda equals 0.
Now in a further twist,
as you probably are aware.
Much later on, starting in the.
2000s. It became.
Clear from supernova observations that the.
Aiming to discover whether
whether the universe was,
on the largest scales,
positively negatively curved or flat,
it became clear that that the most distant
supernovae were retreating from us.
Fair enough, but at an increasing rate.
So the universe seemed to be a inflating,
inflating, being driven by some sort
of of of of external pressure term,
and that was the motivation.
In the you know the 2000s for bringing
this version of Einstein equation back,
gain nonzero cosmological constant a
nonzero value of Lambda no no referred
to as the cosmological constant
which in has the effect was super
children values of of Lambda of.
Adding our.
A negative pressure to what is effectively
a negative pressure to the universe,
which means the universe ends up
expanding at a slightly accelerating rate.
That's all very far from the future.
The point of of saying this is to
think what I said last time really,
that this is, I guess,
which is corroborated by observation.
Einstein thought better of it.
Then she just been again and we've
seen change her mind again to
bring that that that term back.
But the point is that this is that
these steps here of this version
or that version of this version of
that version are by this point not
mathematically but physically and
observationally and astronomically motivated.
And, and and there's a lot more one could say
there about where does this term come from,
quantum gravity or something like that.
And and you know the there are a variety
of rabbit holes we could dive down at
this point but we shall forbear. And.
OK. I'll move you feel swiftly on
unless there any questions about that.
No. OK. So what I did the last last time
was this all leads us to the the point
where space and this is a formulation
attributable to I think it's John,
John Wheeler space tells matter how to move.
And match your speech to curve.
You be able to see that slogan before,
but now you have the,
the essentially all of the mathematical
background to understand what's
actually going on in that slogan,
what that slogan is actually summarizing.
And to some extent, at this point,
the job of G1 is done.
My job of G1 is to give you
the mathematical technology.
That will allow you to go in and
make sense of G2G2 next term Doctor
which is taking over this year.
It's actually about solutions to I,
I sense equation, it's partial solution,
the Freedman, Robertson Walker,
Friedman, Robertson Walker solution,
graphical waves and so on and so on.
And we can do that.
So that's the that's the payoff.
But I can't take you to the
threshold of of that and I'm not
give you any solutions at all.
So we're going to look at one solution
of Einstein's equation in outline.
I just know.
The pretty picture to show you
what an orbit in our in our
space-time is supposed to look like.
I said that. OK.
So you will recall that somewhere,
I think in part one,
I mentioned the idea of natural units,
units in which the speed
of light is equal to 1.
Which essentially is deciding to.
Use the light meter as our source,
as as our unit of our unit of time,
in units of which light light moves
at one metre per light meter or one.
It's not to mention this is merely unitless.
People doing worrying about GR
tend to work in a set of units in
which seat G big is equal to 1.
And that means that the.
Well, we can take the squared equal
to 0, equal to 1 or not, but the.
The big had the value 7.2 to the
minus 28 meters per kilogram,
which, just like CB equal to 1,
is a conversion between seconds and meters.
This is a conversion between
kilograms and meters,
so that masses in these units
are measured in meters.
Which makes sense when you do,
which seems weird, but makes sense when
you discover the functional solution.
Discover that the structural
solution has in it a parameter which.
Has the dimensions of meters,
which is essentially the the event the size
of the event horizon of a collapsed object.
So jumping ahead a bit in
the structural solution,
there is a size parameter inside which
the only time logistics are inwards.
So in other words,
even light cannot escape that there
is no time like there's no you're not
even a null path to greater radius.
What is the radius?
In other words,
the black hole.
So the.
What this far should solution tells
you is that there are such things
as black holes which have a radius
which is proportional to to G&amp;M.
In which context it makes
sense that there is a natural?
And correspondence between mass and the
gravitational radius of of that mass.
So the the mass of the sun
for example is 3 kilometres.
That being the side of the black
hole into the the the the size,
which if you compress the sun
inside would make a black hole.
And.
And you you can divert yourself
by work by working out what your
gravity your own gravity should
radius is and survey small number
well it'll be 10 * 28 times whatever
your weight in kilograms in meters.
And as a curiosity there.
What this also?
A curiosity here is that even in
classical gravitation physics,
the thing that controls the orbit
Newton's gravity potential is g / R.
As you were just recall and it's GM,
the controls the orbits,
the behaviour of of particles in
the source in the solar system
and not G or M separately.
They never appear separately.
