Astronomy 2 Special Relativity Lecture 14.
It was the almost the end of chapter eight. I thought I might
manage to get to the end of Chapter 8 in time, but there was
still a little bit leftover.
But that's fine, because that gives us a an opportunity to
recap the the the flood of quite sophisticated, the sequence of
quite sophisticated ideas that we covered last time. And one of
the key things,
one of the key things of the whole of Chapter 8 is
the the sequence of of restatements of successfully
stronger restatements of the equivalence principle.
Remember the first one?
I was talking about uniform gravitational fields.
The second version was talking about local free falling, non
rotating laboratories, local and national frames and how they
were the thing that we could understand that we already
already in a sense understand from our pre relativistic
physics that's that's basically nuisance laws. And they are the
bridge to the I different way of approaching the question of what
gravity is or how what what it means to be moving under
gravity,
how we go from the physics we understand to the physics we
don't yet.
And the third version of it was a more mathematical version of
the of the whole responsible which said
that if you were
the
that
in general
if you are in a local inertial frame
you're moving under so moving under the influence of gravity
or moving well away from all gravitating bodies
and and physics works as
you can special activity says it does then there's nothing a law
that's expressed in geometrical form. Things like momentum,
energy, momentum is conserved.
Things like, well, let's let's stick with that argumentum is
concerned. That's a physical law expressed in geometrical form in
the form of a the argumentum vector that does not change its
form when talking about a curved space-time. There's no extra
complication that arises when you're falling through a curved
space-time of what would they highly curved space-time
that yeah. So the physics of that natural frame are not are
not any more complicated And that that's actually quite a big
deal because you could imagine that there would be extra
physical features that would happen because you're in a
curved the three time locally curved and they don't happen.
I'm going to do it because it is quite important, but I
appreciate it doesn't sound that big a deal right at this point.
But a quick question,
you don't. You've studied Maxwell. You've probably heard
of, but you haven't studied Maxwell equations yet.
But, but, but what I you you're probably aware of what I'll tell
you now. The maximum equations are the way I I mentioned them
in Chapter 2.
And Maxwell's equations are the way that electromagnetism so
late radio. All these things are described in a format which Jim
Clark Maxwell developed at the end of the 19th century.
And we can write those down in a form which is the right variant,
that is a geometrical form. It's mathematical, equally exciting,
but you can do it.
So there is a geometrical statement of macro equations.
Essentially says that the the direction of the electrical
vector is family variant.
So
to answer the first question first,
I'll say. So you The equivalence was telling us that the maximum
equations that work for
for our normal experience, our known
not living under neutron star experience,
those maximal equations will still work
in
I
in. In general relativity you'll still work in a very highly
gravitating environment without change. There will be a change
appears because you're in a curved space thing.
So if you had a radio receiver anyway, your receiver either a
gramophone or or else you know the the the the the the GPS
receiver on your phone.
If you took that close to a a neutron star and and and were in
orbit around a neutron star, would it still work?
Whose state would still work
was it wouldn't still work.
Have a chat
with that.
OK, having thought about that, let's ask the question again. So
you have a radio receiver and some some some radio pickup for
example. But you're in orbit around the neutron star, so some
highly curved
space-time.
Would radio receiver still work?
Who would see it would?
Who would say it wouldn't?
OK. Well, it would
because the conference,
because the government will say that even though you are in a
very exotic circumstance, you're in orbit around a neutron star,
you're possibly falling into a black hole. We'll come to those
later.
You You're perhaps going to be squished into nothingness
because the black holes, they evade the situation In your
local frame.
You're in freefall, so material, so anything that works in
special activity in any free falling frame still works.
So you're you probably wouldn't be able to pick up anything
because the radio waves that were broadcast from well outside
would be highly blue shifted with the same you pick them up
as you're falling into the black hole. But this thing was still
work
because Maxwell squeeze would be unchanged and that is
extraordinary that that's why the this, that that vote of the
conference is extraordinary because it's telling us that we
already know how physics works, even though it's really exotic
circumstance around a bit a bit of space-time at the very edge
of the universe.
