Protecting Gravitation Vector 13
This is a peaceful lecture. This is where we're all your hard
work thinking about things Start to pay off and you'll come back
to this one, I hope, and think more deeply about the details
as before. Good. I'm coming from
right
as I said before,
but before we get going with that couple of details
1
is that the after some investigation,
the
Echo 360 recording of the lectures are now visible.
I thought they were before, but it turns out that what I was
seeing was what you were seeing. And I cheered them. I carefully,
you know, assembled them all to one place carefully. Shield
shared them with a null set of people
so they were invisible to everyone except me.
Turns out that has now been rectified. So if you go to the
link
in the in the middle to the E360 lectures you should see
is regarding
Now. Also on the Moodle there's a link to our collection of
recordings assembled as a podcast.
That's my recordings. The audio is better for those for reasons
I don't know why, but they don't link to the slides at all. So
it's whichever you you you feel is more useful. If the
recordings equities, the recordings are not visible. I
have had at least one useful report that they are.
If you don't see them when you expect to see them, let me know.
The fact that no one in
but how many weeks is has mentioned to me before that
they're not visible suggest they're not heavily used,
but I am interested in feedback on how useful those either those
sets of recordings is.
Another thing is to reiterate what I mentioned I think last
couple of lectures and also one of the title that posts is that
the material that I deemed to be in scope for the class test is
chapters one to six that everything up to Kinematics
Max is really interesting and there's lots of nice questions,
there's lots of nicely accessible material in there
that's a hint for later assessment opportunities. But
for the class test it up Chapter 6. Kinematics.
Any other questions
about organisational things of that type?
OK, now on to astronomy.
As I mentioned last time, a time before the GR bit is
conceptually
challenging
because it's also
mathematically very challenging. We can't go into very many of
the details, which means that I can't that that there are fewer
things
I can assess in these last five lectures.
Pay attention to what it says in the
objectives, aims, objectives at the beginning of the of the
section because all the things and the objectives as I repeat
are the things I will deem to be in scope for the
degree exam in June. So. So those are the things that I deem
to be accessible,
but they are rather thinner, if you like, in these last five
lectures than in the previous ten, simply because it's hard to
assemble accessible things.
But that should not to tell you from paying the most rapt
attention to these five lectures, because they are, in a
sense, the really interesting bit that the the special
activity
is paying off in.
So enough pep talk. Let's get going.
You remember that last time I was talking about
I? I just got onto
talking about
metrics
and the idea that the the definition of distance inner
space is given by the metric and I mentioned the Euclidean
metric. I think I
the including metric which was
seem not to have written down in a Slade but it's
go back to where I was it's. Well I'll leave you a bit of
chalk just for it's the main straight forward the squared
equals DX squared plus D y ^2 and that's that's Pythagoras's
theorem. OK it's Pythagoras Pythagoras theorem written in
differential form. You go a little bit in the extraction a
little bit in the Y direction. You square the both of those and
the answer is the distance you've gone in Euclidean space,
just by that's theorem.
The reason I'm calling it the metric is because that
relationship between
and a difference, the next coordinate difference, the Y
coordinate and the distance travelled, is characteristic of
flat Euclidean space.
That's the space on a sheet of paper or drawn in the sand.
And the three-dimensional version of that, d = + y ^2 plus
DZ squared is the the way that distance is defined in 3D space.
If you go from here to here, the distance you've travelled is
that squared plus that squared plus that squared square rooted.
OK, and that is completely characterising Euclidean space.
OK, either in two dimensions or three dimensions.
And I I very quickly the at the end of last lecture talked
showed how you could turn that into what looked like a very
complicated way of answering a simple question, the distance
between two points directly from the metric.
And there's other things I could, I could you could talk
about there. But the point is just drive home. That is
this the full story of Euclidean space, if you like. Everything
about Euclidian space is in there.
But that's not the only space you are familiar with that
you're familiar with,
because you're also, as astronomers, familiar with
spherical, spherical trigonometry and the geometry of
things on the surface of the sky, on the celestial sphere.
You remember that from last year,
and there are coordinates just with other X&Y or R and
Theta coordinates.
On the plane
there are coordinates on the sky,
latitude, longitude, or or the the various variants of that
that you know about, something like that. So there's a a
longitude coordinate, and there's a there's a cool
attitude coordinate, and you can identify any point on the on the
surface of the celestial sphere with those with those two
coordinates.
