E2 Relatively Gravitation, Lecture 15.
This is the appeal
this these 5GR lectures. They are sort of the point and the
payoff of the 10 specific lectures, which are
independently interesting but also to some extent lead up to
the GR bits.
So this is the
last of the retribution lectures and we are bang on schedule, so
we'll get to the end without scampering.
And when I got to last time,
I talked about what we covered last time was
various
solutions to Einstein's equation. I mentioned the weak
field solution, which is the limit of small masses.
So that so so or when you're far away from the from the matter in
question
and that
really allows you to to rederive Newtons group love love
universal gravitation. You discover that there's a
potential for a function which is as part of the solution of
equations, and you can find an equation of motion
force in terms of the gradient of that potential function,
which and the potential function is governed by
integration. That's a solution
which is essentially equivalent to
questions
theory of gravity. The next approximation was the
Schwarzschild spacetime, which is the the the full relativistic
case of a single point mass in the universe. And what is the
space-time around that?
We'll get more of the other ones in a moment, but I had talked
about the 1st 2:00 and I had I and and I had emphasised the
importance that what we're doing here,
what the process of solving Einstein's equations consists
of.
It's a metric, that is, that is a set of coefficients of these
DT, DRD, Theta, D, Phi these differential distances. The
metric is the coefficients of those,
and the process of solving ancient equations consists of
finding a set of coefficients for those functional
coefficients which depend on the position of the space-time you
have which satisfy anxiety equation for a given right hand
side which is dependent on the distribution of energy momentum
in the space-time.
And of course, the energy momentum is primarily in the
form of large lumps of matter such as planets and stars and
galaxies and so on.
So that's a recap of of what what we're doing, what we're
doing here.
I mentioned the weak fuel solution and it's properties. It
it was. You can see that if I were zero
with with that that that D Sigma is the DX y ^2 plus D y ^2 + Z
^2 or or whatever. That's the metric of the spatial part. You
can see that if I were zero in the limit of all of of equals
zero. That would turn it back into the Minkowski metric,
but it. But it's clearly not that in in general.
And I then pulled out of the hat because I can't do otherwise.
I pulled out of the hat the Schwarzschild solution. I
mentioned that Schwartz had developed this solution very
quickly in 1916, very quickly after Einstein had introduced
the the field equations in 1950, late 1915. And as you can see,
if that R term would disappear, that would turn back into
the metric of special activity. It would be the t ^2 minus the
art squared minus R-squared the Omega squared. The Omega squared
is just the squared plus sin squared, Theta D Phi squared. So
it's the metric of the surface of the of the area element on
the surface of a sphere. That's that's all The Omega is just to
love that into one place. So if that term R / R weren't present,
this would be just
Makovsky metric. And that's important because that is
because one of the constraints on this being a a sane solution
is that it reduced to the Minkowski metric both in the
limiting case of no mass and in the limiting case of R being
very large.
Because you see that if our very large if. If so a little R. If
the if the radial coordinate is very large
and so you're far away from the central mass,
then big R over little R is small,
much less than one, and this return turns it back into the
Minkowski metric. So this solution is compatible with
Minkowski space-time at large
coordinate R,
which is a sanity check. It couldn't be otherwise,
and this is this is a very important metric. This is. It is
for the ideal idealised case of a single point mass in in your
space-time, but that is a very good approximation to the to
most of the.
Practical cases we want to use General General relativity. So
if you want to design the GPS system or get or the legal
system this is the metric that you use for for the non
relativistic for for for that relativistic corrections to to
to to nutrition theory. If you want to talk about micro lensing
which will come into very very briefly or or or or lensing of
of galaxies by supermassive black holes is this partial
solution that you that you use
if you want to talk about
the most power effect.
That's the non relativistic version of
of gravity that is used practically
in the in the cases where where, where where was relative to.
Corrections matter.
So we have it, but the next thing is to examine what what
what, what happens in here.