And that it gives a big jump into how we understand the what
we see in the in those very exotic circumstances.
So that's why the equivalence principle, that version of it is
such a big deal.
It says we we already know at large chunk of how things work
near gravity
and as I say longer version of that in the in the state. Ohh by
the way, talking about slides reminds me that the I don't
think the order in the election was forwarder, but
I hope the request. I have taken the scribbles from previous
lectures and scanned them and they're in the lecture notes
folder. I'm not sure how useful they are. They were intended to
be useful but they're there and I realised I haven't put the
notes for Chapter 9 up in the lecture notes folder. I will
probably not today. I would have Are we check all over them to
see if they still went bits missing before doing so.
It's one of the things that we
touched on lasting
and
is that
we realised that because of the equivalence principle
we could see space-time tells matter how to move. We can add
the extra
and
bit of physics that
just as when we are standing still in a national frame, we
are moving through space-time in a very particular direction, we
are moving along the T axis, we move along a geodesic. That's
the the natural motion in that initial frame.
But that that that, that geodesic that we trace out puts
very easy trace out because we're just standing here where
our duty is pointing straight along the same axis.
If we turn that into a different set of coordinates then we are
just we. It's the wrong way of saying
it's easy to see what our geodesic is through a
space-time. It's just pointing along our time axis
in a curved space-time,
just as we saw for the the an individual moving across the
surface of the earth. A geodesic is a geometrical thing. It's a
geometrical quantity. It's not. It's A-frame independent
quantity. And so if it is and it is still true that we our motion
through space-time is what happens when we go along a
geodesic in that space-time,
the quote straight line in that space-time. Then we understand
how to move through a curved space-time
because the curvature of the space-time, the shape of the
space-time tells us you know what effect where that you would
what points that you do actually go through. And making this
sound very complicated. This is very clear in my head.
It's basically a long version of space-time that might have to
remove the the the the the shape of the space-time. This is this
is this is the what's being mentioned in those endless
videos of people putting weights on on rubber sheets and and and
and seeing the rubber sheets turn into a curved sheet and
then a a marble roll around in a in a in a in a curved path on
that on that the marvellous discovering the geodesic in that
curved space-time. And that's the point.
I don't want to bang on, bang on about that. Now
what we've been talking about is in all of this is a free fall
under gravity.
So we are talking about, you know, either jumping up and down
or falling into a black hole or falling. After all these things,
we're talking about free fall. But how does this relate to us
standing on the ground or on the floor or whatever and feeling
the gravitational force? Because we know there's a force we can
feel on the soles of our shoes,
and that's where the link back to acceleration comes in.
Because if the floor wasn't there
then we would move under gravity. You can imagine if the
flows under disappeared. You could imagine out of shadow
version of yourself following, falling, falling down, and
you're looking at it and getting it moving fast fashion. Are we
away from you Faster and faster? Accelerating away from you?
Except be careful, it's not the one that's accelerating.
Remember I said that acceleration is what you feel
when you're pushed?
If we feel when the train accelerates,
whereas when you are in a local natural frame, you are not
accelerating in that sense.
So the
that's a ghost version of you that's falling
away with the Secretary, but you it is increasing without a
nonzero second derivative.
It's not that ghost version of this accelerating, it's you
that's accelerating. You are the one that's accelerating away
from what you might think of as the natural motion under gravity
and to the pressure you feel on your feet
is
what's accelerating you.
So this ghost version you is falling, is free falling under
gravity following, following a sort of natural motion strictly
in scapegoats. But you the fraud, accelerating away from
that ghost version. And that's the link between acceleration
and gravity.
So the force of gravity that we feel in our own sort of our feet
is the force of acceleration.