And what that means is that it's
relatively easy to find what GM is and
and and and if you're doing classical
mechanics of the in the solar system
what you the parameter of interest
is GM and it's quite easy to to
determine that from looking at the FMD.
Of planets going around the sun
and you can get, you can get,
you can estimate GM to I think
one part in 10 to the ten.
It's extremely accurate.
Do you to an accuracy where general
statistic corrections matter.
But the only way you can find what G
is is using terrestrial experiments,
such as you're looking at plumbs
next to mountains and so on.
As you will be aware,
we can only do that to about
one part in 10 to the five.
And the way you find what the
mass of the sun is is by finding
what GM is and dividing it by G.
Through the mass of the sun in
kilograms is obtained by GM over G
and has the error of G which 10 to
minus 12:50 and 10:00 to the five.
But the error uncertainty of the sun's
mass in meters is essentially the
error of of the of the gravitation power GM.
The mass of the sun in meters
is known to about 10 * 110 to
10 kilograms .10 to the five,
so there's a nice inversion of
what you might expect there.
Sorry, that's all in a
big parenthesis really.
Moving on the. Point here is that
we're now going to look at the.
Solution of intense equations
in a particular approximation,
namely the weak field approximation,
the approximation of.
Small masses, so small central mass,
something planet size or star size.
Milk that just is isolated
in in the in the universe.
Or or equivalently the approximation
where you're looking at the solution
for a mass but you're you're quite
far away from the mass of the masses,
so 2nd order terms disappear.
And then we do that and I'm going
to go through this in in outline
rather than in in line by line
detail is we approximate the metric.
By the Minkowski metric.
Which is minus plus plus plus
diagonal plus. A perturbation.
And the point here that.
This is a perturbation.
H is small in the sense that
the magnitude of all the values
of H is much less than one.
So each squad is is is ignorant.
Now that's a matrix equation
and not a tensor equation.
But it turns out that.
For reasons which we could expand on,
but might make expander if we
have more time at the end,
this can be treated as if it were a tensor.
And what we can then do is. Reexpress.
The point I want to make here are that.
What you're doing here is essentially
changing into coordinates in which the.
In which each can be regarded as
a tensor now, and there's a couple
of ways of thinking of that.
One is that you are making a particular.
Particularly our particular coordinate
transformation which is constrained
by the the constraint that this be
small or you can regard this and
this is quite quite a productive way
of regarding of thinking about it.
You could regard this as being as is
it asking. About the behaviour of.
Tensor each in a Minkowski background
as fluctuations. Unlucky background.
The point is that this is.
Using that you can then calculate
what expression is for the.
A connection for the Riemann tensor
in terms of H as opposed to G.
And then express Einstein's equations,
which are of course obtained from
the room tense contractions in terms
of each and and because you then
at that at that point are dealing
with something which is small,
where second order terms can be neglected,
that becomes easier to solve.
And the solution is look at this
just to get the terms right where
each the as a quasi tensor is.
Diagonal. Each nought nought
H11H22H33. Yeah. All of these.
Are equal to. The same.
5 Phi.
I'm plugging this back into the minkovski.
Metric using the copy metric again
the that means that our solution. Is.
Diagonal and minus 1 + 2 Phi. 1 -,
2 Phi, 1 -, 2 Phi, 1 -, 2 Phi, or.
With an interval of.
That's good. Plus one minus.
Just checking up to make sure
you get the signs right.
Where the Sigma there is the.
Is the the the spatial sector.
And.
So that that's our,
so that's the the metric in that
low mass weak field approximation.
And I've. Mr Bit here which is
fiddly rather than hard and and
and and shoots for example goes
does go through it step by step.
It's not terrifically edifying but
it's sort of reassuring that it's
actually quite a short calculation.
Roughly I think even they're part
of what he says is if you then go
through this and and and work out
what are the components of our are
in in details several pages of
algebra but it's not hard algebra.
Just turning the handle.
Umm.
So that's all very nice that's that looks
like pretty and jumping ahead because we
have time and it's quite interesting you
will discover that when you look at the.
We discovered this partial
solution next semester use.
You discover that this ends
up being the, which is the.
Exact solution for this same
problem of our single central mass.
You get an expression for the for
the for the metric for which which
is is equal to this to 1st order.
So in this case we have obtained
this by demanding that well,
but by building on the fact
that each is small.
We can recover this as as the low mass
limit of the Schwarzschild solution by.
Depending on Phi being small
at that hand, yes.
I second. What these fees?
Physical significance, good point,
good point. This ends up being.