Now as you can see,
when when little is big, this turns back in and I've just said
I emphasised little are as big, this turns back into the
Minkowski metric. But when little gets to be equal to
bigger, so when when you come down, you shrink our down to the
point where it's equal to this. There's this are then the second
term, the term on the the 2nd
a coefficient of the second term, this one
1 / 1 -, 1. It blows up
the singularity there,
and for a long time it was thought something very exotic
happens there. The space-time gets infinite at that point. Now
to the long. While
a surprisingly long while, possibly before it was realised
that that was just a coordinate singularity, there's nothing
actually happens there
that you could detect from a physical experiment.
You know that that's not obvious that that that that that's the
case. And it was highly nonobvious people to people like
Oppenheimer and Einstein. It it took, it took a good amount of
work to find a set of coordinates which what didn't
have that bad behaviour at that at that point.
So there's nothing strange happens there. And if you were
falling into this fall, falling radially inwards
as you as you cross that point a little I equals bigger,
you would do the same thing. You just be you still be threefold.
So all of the the the the you, you, you, you have the freedom
attached you the frame of which you are not moving would still
be in a national freedom. And there's no experiment you could
do that would tell you you were at that that special place your
local experiment. You could.
And that's and that's an important distinct,
so that no, that inside your box
there's nothing you could do that would let you know that
you're crossing.
So does anything happen there
at another? Yes,
and there's some sample values.
What happened there is important
and now the visualizer isn't doesn't want to turn on, so I'm
going to have to do this on the board.
I said
that
as you cross that regional distance R equals big R
you are still as far as your concerned and you are still in a
natural frame and nothing happens. So what's the?
What does it look like? And monkey diagram. When that
happens
in that diagram,
is that visible at all
And you would find out the pain
better.
And so this is your. It's called the X prime
T frame, and if you are at rest in that frame
then your water line is going to be a long the TM access.
Yeah,
as usual.
And in that in that frame, the
no lines, which are the water lines of something moving at the
speed of light are diagonal, as we saw back in chapter four, I
think. And that demarcates the space-time into three regions,
the future, the past, and elsewhere
to the future. It all of the places that you could get to
from this event here by moving at less than speed of light.
The past is all the pieces you could have been and got to this
event.
Moving lessons, be late and in the elsewhere are all the places
that are space like separated,
which you can't get to moving at the speed of light or
or or or less, and which could
and put your such that you could find A-frame in which an event
somewhere in the elsewhere was was simultaneous with the event
at the origin. So you could find A-frame in which those are
simultaneous. OK. There's space like separated
and it is a feature of the Lorentz transformation that if I
now ask, OK, that's that's the the motion
in the
in the frame
that's attached to me that I'm stationary in. So what does that
world line look like in our frame in which that frame is
moving? So in another frame
X
T
and I'm not telling you anything, should surprise you
that water line would be constant speed. Look something
like that
because that's just the where the TPM axis ends up.
But in this room as well
the no lanes are still at 45°
and this world line,
it's still insane
those and and you can't find a there's a Lawrence
transformation that would get you from this frame where you're
not moving
to A-frame in which you were moving faster than the speed of
light in which that world lane moved outside of those of that
light gone
OK. So that's a bit of special activity,
and it is a feature of the Lorentz transformation,
those diagonal lines
trying to do diagonal lines.
OK,
but that's a feature of the Lorentz transformation in
special relativity.
In
the special metric,
it's a little different,
so let's look at this the right hand side here. First of all,
that's intended to represent the
the
Bitcoin
of the observer who is
free falling, who? Yeah, we're moving purely under the
influence of gravity. Notice that the that's the radial
direction, that's the the the time, the time direction. So
basically and that's the the late cone of an observer who is
in freefall moving in some direction rather.
But when we turned that from
their frame
into the
into this frame
in this metric
the the, the, the right cone that's 45 years here ends up not
at 45 read but a bit a bit a bit a bit twisted
but in a slightly different direction
which is you know OK that's that's that's fine. It just
looking a bit strange. But that dash lane is still a viable
worldwide for that observer. It doesn't go outside of the lake
on the fact the lake was a little twisted. It's fine,
but as you move further in toward the centre, toward lower
R
that rotation, that distortion ends up being more pronounced.