But one of the other things you're taught in first astronomy
is how to get
distances, calculated distances between two points on the
celestial sphere. And what you don't do
is take the if you've got two points which are separated by by
D Theta and and and and and and D Phi. What you don't do
each calculate do theatre squared plus D Phi squared and
squared for that, as you recall, because that's not what that
give you the wrong answer. On the celestial sphere
do I have a? I don't have a a. But what you do have instead is
that the the s ^2 is equal to
are able to speculate memory the Theta squared plus
sine squared. Theta D Phi squared
I I'm
how will they? Haven't they? Haven't they?
Yeah. So that that's the distance between two points on
the celestial sphere,
and it's not the same.
I'm not telling you you don't know here, but I am pointing out
that the difference between the geometry of things in flat space
and the geometry of things on the celestial sphere is all in
that.
How does that manifest itself?
It manifests itself in things like this. A spherical triangle
on the surface of a sphere.
OK, you remember spherical triangle. You go from that, the
other three points on the on the sphere, and you take great
circle here, here and here.
And you can do things like ask what are the internal angles,
Look at the internal angle of this triangle and ask things
like what's the given two of these angles, what's the third?
You remember that sort of stuff.
And the rules for that special trigonometry are not the rules.
You learned about triangles when you're doing soccer too, and all
that sort of stuff in school.
And some of the things that you are familiar with from plane
geometry are not true here.
So for example, the on plane geometry, the internal angles of
a triangle add up to 100 degrees or π radians. You remember that
and that's not true here
on in plane geometry the the the area of our
of a triangle is given by some something to do with the the
internal angles and that's that's a different thing here.
And what we again are.
I would like to have more slides on more equations than this.
I'm not gonna write it up, but the
there is an expression for the
Yeah, there's an expression for the deviation
between the sum of those angles, AB&C and π radians, which is
a complicated thing. We know that's the point really having
to say because just a bit of a mess but it's something you can
work out
and it depends on the
the length of the side of that triangle AB, little AB and C and
it depends on the radius of that of that of the sphere there now
in
celestial is is is special trick you you've learned the celestial
sphere is deemed to be of radius one.
But in this case we're going to say it's it's of the the sphere
we're talking about is of radius R
and we can do our calculations and get and learn that the
trigonometrical things on stuff this year. Just to to remind
you, this is the surface of the sphere I'm talking about, not
the sphere as a whole embedded in 3D space.
And an important point
is that although we're looking at that
from outside, we're looking at that sphere from outside We're
looking down on the sphere, if you like,
from our position above it,
we we could do this, the same calculations on the surface of
the Earth and discover that the Earth is round so we can do
things like add up the one, can you do that? I'm sure surveyors
have to do this sort of calculation, at least at some
approximation. We can draw lines on the surface of the Earth,
carefully measure the angles between the three angles of that
triangle, and discover that you add up to 180 degrees. They very
nearly do,
but not quite.
In other words, all this geometry that we're doing on the
surface of the earth of surface the sphere,
is intrinsic to the surface. It's not just an artefact of us
looking down on it from at the motor side. It's not just an
artefact of the coordinates we're choosing,
it is intrinsic to the sphere and we can learn about it
without stepping outside.
Is an important point. So some ants could live on that sphere
and work out geometry and discover they were on a sphere
of radius R.
Just as we can make suitably accurate measurements on the
surface of the Earth and discover were not on a sphere,
we are on fear. I'm not on a plane
and this can also be fairly obvious in some circumstances
mean that this is probably going to be fairly
fine calculation. But if I do something like I'm telling you
have made a decision to the board a lot today
I had planned I'd expected not to have to write anything but I
shouldn't be much angle. And you can imagine our
but sort of the Earth, sea
and the equator.
And if we took a start at the pole,
Andrew,
a great circle down to the equator
and another great circle down to the equator,
what we would see
if you get 2 right angles there
and an arbitrary angle at the top. So obviously
the the internal angles of a triangle on the surface of a
sphere don't add up to 180 degrees.
Is very clear
and obviously
the circumference
on a on a on the on the sphere. Here a circle on the plane,
a circle in the set of points which are a constant distance
away from ascent from the centre. That's how you define a
circle.
OK, on the surface of a sphere,
the circle is the set of points which are a constant distance
away from, for example, the pole. So the equator is a circle
on the surface of a sphere because all the points on the
equator are
10,000 kilometres away from the pool,
obviously.