Remarkably enough, what comes out of
this is that if I just the numerically
the same as Newton's gravitational
potential. So using gravitational
potential pops out of this.
And exactly the place you'd expect.
And or or. And it it turns out that this.
Readius.
It's 2.
The the radius two GM.
Turns out to be the that that
that this, which is basically.
You can see the two coming from there.
That radius is the radius which
I mentioned, which is the.
Size where the structural
solution gets interesting and
where the black hole appears.
So the size of a black hole is.
Dependently it directly linked to
this GM parameter which comes up
which appears just as this potential
factor in in in inside the metric.
So that's very nice.
But what can we do with that well?
We've been half the thing what we we
have worked out at this point that.
Are are.
A solution for instance equations.
The next thing we have to do is workout.
How do things move into that space-time.
And we can do that by using
the geodesic equation. Uh.
Right.
So. Point.
So the geodesic equation
we've seen versions of of it.
But if you look back one of the versions
that the sort of prime Prime primal
version of the judges equation is,
is that one which is the one saying
that as as you parallel transport the
tangent to a geodesic along the geodesic,
it stays tangent to the geodesic.
So that's the mathematical version
of walking in a straight line.
And the the part you you, you,
you draw out by walking straight
line is as you desire.
That's not particularly convenient,
but let's instead recall that the.
We can talk about the full momentum
of an object as just being the
mass times the that that this.
The the the full velocity.
In this case, we're taking the
full velocity to be the full
velocity along a judic, so this.
Judy equation turns into an expression
involving the. Momentum of an object.
And then the. But that's the
geometrical version of it.
You would recall that the
component version of that.
And get everything in the right place.
And choose their indexes
that I am consistent with.
Who's that? It will evolve in
the covariant derivative of the.
Of the vector P so that's just
the component version of this,
which is is the A scaling
of the duties equation.
So asking what are the what?
What are the are the is the the field
of P vectors of momentum vectors which
satisfies this equation and thus which.
Indicate.
The judaics in this space
tech in the space-time.
And that in this space dangers and
because this covad derivative is
picking up curve the the the way
that the the components change as
you move around the the space.
So the curvature if you like is in
the coverage of the of the space
we're looking at here is in that.
You could be derivative.
And now so that's exact.
This this far, no.
Because we're interested in
the weak field solution,
we're going to take the another
weak field approximation,
which is to say that things
are going to be moving slowly.
And what that means is that for
the momentum 4 vector of the
geodesic we're interested in.
The. Time component.
It could be much larger than
the spatial components.
So things are going to be
moving through time faster than
they're moving through space.
They are moving slowly, in other words.
So what this? Implies is and again
keeping things neat P alpha P mu
comma alpha plus gamma mu. And.
Alpha Beta P Alpha P beta. Equals zero.
All I'm doing there is simply.
Breaking that out in a slightly
longer version, but if.
The 0 component of these vectors
are much larger than the.
And spatial components,
then we can then discard all the
spatial components in that sum.
So the only terms that
will survive in that sum.
Are going to be.
The 00 times. And similarly if the
only term that survives in this sum
here over alpha is the zero term then.
The. And and given that.
The.
P is equal to gamma M.
When in M visa.
No comma M1. The. The momentum for
vector is proportional to the gamma the.
Matter of the object being moving
plus this one V XYZ vector here.
The that that means that the.
And at low speed, regardless small,
gamma is 1, the zero the the 0
component is just M so it's M.
DP mu by D Tau that survives.
So what we're doing here is this current
derivative in this approximation.
I should probably see.
OK, now what we can then do?
It go back to the metric.
Here. And do the things that were were,
you know, I hope, fairly well rehearsed
that calculating the Christoffel
symbols corresponding to this metric.
And we find most of them are zero.
I just saw from the case and there's a
hydrogen symmetry and the ones that are not.
Are gamma 000. Which is equal to.
5 comma 0 plus terms of order Phi
squared. Which number is small?
And gamma I-00. Which is.
Comma, G.
Both genes evolving fine before.
OK. And what if we then look at
this component by component,
we find that therefore M.
DP naughty by detour. Yeah,
I can't, right? Plus gamma.
Comma 005 comma 0. P nought squared.
Which is just. MDP nought by D Tour plus.
I'm expecting to see our.
The term here. No, it's good.
Ohh yes plus m ^2. Have. Yeah, squared.
Phi comma not equals 0 or.
DP nought by D Tau is equal
to minus MD Phi by DT. Toll.
And what that is saying is that.