So once you're quite close, that thing is is is turned over a
bit.
So the the set of possible
future, the set of of trajectories from the point here
is still has to see in that league. But it's. You can see
it's it's it's somewhat constrained and the time you get
to r = 2 GM, this functional radius that's turned over to the
point where the light cone is
vertical in this diagram.
No,
I found it. The the the the person in the in the in the in
that frame is concerned it's still there. They're they're
Mitkowski, Diane still looks like this. They're they're past.
The future still looks like this. They can only move in the
in in the future cone
the the the only possible trajectories are ones which are
in the future code. But because of the way that this has changed
in these coordinates,
the future
is entirely inward pointing,
because there's no worldly going from this point here.
The new world lane that stayed within that light cone that goes
to increasing our
in other words, the future of that observer at that point is
inward pointing
and as you go further in, it becomes even more extreme.
So there's nothing that feels different at that point.
It still feels like you're in inertial frame. There's no local
measurement you can make. But your future is very different
and rather bleak
because at that point there is no way
you can get out of the black hole if you're in freefall
and
so so and that point, that radius there is the, it's called
the event horizon, it's got, it's the it's the size of the
black hole. The black hole itself is I think a larity at
the very centre.
OK
so so the black hole if you like is an edge of space-time which
is at R = 0 and I said ohh mathematical bets are off there
because it's it discontinues.
But the the size of a black hole is the size that this radius
here or two GM which as I've said for the Sun,
if the Sun were compressed into a radius of 3 kilometres, just
three kilometres, it would be a black hole. If the Earth
compressed into account the figure, it would be a black
hole,
and so that's the the distinctive feature of black of
black holes and other things you can talk about with black holes.
The fact that there are tidal effects, if you remember
way back in I think Lecture 2 I talked with an example of two
people falling toward the centre of the Earth and they are
a separation between them
because they're they're they're falling except they are they're
moving, pulling into gravity. So as far as they're concerned
they're not accelerating but the separation between them is
changing as time goes on. So the the and that's a title effect.
It's not a a local effect. It's purely an effect discernible
with a a measurement over an extended non local part of
space-time. And another sort of tidal effect is that
in the high gravitational the rapidly changing gravitational
field, if you like. Here, if you were falling in feet first, see,
your feet would be accelerated more than your head would, and
so a tidal effect. You'd be stretched out
and put it and and and and that's for accretion discs
that one of the things that happens in accretion discs, it's
partly because of the the the the extreme tidal effects on the
creation discount black hole that cause all of the some of
the physical features there. Just looking at the light cone
on the slide,
Yeah. Does that mean that because it like sort of like was
taught so much that it comes above the access, does that mean
that your past could have been in the future?
Right now that's a very good question and we're gonna get
rather evasive answer.
Um, because inside the black hole the
and
the the the. The problem is that the
so speed access and the time axis sort of swap over
and so the the the the space axis becomes your time like
direction
in in its own way. So that
at this point you can have a discussion of what does it mean
to move forward in time. But in inside the black, inside they
went horizon black called moving forward in time means moving
inwards in radius. So the radio direction becomes time
in a way.
And so
I have never found a satisfy our intuition for this picture for
this entire satisfactory to me or what's actually happening in
there. I think it's something that is requires careful
thought, but it it does get quite exotic there and things
like that you could have been. So I think that that time
becomes a sort of Space Flight direction. So you'll watch
Google backwards and forwards. You know Mumble, you know,
And that's not a very satisfactory answer, but I don't
really have a very satisfactory answer.
Just only you're saying there about the the time Max coming.
You're like spatial access that effectively being made the
bottom, like the singularity is just always in your future. Yes.
So you you never get to the singularity
and I you understand now keep on opening Rd, never and never
never get there. And we've talked. We've talked here about
the what happens to you as you you're going in. There's a whole
separate thing we could talk about, about what someone
looking at you sees.