But that means that the
and and the circumference of the Earth is 40,000 kilometres.
So the
there's a conference divided by the diameter
is 2,
not π. It's two.
So even things like the ratio of the circumference of a circle to
its diameter is quotes wrong on the surface of a sphere. So you,
you, you, you've been looking at it. You're familiar with the
game,
you are familiar with the non Euclidean geometry. You just
haven't been think of it in those terms.
So, so want you to you know hold on to that thought.
And now here, as is probably fairly obvious and I could be
illuminated by by looking at the, the detailed expression for
the internal angles and so on. But we're not going to be
it's clear that as that radius gets bigger
this will that this triangle will get closer. If these
lengths,
you know an bits of string lengths on the surfaces you
don't don't don't get bigger as well as this this radius gets
bigger that will become closer and closer to our plane. I mean,
as we look around here that the Earth looks flat and locally
it's flattish,
so the the amount by which
the special trigonometry of the surface of this fear differs
from plain geometry,
It's governed by the size of that radius.
So there's a parameter
which tells us how much we've gone wrong from Euclidean
geometry,
and in this case the parameter is obviously the radius.
What turns out to be a useful
and re rephrasing of that if you like is a thing called the
curvature which in this case is just one over the radius.
So as the the radius gets bigger the coverage one of the readers
gets smaller and the cover should moves to zero. You're
arbitrarily large radius the the thing becomes you know non
curved and and more like Euclidean
geometry. So
to summarise that you are familiar with the a non
euclidean geometry,
it's fairly obvious that there is a a parameter, in this case
the radius, which characterises how much that deviates from
nuclear geometry. And as the curvature, in this case one over
the radius, gets smaller, you end up going back to flat space
just implanting those words in your head.
OK.
And that curvature is, as I mentioned, a property of the
space. You you don't know how to be looking at it from outside.
You don't have to have a radius for that curvature to be
present.
So that is a sphere as a surface with constant curvature because
the fairly obviously the the radius is the same all over the
sphere
if you smell like an ellipsoid, so like a like a a squashed
and
squashed sphere. And the curvature is is fairly obviously
different at different points in this. In like a rugby ball, the
curvature is less in the middle than it is at the end, so the
curvature can vary over
over the over the space. So different points in the 2D space
at the surface of that of that rugby ball have different
curvatures
and you can track the as you move around you can you can you
can see you can measure the curvature being different,
different different points, and discover the way that that that
that the the geometry is different at those different
points,
those of positive curvature. You'd also get spaces which have
a negative curvature,
and that's not so easily interpretable in terms of
radius. But
believe the algebra followed through and and you end up with
a curvature here, which is conveniently characterised with
a negative number. And that also makes sense though
because here if you start, if you look at that point in the
middle,
call that the pool see
and look at one of the
and and head out to one of the pick a a a radiate head, head
head out along it a bit that's that's the radius of of of a of
a circle. If you trace that
circle around
I particular radius, see you pick that one, see and and just
follow it round you'll see It's fairly obvious that the length
of that will be more
than π times the diameter.
When the sphere are a a circle was less than PI10 diameter,
here a radius. A circle has circumference which is more than
π times the diameter. So the geometry on this is also
different.
It also intrinsic to the to to to the space. It's not an
illusion caused by looking at from an infamous site,
and it's characterised by a coverage parameter, which in
this case is negative,
and the the culture of that speed is constant, just like the
sphere, except sort of the other way around.
How that feeling so far?
Any questions?
Thank you.
So that's
spatial, spatial, spatial spaces.
But the space that we are interested in
is not one of these. Is not a space. So these are, if you
like, variance of this sort of space, A space looking like
that. The space we're interested in is not that, but instead a
space
of. Minkowski speaks
where the interval, the definition of length, the
definition of how far apart two events are. So there's an event,
there's an event. They're separated by amount of space and
amount of time. How far apart are they? We've learned that
it's this quantity
that characterises this separation between 22 events,
and that's the thing that is our flat space, if you like, in this
context. So just as we can start playing games with the different
different coefficients in front of of this metric and get things
like sphere, spherical trigonometry,
we can discover that there are variants of this
which are also curved spaces.