And and that looks right back to the very beginning of of of
Chapter 8 where I talked about the the, the, these objects in
the box and the rocket acceleration of the box and
people and and the objects being pushed to floor and being
accelerated by it. So that is actually a key notion,
although it seems like there's a rather simple thought
experiment,
and that's a profound point.
And um,
which is worth thinking about as you as you as you walk home.
And a final point there is that there are
if you go to the library and look at
general relativity textbooks. So look the the body of this is.
Get mathematically complicated quite quickly, but the
introductory chapters of these textbooks are often very good
about
putting these ideas I've talked about here into into different
and portable water laminating sequences.
So look and Rinder, which I I think I've mentioned a couple of
times as a more advanced general relativity textbook, is good
about the fundamental ideas and the sequence of ideas. So if
you're if you're if you're puzzling into that and you want
to be prompted to think more about it then
the introductory chapters of Advanced Books are accessible.
And so that was I quickly recap what I I got a debate. Are there
any questions about that, about that sort of stuff for the final
section of this
of this chapter?
What I was talking about the end last time was Poisson's equation
and the analogy between that and the equation we're going to go
into There. I said that this was the
that's the fire. There was the gravitational potential,
and that's just the thing that you're familiar with from your
your knowledge of of Newtonian gravity. It's how much
gravitational potential is that this height, at this height, at
this height and this height, and so on.
And the second derivative of that
is constrained by the amount of mass at a point
R.
So in the typical thing that we're interested in, there's a a
lump of mass at the the centre, the sun see, and the the the the
gravity potential that results from that is a solution of
Poisson's equation. It is such that the 2nd derivative of it is
zero everywhere except at the centre.
OK, and a solution or personal equation
consists of identifying what what role is the daughter of the
centre and then mathematically you know, solving that equation
to find what Phi is and you get a A1 over our potential from
which you get the an inverse square force which from which
you get nuisance gravity.
So that is where Newton gravity comes from.
And I mentioned that
going to Einstein's relativity, Einstein's version of gravity,
we you know with a version of that
we can't talk about
mass
quite the same way, because as we know from Chapter
7, the dynamics chapter at mass is just the sort of the time
component of an argument vector.
So what we have to consider instead is a more general
object.
I talked to an energy momentum tensor and and and talked about
the a tensor being the same as the the stress strain tensor.
Which allows you that to say, to say things like what's the the
component of force and the extraction across a plane
through it, through this, this solid object and so on.
The environmental tensor talks about the the flow or the
momentum flux across a surface. It talks about the energy flux
across the surface into the future.
So it it talks about the energy and momentum, or the the energy
momentum inside a box if you like, inside an object and
waving my hands you're quite literally as because I don't
want to get into the details of what that is. But the point is
it is the the relativistic analogue of that matched term.
So the right hand side of the equation we're we're gonna be
looking but we we can't study but we're gonna be quote is a
mass related thing in the same way
towards the left hand side
if you
I I have repeatedly quoted.
Things like the
the s ^2 equals the X + y ^2 as the differential version of
Pythagoras theorem.
I've talked about the
describe minus DX squared, a sort of differential version of
right
of the of the interval and has said that this that both of
these are are metrics
in the sense that they are the the things that define distance
in the respective spaces.
With this
now the
the the thing that's called the metric
is
from maps so that for example also in
spherical pullers that would be the r ^2 + r D Theta squared
plus
sine squared, Theta D Phi squared.
Just going to quote that just to say that that that that's still
the metric of of of flat 3D space but it looks a bit more
complicated because we could we're doing it in in in in
spherical Polaris. But the point is you can see there are
crawfish is multiplying the various,
you know, one outward outward change word, multiplying the
various differential elements there and the
those are simple versions of of a matrix essentially
are another tangible object which is called the metric.
Again
elating some details,
the point is the metric. This definition of distance d ^2
is the thing that corresponds to the
potential in
Poisson's equation
with with the metric is the thing that defines the shape of
the of the curved surface. Remember it was because we had a
a metric like that on the surface of the of the of the
sphere that we that we we end up discovering spherical
trigonometry with with with these different rules about
internal angles of triangles.