Because what someone looking at you sees is
light coming from you
and so some some light emitted. So see you. You're here
and managed to send a a sort of goodbye forever, sweet world
message out of but but but but by pointing a laser you know,
directly outwards? That's good dispute. Later it would get out
moving the spotlight if it was sent from outside the event
horizon,
but that then has to climb through a huge gravitational
field in order to get to the observer.
We are here
and so it would be hugely red shifted and the the time
and another one of the effects that we could talk about if we
had more time is what you know related to the idea of of of red
shifting light is how much would time slow down as you watch
someone
falling into a black hole. So you would see their watch tick
slower
for for a number of reasons,
but you would never see. So one of the things that
to reflect on it, you would never see anyone cross into
across the event horizon.
Someone falling in
just falls in and they cross the road and they don't locally
notice. But if you are looking at that from our side, you would
never see them actually cross because the time that you as
you're watching there watch, you're looking at their clock.
You'd never the time was fluent enough so that pertained to get
the crossed the event horizon. It would have slowed down
infinitely from your in your measurement, your observation of
them.
So all sorts of exotic effects what you what you're talking
about here. But that event right? The thing is the the the
key one if you like.
Umm,
so good. The other thing that that can happen here is let's
not worry about the
let's move back a bit from the exotic behaviour at the centre
and concentrate on things a little bit further.
So you can imagine
that that's just the RT plane of this space-time diagram.
There's also the the the Theta and Phi coordinates that are
suppressed here,
but
amongst the SO so so we've talked about the the metric.
Once you put the metric you can talk about geodesics, and
geodesics are the trajectories that are
a body in free fall could potentially follow, so the the
the. So if if I if I'm if I'm on the surface of a sphere and I'm
facing in this direction, then there's a geodesic that I I will
fall follow if I just head off in that direction. So depending
on what direction I'm I'm facing in on the surface of a sphere,
I'll pick out a different geodesic.
Really, obviously, but there will be geodesics. But the
initial conditions,
what direction of fitting we'll select which duty it is.
Similarly, geodesics in our our
which includes time.
All the possible motions from a particular point or point of
water lines will be geodesics. But which one you follow, in
other words, how fast you're moving and in which direction
depends on the initial conditions,
but they're all geodesic
amongst the eugenics
that that happened in this space. Like this is 1. Where the
where you moving mostly in the
and
find direction for example and that what that will look like is
a spiral on that in in that space-time as as time goes on
your your your
you you you can you can imagine an Arteta plane
and a time axis here and and and then one of the geodesics is
something like that in a spiral and that is basically an orbit
that's when orbit looks like in the
because in the in the in the in the space-time.
And if you work that out and look at and do the calculations
you get that what that duty looks like you discover that
dudzic traces out something which looks very like and lips
of course because you have to reproduce the the the results
that you get from sodium analysis the the the two body
problem so you get something which is very like an lips
except
that is an ellipse which slowly processes
over the course of time as so So this this spiral isn't quite a
spiral it's slightly elliptical spiral and the axis rotates as
time moves as it moves up the the T axis
So the orbit processes
no step back a bit
and
and I think I have
it
and and the amount of procession
ignore the second line there for the moment the amount of
procession how much the direction. Of the semi drags is
moves per orbit is a function you can you can work it out you
can calculate it it depends on the mass the semi draxis and the
the eccentricity.
Now the nearest planet to the sun is Mercury and it was well
known as long we well known that the orbit of Mercury's processes
it processes at
574 arc seconds per century.
OK so not a lot but it's you know straight forward and
measurable
and almost all of that is explicable by the gravitational
influence of the other planets and solar system
you know 99 point something percent of the masters or system
is in the sun but the other planets are not are not
negligible and the affect the each other. So solving the
equations of motion of the planets in classical celestial
mechanics is
complicated but doable. And
by doing things like that people were able to predict that there
must be a planet beyond Uranus and and and and who was said, if
you look there you're bound to find a planet and and Neptune
was found so that that that works. But in that calculation
there remained A stubbornly unexplained bit of the the the
procession of of Mercury.