The. Reach the the the.
Change in the energy.
Of this particle.
Is proportional to the change in time.
Of the potential,
and we given that there isn't more
mass certainly appearing here,
that's going to be 0.
In other words,
that's saying the energy is
conserved as the particle
moves along the geodesic.
Through phoned one of the relevant.
Descriptions of the motion.
Now looking at the special one.
Uh. And looking at. This one here.
What we then discovered
there is that DPI by D.
Tall. Is equal to minus M.
If I. Comma I which is the I
spatial derivative of the.
Of this potential, which is
just a funny way of writing.
The Richard change of momentum.
The force. Is equal to minus. Gradifi.
Which you will recognize as the equations
of motion in Newton's gravitational theory.
That the the particle moves in such
a way that the rate of change of its
momentum is directed along the gradient
of the gravitational potential.
Which is very gratifying
because this means that that
the low energy approximation,
that low energy approximation
of of of Einstein's theory
recovers the manifest successful.
Theory of gravity that Newton
developed for from starting from
a completely different place.
So.
I think I I've I've missed
these quick questions.
Those are in, in, in, in, in the notes.
So I think that's that's basically budget
and and I think that's a a remarkable thing.
I I do know what know what I may appear
to the world but to myself I seem to be
more like a a boy a boy playing on the
seashore and diverting myself now and now
and then finding a smoother Pebble or a
prettier shell than ordinary with great
ocean of truth lay all undiscovered before
me there's a certain rejection to that.
I think on Newton's part he
knew that he had done great.
Things and found beautiful,
mathematically beautiful
explanations of what happened.
But there was much more to find.
It took 300 years to find
something that was better.
But as we've discovered,
what he did is contained
within a later theory.
And that is essentially it.
I we were ahead of time.
I've slightly spun out by talking about
spatial solution and G2 and and and so on,
but I think it's actually a first that
I've managed to get to the end without
galloping through the last lecture
in a something of a mild panic but.
That we've got here,
we've got technology,
you've got one solution and you have
the browser uplands of G2 to find all
sorts of other solutions next semester.
And so we we might as well stop
there or we'll get questions.
Questions. But we can do.
If we want to run off, that's fine.
If if one turns into an impromptu
supervision question session,
question session or chat session,
then that's fine too.
How about it? Question over there.
Can you say where's tricked
ourselves to the motion by N
relativistic particle we have this,
where does this come from like.
So uhm. The question there is um.
This approximation.
Why that drove that approximation?
Is basically comes from.
From this. So you may recall,
you may not recall that when you
talk when a special activity,
you talk about the relativistic.
Velocity.
It's it's a form momentum
which includes the the.
Let's not talk to the
momentum rather than velocity.
The full momentum involves the spatial
momentum and the time component
of the four momentum which is the
energy of the particle and you
discover that the low speed limit
of that what that in in in the
frame in which the word is that.
Does that look sort of familiar?
Have you seen something like that?
Before or vaguely enough that you
believe me that that's the key, right?
So that's that's the key and and
the low speed in the frame of
which the particle is not moving,
you discover,
good heavens,
that the the 0 component has is
gamma M which doesn't go to zero as.
As the speed goes to 0,
the the 0 component energy is gamma M
or in physical units gamma Mt squared?
Or will be SU equals MC squared.
That's where equals MC squared comes from.
But in this case. The.
This time component is always gamma
M but in the case where you're
looking at particles which are moving
only slowly and by slowly meaning.
Much less than the speed of light.
Then. Each of these spatial components
VVZ will be much less than one,
much less than C.
And so these people,
and that will be therefore true of the.
Overall momentum component,
so the spatial components.
Will be small more than the energy
components simply because the energy
component is primarily the particles mass.
So in the case where essentially
all of the particles energy is in
the form of its mass as opposed to
its mass and its kinetic energy.
Then we can solve this in in that
limit to get Newton's theory.
So that's telling us that Newton's
theory goes wrong.
When things move at rustic speeds,
which is terribly surprising.
And what that means is things go
wrong when things were rustic speeds,
a because they're moving rapidly
and B because there is a component,
there's an element of of energy. In the.
Simply by virtue of the particles motion.
Which, which is which will grab,
which gravitates.
So the the the the.
The energy that's in our particles motion.
The instruments in particle motion,
it was Einstein's theory.
It's Andrew Mentum, the gravity.
It's not mass.
And that is a thing which does not.
It was something moves faster
than it gravitates more.
And that's that's completely
alien to Newton's theory.