So jumping. So this is all
seeing that
to make plausible the observation
that the thing that ends up being.
Instead the equation is
and Andrew Mentum tensor,
which is a way of encapsulating the amount of energy momentum in
a in a box. If you like please. And another thing, sort of G
big.
Which involves the 2nd derivatives of the
the
of the coefficients of the metric.
I am missing a lot out there,
but but the the the key thing is this in a way looks like
Poisson's equation.
In both cases you have a thing on the left, on the right,
sorry, which is related to the mass,
and I think on the left which is related to a second derivative
of the shape, if you like.
In that case the shape is just a potential. In the case of
Einstein's equations, the shape a second derivatives of the
metric of the coefficients of the metric.
There's not much I can say about that before, without going into
way too much mathematical detail that that was occupied for
another few months.
But the the key thing is that analogy is
it is real. It is about analogy and to point out that this
equation, this is what I say published in 1915
plus definitions were GS because it's so that that expands to A
to A to a big set of equations. But it's basically a simple
idea, and that is another of these physical statements.
You can imagine that being otherwise, it is mathematically
reasonable for that to be otherwise. And there were some
other alternatives that Einstein tried first to think, oh it's
going to be this is going to be the physical law that's going to
explain everything and it didn't work out.
So over the course of of some years and months leading up to
the end of 20/19/15 he eventually discovered that the
there was a way of of putting things together that identified
A quantity of G and quantity T which for which this equation is
equality
explained everything. I would have discovered how it explains
things in in a moment,
So there's not much you can do with that fact, but nothing you
can do with that because maths. But the the reason I'm showing
you is simply true that it that in a sense it's quite simple,
but in a sense it's also very hard
because these are equations linking multiple components of
these matrices. So those are both
that that that equation there represents 10 simultaneous
second order differential equations and that's not easy to
solve
and they have not been solved in general.
What has happened is people have solved them in particular cases.
Particular special cases, particularly simple simple cases
could discover those.
So no general solution of that exists,
only
situations only, only analytic solutions of special cases
critical and.
But the point here then is this allows us to complete the other
half of the female slogan
speech themselves. Material curve
The culture of the space-time government. The geodesics in the
space-time and the geodesics are how we move through things
and matter to your speech temperature curve.
The distribution of energy, momentum in space governs. That
is what governs the shape of that space-time.
So the two things linked together
and and that is the key insight
of general relativity if you like
masters in the details. But that is that is the key into and and
and and it links very smoothly to special relativity.
So one of the of of the questions that came up I think
I think on the, on the, on the on the padlet was why studying
special video. I'm not sure if that was how it was explained,
but what would somebody if the application, you're special
activity, and there are applications, special activity
within particle physics, You can't design CERN without
knowing lots of special activity.
But the other big thing, that very important thing, and The
thing is very important for you as astronomers is to is that I
want you to have a very clear picture in your head of how
smoothly special relativity turns into general activity.
Specialty is the the key link that that goes from the physics
that we understand to the physics that we don't yet.
We'll come back to that picture later,
OK?
On to Chapter 9
and breathe.
Are there any questions about that that certainly caught
anybody?
There were plenty of questions as you as you mould this over in
your head and so I do ask those, But these are some of the most
exciting ideas in physics of the last of the 20th century. And
the thing that's more exciting than these is in 21st century
things like spring cleaning and quantum gravity and stuff. But
this links directly to directly to that.
So this is all a classical theory.
OK, moving on.
So in this chapter there's the final chapter and we'll have a
lecture and a half to get through.
I'm going to just talk about solutions to ancient equation.
So I I quoted intense equation there and said details
complicated,
but a solution to Einstein's equation is when you you you
take a configuration of
of matter and energy, whatever on the right hand. You can write
down fairly easily what the right hand side of that equation
is, what the energy metric tensor is,
and then the key to finding what is the metric,
what are the coefficients of the of all these coefficients of the
metrics, such as the? What's the factors in front of of DRD,
Theta, and Phi? What are, what are those coefficients of that
which satisfy itens equations? And we've got that. That's a
solution to
equation. It gives you a space-time which is curved in
the right way.