And Mercury was processing at 43 arcseconds per century. There
was unexplained by this process
and the thought was perhaps Newton's gravity is wrong. Which
of course is the case.
And if you work out what that procession is
that that
particular from from this relativity correction, calculate
calculation, put the numbers in, what you get is a procession of
that orbit of 43 arcseconds per century, exactly what was
observed.
And that was one of the that's one of the classical tests of GR
that GR does predict, or this financial solution does predict
exactly the missing procession of the the orbit of Mercury,
you know for three hours seconds is again not much. For three
seconds I think it's a. It's a metre
4 1/2 kilometres away so it's not much but it's detected
and would long detectable before general activity. So this is a a
classic case of there being an anomaly which and explained
their new theory comes along and says Ohh I can explain that
and we could talk more about that
but we won't.
So what that means is that OK, you can have orbit,
but you also have
and as we learned at the beginning of the of the of of
chapter chapter 8, we also have light being bent by a
gravitational field,
and you can come up with fairly based heuristic measurements of
how much that is. But also just by looking at the small solution
you can work out how much light ray is bent as it goes past an
object. And here, for example, there's a a star A
an array of light which comes from. The star will be bent as
it goes near a mass
in such a way that when it arrives at the observer we see
it coming from a star
the the the star appears to be a position B there.
So the star has moved its position
because of the presence of because the late we had gone
past a large pass
and
I think in and in 1919
through just after the First World War Eddington and Dyson
an expedition to Principe in
UM,
Brazil. And
no no found that precipitate is is an island off Africa and
another one with Brazil. I can't remember where it was, but the
point because there was going to be a total eclipse of the sun
at
over there, visible in those places
at that point. So what they did,
they took, did careful astrometry, got the
put, confirmed the positions of of of, of the stars at night.
Then during the eclipse the next day, when of course, they when
when the sun and the moon were between what proof of the sun
was between them and the stars. They remeasured the positions of
the stars and found that the stars were coming, were in a
different position, observed to be in different position from
what they were at night when the sun wasn't. What wasn't there,
of course,
had away from eclipse, because the Sun had to be occluded by by
the moon to make the observation,
and with a very fine, very fine, fine and difficult measurement.
And there's a lot more one can talk about that particular
measurement. It's very interesting historically,
but they found a TV ad, a deflection of exactly the right
amount.
Another the the second classical test of of GR that the light
from the from the start was indeed been deflected at a
passed by the mass of the
SO
and now
that that was an extreme, a fairly extreme measurement which
had to wait for an eclipse to happen.
But if you're a radio astronomer now,
radio radio telescopes have extremely good angular
resolution, and they don't care about the sunbeam being up, and
so this deflection is a routine experimental
detail when you're making visual observations. If you're making
radio observations near the the you near the limit of the sun,
then you have to correct for it routinely, otherwise you're sums
won't add up.
Another place where this happens is if you have a.
If that mass there is
the Galaxy,
and what you're looking at is a quasar on the far side of it,
then
that guitar will appear to be a different place and a different
shape
from what it would be if you're looking at it as
straightforward. So you can detect the Galaxy in the
foreground by looking at the distortions in the shape of the
equator in the background. So again, a routine observational
effect which is purely realistic.
OK.
And
there's the the figure,
no,
I'm not going to talk about that. We're gonna talk about
Shapiro Delay and the change in
frequency, but we won't because that extra detail there.
The next
possibility is that the next solution we're going to talk
about is our
I didn't Amic 1.
In both weaker Solution and the Charter Solution,
the setup has been amassed in the centre of the universe, but
you know, amassed by itself in the universe. And what's the
shape of the space-time around that?
But the
Einstein's equation, as I mentioned, is on the left hand
side
term involving the 2nd derivatives of coefficients of
the metric, and the right hand side are term which is the
which characterises the
energy momentum
at that point.
But if the right hand side is 0, so if there's no mass at a
point, you can still get a solution to
to that question. It's and it's a wave. A wave solution.
You can still resolution which is.
Possible
I've I've lost quite a lot of thought and and a solitary
solution.