I just you cannot be there and your theory.
That's why in a sense this this
has to be the the limit in which
Newton's theory will pop out,
the case where we're
ignoring the gravitation,
the gravitating influence of kinetic energy.
That's the physical interpretation of.
So this is the. As I mentioned,
this links to this partial
solution in the sense that.
There. This metric here is derivable
as the as the little file limit
of the partial solution. The.
As far as your solution is the exact
solution to an approximate problem.
In the sense that it is the
solution to the approximation where
the universe has one mass in it.
And that's not actually true there.
There's more than one star in the universe.
But in certainly in our environment there
it is a very, very good approximation.
And it's the solution that is used
for essentially all of the relativity
corrections to things that GPS,
to things like precise timing,
to things like the deflection of radio
waves by going near the start that,
that, that the, the, the,
the sun and the Eddington Dyson.
Observations of the deflections of the.
Ohh of um.
Dilate as it comes past the sun in Eclipse,
which you've heard of.
Yes, eddington.
And perhaps I didn't mention that
and well I didn't mention though,
but I thought yeah it might be
that that that that sort of
normally comes later than the,
the what I'm seeing but earlier
than G2 because amongst the the
the effects of this of this.
Solution to.
GR Well,
if you remember back in in
lecture one beginning in part,
one of the things we discovered
was the coolest principle tells
you that that that light must
bend in the gravitational field,
that the wholeness of things falling down,
down lift shafts.
But we didn't calculate how
much that that deflection was.
Now you can calculate it.
Based on the. Gravitational.
Red shift of our particle.
So one of the other things we
mentioned in part one was the
idea that as our photon.
Claims through graphical field it's
frequency changes and to the extent
the frequency is are a proxy for a clock,
a photon oscillation is proxy for
clock that is telling us that time
moved differently at different
at different gravitational
potentials and from that.
You can deduce.
Through a few a few steps,
but not too many,
that there will be a particular
deflection of Starlight as it
comes a past a gravitating body.
You can work out what the
selection will be purely from that.
And you can also use this solution
this approximate solution.
The this week full solution
of Einstein's equations.
To work out what the deflection what what,
what, what the geodesic or
photon is as it goes near a mass,
whatever near counts as and
you discover it's deflected.
Of course you know very way
better that it didn't happen,
so there's a deflection of
light as it goes near mass.
And you can calculate what the
angle of that deflection is.
You discover is twice the angle
that you got when you use only
the gravitational redshift.
Which and. And so Einstein and Eddington.
Got to the first answer first.
I think Einstein got the I think
it would be representing 13 or
something that he worked out how
much the deflection would be based
purely on on on the gravitational
redshift and that was the prediction
for how much that affection would be
and it was only after in about 19.
16 or 17 I think when the when this
solution was was available to work
out what the deflection would be
based on the field equations and so
there's going to be a reflection.
The OR the prediction and how
do you find that deflection?
You what you look at star at
stars as the as the lake from the
moves near a battered body.
There could be massive
body in the neighborhood.
Yes, there is the sun.
Unfortunately you can't see the stars in
the daylight because the sun's very bright.
So you wait for an eclipse.
And conveniently,
there was a total solar eclipse visible from
some parts of the of the planet in 1919,
so just after the First World War.
And.
You know, I could go on the
story for quite a long time.
With all sorts of layers of of interest,
but the short version,
the focusing on the on the physics version.
Is that it was a fairly prediction
at this point of GR that there
would be deflection.
And through the next edition
mounted by Edison,
who is the head of the
Cambridge Observatory and.
Herbert Dyson. Some Frank Frank Dyson,
who was the director of the grand jury,
and they put together the equipment you've
scattered because of the First World War,
but the equipment to make an
expedition to Brazil and the.
I don't want to keep Verde islands anyway,
somewhere in the southern Atlantic where
where there was it was going to seasonality
and long observational story later the the,
the, the three three possible outcomes of
that of the observation were no deflection,
which is what in the sense that the the
the intoning, they would see the what
was called the Newtonian deflection,
which was the the one that that
Einstein and Co had produced based on
purely graphical redshift and the full
Einsteinian deflection which was the.
And they obtained from this,
which is twice the Newtonian 1. And the.
Observational but opposition nightmare things
which were supposed to work didn't work.
There was rain though, you know,
in the field, literally covered in
mud in the 10 minutes of totality.
But they did manage to exclude the the,
the, the reflection case and.