OK,
the so the objective here are rather hand waving ones, you
know give a qualitative account and and and so on and so on.
What we're going to cover is
the weak field solution, and the weak field solution is the case
of the the extreme case where the engine intends to. The right
hand side is is small
and as you will be familiar from other basic but your education,
extreme cases such as you know, small mass are solvable in a way
that the more general things are.
We'll discover the special space-time
which is extreme in the case that there is only one mass in
the universe.
That's another extreme case which happened to be solvable.
Will briefly touch on gravitational waves, which are
interesting because they are source less
solution in the sense that gravitational waves don't depend
on any masses being there to propagate. They're all about
gravitation wave propagating through an otherwise empty
space. Time, and we'll briefly touch on if we have time, on the
solution of the of Einstein's equations, which refers to the
entire universe,
which is another extreme case.
So we can see a little bit about all of these things,
no?
Here's another metric,
and this is actually similar to what we've seen before,
because if you
ignore that bit, which I'll come to the moment, that metric there
is just DT squared minus DX squared and 100 X squared
written D Sigma squared and D Sigma squared is. I will
occasionally refer to it as
is DX squared plus dy
squared plus.
These are just the the, the, the, the the distance element of
Euclidean space. Texas, that's flat space that's what would be
segment. So without that there that's just the metric of
of special activities Explained it. DT squared minus DX squared
basically.
But if you think back to the beginning of Chapter 8,
one of the of thought experiments I talked about was
this idea of our a mass following, following being
turned into a photon and and writing again. And we were able
to deduce that there was a
are our redshift which happens to be found at that point.
And
I showed you, I I I fairly quickly. I showed you that you
could characterise that red shift by the gravitational
potential.
A Newton gravitational potential. How? How? How high
above the the mass are you
and
you know little did little little detail
you can
and
I'm trying to use the free It can be shown that as little as
possible but I'm not going to succeed very much in the in this
final section.
So I think that I haven't expanded that in the notes and
I've just said it. It can be shown that
that the the way that that that observation of a red shift
coming through a gravitational field can be held together is if
the distance between that's it. You remember the Shields photon
thing that I've briefly mentioned we had. You had two of
four
diagram with two photons going vertically upwards after a delay
and we discovered and I
very quickly mentioned that you could see because of the red
shift of the photon climbing through the field, that the
lengths times between the end point of those arriving photons
were and the departing photons were different. Blah blah blah,
doesn't matter. The point is you can justify this as A
at a metric
for the space-time where this
function Phi this coefficient in front of the the DTSA component.
Is this recognisable expression here GM over R?
And this is in natural units, so this ends up being a very small
number in these units, so this is 1 + a small number. In other
words, this metric is very nearly the Minkowski metric,
but with an extra term here, an extra small term here which
allows you to.
Slightly deviates from that
and
no
the
there's movement could see about that, but I'm going to move on
to the next to the to quickly to the next. The the reason I
mentioned that is because you can go from just to. Note that
you can go from essentially the
the arguments in the
the beginning of chapter 8, the regift argument and the shows
photons arguments to that metric fairly directly. You don't have
to use any general activity I and and.
I'm not seeing any more of that.
What I am going to see instead
is that. But what? But the point of that is that all of that is
that you don't use Einstein's equation in that you can
slightly clever way get teacher tiptoe into talking about curved
space times without actually using items.
If however you start with ancient equation, the G equals
Kappa T
and look at it in the extreme case of
a very small mass.
And a small mass here is something less dense than a
neutron star,
so a star or a or a Galaxy or something like that. So it's a
small, a small mass,
and if if in that case a Galaxy probably wouldn't have it was
dark, but it's something something anonymously small,
then you can do. Then a lot of the of the mathematical
complication at that point falls away and you end up with a
aversion,
a low mass version of Einstein's equation, which is solvable.