And our solitary solution exists and and that solution is
the solution of a wave equation and that and that is the
gravitational waves.
What do gravitational waves look like?
Something like this.
So if you imagine that the
that that that's a two-dimensional space, A2
dimensional space for example the top of a drum for example,
then
that there there are various solitary modes in in in that
drum skin in in that in that surface and and that that that
that illustrates 2
phases
in a portable oscillation.
But if you were at.
If you were on on the outside edge of that,
you could walk around the perimeter and get a length for
it. You could you could you could measure the size of the
circumference of that
of that space. And if you walked across from why
one way to the other, why this bottom case? What you see is you
you you measure a diameter which was 1 / 2π times the
circumference.
But if you walked instead from X to X,
because of the distortions in in the space you'd be walking
longer would take longer to get from X to X than to Y. So the
distance across that the diameter of that space would be
larger in that direction than in that direction. If if you were
thinking about it, you would think the cleanliness and the
lips,
but then at a different phase of the oscillation it would be like
at the top. And now the distance from X to X
in the top diagram is less. The distance from
quite away is more
so.
In both cases, the separation between these two positions has
changed
without acceleration,
so observers at excellent at X&Y in these two cases have
not accelerated. They don't feel any movement,
but the separation between X&Y has changed
and that is a
right
hold on to that thought for a moment. So that's that's roughly
progression we've looked like 2 points in
in a speech team as a gravity wave goes past the separation
between the changes without them moving
to to to to the speech that there's more space between them
without
demonstration.
Can you observe these?
They would be they observed indirectly before they were
observed directly.
And this is the diagram of the
A a binary pulsar called the whole tailor binary
PR for the British can't remember and
it being a pulsar, the
orbital frequency of the two stars in the binary. We're very
well characterised but over time
the period
changed,
it slowed down.
Why was it slowing down?
A number of possibilities exist, but
if you
plot
the PD shift over those years 1955 to 2005
and ask what would the how much energy we would be rotated would
be emitted in the formal gravitational waves by these two
accelerating masses operating each other. And you plotted that
on the same diagram,
it worked rather well.
Those red dots have error bars
with arabad are the side of the of of the line in in in the
graph. So that is a magnificent bit of experimental observation.
The observation is perfectly match the what what the change
in the period would be if this binary were emitting
gravitational waves.
So that's an indirect but very convincing account of what
evidence for the existence of gravitational waves.
But going back to this,
I mentioned that here what you have is if you had test masses
at these four points from the edge
and measured the distance between them
just by whatever means,
then as a gravitational wave went past, the distance between
these test masses would change without the test masses being
accelerated.
And what that is, is a description of
an interferometer.
So here you have a laser
being fitter,
being reflected from a mirror
and possibly several times. And we we observed and that,
ironically it turns out, is the same setup as was used in the
Michael Morley experiments, which were one of the famous
null results which were one of the puzzles leading up to the.
Development of or special activity in the late 9th
century, but it's also the the layout of our interferometers,
such as the Legal or Virgo or Geo 600 gravitational wave
interferometers which were.
Built, Built over a number of decades in the US and in Italy
and in Germany.
So these masses May 123 and four are
just suspend it the the, the, the, the, the, the quartz
mirrors that sort of size. They're big things and the
suspended very carefully so that they are not vibrating.
You measure the distance between them by shining a laser back and
forth and looking for interference fringes.
You you'll have heard about this and
it would be formed I'm sure. And this is an interesting
experiment because the this is the the the change in the what
you're measuring is the change in the distance between these
two masses, which
because the the the pure suspended they are not
accelerating. And that change in difference change in distance is
around 10 to the -19 metres
through 10 thousandth of the of the diameter of a nucleus.
So it's not much
so and it was in 2014 and a lot of the the, the and and Glasgow
played an important part in this. Glasgow particular
contribution was the was very much the experimental
details of this. I think Glasgow is big on the details of how the
how these lasers work in her condition and on on how the
these test matches are suspended. So that Glasgow's
contribution to this very experimental part,
like we need a lot of of data analysis for gravitational
waves. And that's the other the other end of the of the whole
sausage, if you like.