Arguably and correctly exclude
the Newtonian version,
and thus confirmed by direct observation
that the deflection was what Einstein,
Einstein Field equation set,
and Einstein became a worldwide
celebrity and so on.
And there's also a footnote to
that story which are fascinating,
which I might put something and pass
on to you because it's interesting.
Before we start off with this,
ohh yes,
but the the point is that that's
an approximate solution,
but the smart solution is the exact
solution to the same problem which.
Can be approximated and the
smart solution because the metric
would be approximated by this.
So that was a very long answer
to a question which I've slightly
lost track of, but I other more.
You have a box?
Yeah, we know that photo.
That will be the gravitational well,
yeah. Four, yes, so, so.
Right, so the question is why do
why are photons deflected by this?
And the answer to that is.
That this is the. What we have here is the.
Well. That's the equation which solving.
Uh, what? Asking what is the?
Geodesic traced out by a Momentum 4 vector.
Now we motivated here by by describing the
momentum of a massive massive particle, but.
I don't think we covered that here.
You can also talk about
the momentum of a photon.
Even classically you could talk
about the momentum of photon.
How much I as all the momentum
of an electric field.
As an electric field interacts
with with matter, it will transmit.
Momentum to it, in some cases through
the Lorentz force law and so on.
And that you so you can talk about
the momentum of our classical field
and if you think of of the quantum
mechanics you know you will know
that the photons have have have 4
vectors they they have, they have,
they have energy and momentum and
So what we're solving here is the.
Judic. Of the momentum 4 vector.
Something irrespective of what
the momentum 4 vector of.
So for a massive particle it'll be the
mass times the four four velocity of that.
Of that object for a photon that the
the previous didn't really mean much
in the matter 0 so we have a different
definition of what the full mentum is.
But it's still that we're,
we're we're solving however in.
In this expression for the.
The potential Phi.
The mass here is the mass
of the central object.
That's the mass of your star or
your planet or or or whatever
you're talking about.
So does that sound evasive or
is that the does that cover?
Because of the.
And this this mindset here.
Of the final final. Well,
I think that's that comes just because.
This is our a rewrite of.
MPIBYD. Tall plus.
And I think the whoops,
we're just rearranging that equation.
That's the geodesic, that's the
space part of the geodesic equation.
And so just rearranging that it's where the.
It's where that when you're saying. Appears.
It had, but it worked. A question there.
Lucky.
Because we go back to.
Yeah. And. Each. Here in detention,
because there's no reason why it should be.
All we've done here is, is, is.
Write down the. Component of the metric
in a particular frame. Being the.
Components of the of the Minkowski metric
plus A+ some other other components.
So this is just a matrix equation.
So there's nothing.
We're right in that and not care and not
make any constraints of what each is.
Each could be as big as because we're like.
We are, however, choosing.
The the point of doing this is that we
want to see these are perturbations,
so we want to say these are are small.
But that's not attention thing to say.
You can't really talk about attention
in that context, attention being small.
So we couldn't write G equals ETA
plus H and say each is small.
Because that doesn't really mean that
that that's not a sensory thing to say.
If you're like,
there's a better way of expressing that,
but we.
Are saying we're making this constraint,
this constraint as a matrix constraint.
So it's true, it's this approximation is
meaningful only in a particular frame.
So yeah so that's basically this
is a frame dependent approximation.
It's only in one frame in basically
the local inertial frame that we
that that this we can talk about
these components being small.
And it's then involves a bit of
stepping back and I think about it to
discover that when you turn the handle,
you can review this as a tensor
on a on a on a flat background.
So I think that, that, that,
that,
that the basic AHA is that that
approximation,
that those those pair of things
is meaningful only in one in in,
in in a small set of coordinate choices.
Namely,
those which are which are almost minkowsky.
And and and the there's a dangerous
bend as a section there which I think
was added after veteran requestion
what one of the year which talks of
which goes into more detail about
that which talks about what how what
you're doing here is either talking
about tensor on a flat background or
you're talking about age choice here.
And if you've done a quantum field
theory as some of you will have
done you and certainly if you've
done classical electromagnetic.
80 and the notion of the wrench gauge.
You will discover that there is a
the notion of gauge fixing being
engaged choice essentially the
the exotic mathematical version
of choosing the right coordinates.
If you choose the right coordinates
then you can do all your calories
in a particular gauge.
Where things make are simple and
this essentially therefore engage
choice in those terms.
And that is time up, I think.
So I have the usual, usual second
officer tomorrow and I think we
have a supervision a week on Friday.
So I meet you some of you
then goodbye and enjoy.