And it's and it's solvable in the sense that you can find
a metric which which satisfies that equation and that
that solution is this one. It's called the weak field solution,
and again you can see that
if I ignore bits of it,
that's still DT squared minus DX squared. So that's still vague.
That's still close to being the Minkowski metric.
So the the, the, the the space-time in our around our
a small mass object is still. This is seeing basically that of
special activity
but with some perturbations from it
Which and this and this pops out from pops out from Einstein's
equation
which have this form this GM over R which is the potential
that you find in using gravity
and this if you like is the first Test of general activity
and and so I think Einstein had a happy day when he he he he
worked this out
that if you go from Einstein's equations
take the low mass limit of them
then what you get is an expression which involves
something which is very clearly identifiable as the interim
potential. And if you then turn the different handle
and ask OK what are the geodesics
in this
in in this space-time do I have a picture of those Yeah ohh
those are the those those are the the other dude the equations
of motion for the duo D6 OK does that help us. It does help us
because those that that that P there is the energy momentum
vector.
If you remember, that's the thing with ease in the time
component and the spatial momentum in the spatial
components.
And what that first equation is saying is that the rate of
change of the 0 component of the energy momentum of our test
particle
is proportional to the rate of change of Phi with time. And so
if if the the the the the the central mass is a star we're
talking about isn't changing mass which it won't be
very high stable to 0, so there's zero component of the
energy, momentum of our test particle will will be constant.
In other words, energy is conserved in as things move
around in this space-time,
and this
second equation, which is I in the in the in the spatial sector
is a way of rating F = -, M
del Phi,
which is Newton's law of universal gravitation.
In other words,
the the low mass limit of isange equations
is solvable to give
a metric to give a a space-time which is curved slightly curved.
Because again Phi is much smaller than one, it's slightly
curved in a particular way.
And when we ask how do part, what are the geodesics in that
space-time? How do particles move in that space-time? We
discovered they move in the same way. Excuse me that Newton says
they
So we have invented Newton's law of universal gravitation.
So this love universal competition is a pretty direct
consequence of one of the simple simplest,
one of the simplification #1 of Einstein's equations.
That is a solution that maintains equation,
and that is pretty mind blowing I think.
And that is one of the first tests of generativity. That's
one of the first reassurances that you're onto something with
that, that that Einstein's guess of G equals Kappa T
That was right at her physical state,
which just meditate on that just for for for for 10 seconds, I
think. How wonderful that is.
Key points.
So that was the.
We've looked at 2 simplifications there.
We looked at the very first thing I mentioned, which was the
slightly partial missed out. Next time the a version of what
we are getting
a curve to be saying which doesn't use instant equation but
is fairly fairly directly follows from the thought
experiments we had at the beginning of chapter 8 And it's
one thing the simple case of very low mass and non
relativistic motion. I should I should also mention gives us a
nice agree which is solvable feel straightforwardly to give
nuisance laws.
But the other simplification that we can make to a scenario
is not that different from that weak field solution. It's to see
what happens if we have only a single mass in the universe,
only a single massive object in the universe, and the universe
with single message. What is the shape of the space-time
Miranda
and we're going to not talk about things moving at non
rustic speeds. What we'll we'll talk things moving as fast
rewards. We won't have that restriction that was that
restriction is implicit in the week. The previous weekly
version and
I said equations were actually published those as equations in
1815,
in 1916. So only a year later
Carl Starshield
vote solution to that other simple case where there's only a
single mother universe. And there's all the more impressive
because Churchill was an officer in the in the army and was in
the middle of the First World War. So there there were
literally shells going overhead when he was trying to work this
out. You know there's a variety of ways of distracting yourself,
but solving 19 equations is is one of the most stylish ones, I
think.
Anyway, Schwarzschild discovered that this was a solution to the
twins equations in that special case of a single single mass,
and this also
if you were to take R to be 0,
you see this also is DT squared.