And as I say, it was in 2014 that this was trevally announced
as a as amendment, a direct measurement of the existence of
traditional waves, which is rather beautiful.
OK, Neil there.
Umm,
so I won't put for questions, just go straight on another
metric,
and this metric is
mathematically inspired.
It's the answer to the question What is the most general measure
you can have which is homogeneous, that is the same
everywhere and isotropic, that is the same in all directions.
The idea being that where we are now isn't special, it's called
the Copernican principle.
OK, and
details the details omitted. The more general metric that you can
find that has those probability properties is the Friedman,
Levitra, Robertson Walker metric FRW
which is that is that one.
And you can see that on the this part here you've got that the
Omega that's the the angular angular bit and
a different coefficient in front of the Dr term. And the the the
radial parameter R is scaled so that Kappa is either -1 zero or
plus one for different different cases
And that E prompter. The coefficient sitting out at the
front is an overall scale factor, which
is time dependent, time dependent, but not dependent on
anything else,
and you plug that into the.
Thanks to Einstein's question,
turn the handle not not. Again, not trivial, and what you get
out is a solution for the universe as a whole,
which has the property that a dot
is not zero and it a double dot is not zero. So the 1st and 2nd
derivatives of A are not constrained to be 0,
and that solution was thought to be impossible. You can't have an
expanding universe. That's clearly silly. And I'm saying
then that's OK. Perhaps my part of the island equation is wrong.
Well, another term to it you can add another term which doesn't,
which is plausible, which has a a constant in front of it times
the details that matter. But there's a constant in there
called big Lambda logical constant. When you solve that
version of the Einstein's equations with this metric, you
get you can you can pick the constant Lambda.
You know, another white, unmotivated way so the universe
isn't expanding.
And that was fine. That was good,
but then the Hubble expansion was detected. It appeared the
universe was actually expanding and so the IT became unnecessary
for that extra term to be in in any sense equation
and and I I think also his biggest blunder Oh my God from
Queen.
But subsequent to that it turns out that the that the
that me will be
rule for that that that constant which which adds what is
effectively a negative pressure through universe details essence
to follow. And to that that time has come back in as a plausible
bit of physics which explains the observed behaviour of of the
actual universe. And the last thing to see there is that what
that that that that that parameter E the overall scale
the overall size of the universe is independent. It's changing,
it's increasing.
And what that means is that if you track the universe back,
there was a point where that A
was 0
universal 0 size
and that is and and then really, really playing forward from
there you have the universe expanding from that point. And
this of course is the, the, the, The Big Bang. But The Big Bang
is a
E dot being positive
and a dot and a being zero at some time,
which is a big deal.
But the
last, the very last point to make in the 20 seconds remaining
to us
is that
at
that initial time
universe was very small
obviously.
And that means that it was high. They highly curved,
the extremely curved
to the point where and. And part of the nonlinearity of
Einstein's equations is that there is also that the the the
curvature of space-time is also a source of curvature
that that's one reason why institutions are very hard to
solve.
And and what that means is the energy density gets very high
when the universe is very small, to the point where the energy
density in curvature
is big enough that you can get particles being created from the
vacuum. At that point you have to worry about the quantum
mechanics of space-time
and that it's quantum gravity and that is still very much up
in the air. But that's the that that intense is the the next
step after this which is not beyond the scope.
And that is the end of of the 15 lectures. There's an inspiring
remark from the great ocean of truth that all discovered before
me. But and so we still know there's still vast quantities of
don't know and the quantum gravity is a large chunk of
that. But from
starting with the two axioms of spectral activity 15 Electrical,
we have come a very long way. I've there's been a lot of hand
waving in the last five lectures a lot of it can be shown that.
But I hope that I have connected the GR stuff to the special
edition stuff well enough that you have some idea of the shape
of the mathematical ideas that are that make GR the the
description of gravity.
We'll stop there,
I'll enjoy it.