Um minus
at the artwork,
minus R ^2 D Omega squared. The Omega squared there is is
is this the Theta squared plus sine squared D Phi squared?
That's the the the the the area element on the surface of a
sphere is what is all the Omega is there.
So you can see that if R is zero, that also reduces to
Minkowski Minkowski metric,
so that also is compatible with
special activity.
You can also see that in the spatial sector, that's the D
R&D, Phi and the Omega. So that's that's on the surface of
the sphere, that's the radial component
that the that they are no longer flat.
So this is spatially curved as well as curved in space-time
and other things that for that are worth pointing out. There
are
ohh yes that, this
and the reason I haven't written up.
Ohh, yes. And and this
them.
There's this big art. There is
parameter which has the same which has the same dimensions as
a radius.
So the the, the the the. The radial scale here is governed by
this parameter big R
and that
that that that are is called the short short radius.
What is that parameter R?
But in physical units, it's two GM over C ^2.
So again, there's something reminiscent of
Newtons gravity
in there,
and that really is there is the radius on which this functional
solution
deviates from
from from a flat that that that characterises the the the the
length scale on which the spatial solution deviates from
flat space.
What number does that produce
if you put in the numbers there for that? That G, by the way, is
not confusingly isolated intensive, but but but but the
but nuisance constant use constant coverage of of of of
gravitation. You put the numbers in there with, for example, the
mass of the Sun.
Then the structural radius of the Sun is 1 1/2 kilometres.
So
if the mass of the sun were compressed into something
smaller than 1/2 kilometres,
because it would be like who and and and and that's what we're
about to come to. But
that that that radius are we 1/2 kilometres and
whenever art was bigger, was much bigger.
The ratio between that slash rates are and the regional
parameter little R is how much this deviates from being just
flat. Space-time flatmate office space-time. So out here
when we are 500 light seconds from the sun,
that R here is much bigger than
the the 1/2 kilometres.
So out here, space M isn't curved at all really,
basically not covered at all.
It is but but it but it it's very, very small. It's small by
a factor of 1/2 kilometres. Over 500 light second
means not very much
and and and the I can't offhand but the the special region of
the Earth is or you can work at your own spatial radius
if you wanted to
and
and I'll also mention just in the last few minutes that those
are
some weight of of of writing down
the
some sample version so that the
GM is that the coverage of constant times the mass of the
sun is that or 1 1/2 kilometres in natural units
and
that number at the top there 1.327 at 10% twenty actually had
a lot more decimal places
add to it because it can be measured quite very precisely
because you can make very precise measurements of things
like artificial satellites in orbit around the around the sun
and make and measure that very carefully.
So you can characterise that's called the gravitational
parameter of the sun, and this is a Metal Gear. Here we're just
that parameter. You characterise that very, very precisely. So I
think to sort of like
8 significant figure, 7-8 significant figures,
but then to look at what the mass of the sun is. What you do
is you take that gravitational parameter GM and you divide by G
How do you get G? You make experiments on Earth with things
like torsion balances and so on, but the
those
figures for big G, the gravitational constant, you know
it's a hard measurement to do, so you only know those to to to
to perhaps three significant significant figures.
So you're dividing this quite accurate number by a number
which is not only three significant figures. So you get
the mass of the sun in kilogrammes
to only three significant figures.
So we know that this factual radius of the of the Sun
GM
to about 8 significant figures.
But we know the mass of the Sun kilogrammes. So that's the mass
of the sun in metres is what we know.
But to convert that to the mass of sun in kilogrammes we have
divided by number. We don't know very well. So we know that the
radius of the sun the matter sun metres better than we know the
start. Again, we know the mass of the Sun in metres
better than we do with the mass of the Sun in kilogrammes,
and that's a striking thing I think.
And and and that's the the official radius of the of the
Earth there.
Before we go, I think next time that's a good place to stop and
next time we'll pick things up until you stop active at
Blackpool.