But there's spaces where one of the dimensions is in time. And
in purely mathematical terms the the the key thing here is that -
because all of the spaces I was talking about up to this point
had all plus signs in there. So that in mathematical terms
that's the key difference between that and the the the the
sort of Minkowski space times that we've that we've been
talking about.
Thank you.
Moving on.
So creature is, is the thing that you've remembered from
that.
The next thing we have to talk about is
that I cheerfully talked up to this point about going going
from the centre of, for example, that diagram and heading out to
a particular radius.
Here I talked to start at the North Pole and heading out in a
straight line to the equator.
And you know what a straight line is on the surface of a
sphere. It's a great circle. You you remember that from last year
that that's what a straight line means in a curve speed in a in a
curved space.
Why a great circle? What's special about great circle that
makes it this the the obvious straight line? In that context,
the question turns into well, what is a straight line anyway?
No,
it seems fairly obvious what a straight line is.
I go up here to there. I don't wiggle.
I I don't do that,
OK? I just go from here
by the shortest path.
So in Euclidean space and indeed on a surface like that, a
straight line is the shortest path.
So they get if if if. I tried a a bit of string to the table
like there and pulled it tight.
This string would trace out a straight line
between there and here and
that's. It's a straight line by definition. That's what we mean
by a straight line in
in geometrical terms. And similarly, if I tied a bit of
string to the North Pole, went down to the equator and pulled,
what I would get would be a great circle.
OK,
but that's not the only way you can define a straight line.
Because the other way that I can talk about a straight line
but
search for beer
is to say, OK, I'm facing in that direction. We're closing my
eyes and just walk in a straight line
and where do I end up?
So I'm not going to go, I'm not going to turn any corners. I'm
just going to start at one place and move, move one, put 1 foot
ahead of the other and keep moving.
Think as far as I'm concerned right here right now is the
straightforward direction,
and in this case I trace out the same same path
and in that case I treat it the same path. If I start at the
equator,
face north, and just keep walking, I end up at the North
Pole.
I don't end up anywhere else. There's nowhere else I can go
because because what I I do now I I trace that out is I follow a
great circle.
So in these two cases,
the straight line is both the straight ahead that that that's
straight ahead.
Street line is the.
IT matches the shortest distance between two points. Straight
line.
But also if I start at the equator
and headed South
and just kept going, I'll go. I'll go past the South Pole, go
up and come to the North Pole. Again,
I go that that means I go the, the, the, the. The route I've
taken from this point to the North Pole is, at that point the
longest route I could possibly take in a straight between those
two points of looping around myself or anything.
So that means that the straight line in that context is the
extremal
path between 2 two points.
And these two things as I say match up.
But the important distinction between them is if I talk about
the type of string to the table leg and pulling it tight,
I have no sense pre, pre chosen where I'm going to go
that that that's intrinsically a a fairly global definition of
straight line. Whereas the definition where I say OK, I'm
just going to look at my feet and walk straight forward and
making sure that my feet move in in what is, as far as I'm
concerned right now, a straight line, that's an entirely local
definition of straight line
and and and and that locality, that localness is key, could be
we we want our physics to be local. We don't really. One of
the things that Newton didn't like about the law of universal
gravitation was that it depended on an element of non locality.
It depended on the the sun
setting up this structure of gravitational field all through
the universe and everything in it. Then, you know, paying
attention to that, but but paying attention to a thing that
was a very large distance away
and we don't really like that. I mean, and Newton didn't like
that because Newton wanted a contact theory. In other words,
that that the only things that would affect something were the
things that were touching it or next to it.
And so Newton was disappointed
that the theory that that worked was the one that worked
and we are still just and we are disappointed about it as well.
So we want things that are local only. So the fact that we can
define straight line in terms of of, I don't care where I'm
going, I'm just looking at paying attention to my feet and
walking forward, that's a good thing,
the locality
important word there.
And so
there's a quick question which is useful to meditate on in a
bit because there's quite a long, a lot of chat we could
talk about. And that's just an illustration of the idea that to
go from one place to another on, for example, the surface sphere,
you can do so by just following a straight line and what you end
up tracing out is
well as geodesic.
It's what it's called the sorry that's the keyword here. The
process of doing so, the process of of of just walking forwards
media trace out a geodesic and the the the thing that that that
usually is the thing that you you, you, you trace out. When
you do that process,
it had also
a line of extremal length. In
geometry is with this signature it's it's minimal length. In
terms of this signature, it's maximal length because you if
you think back to chapter 5 and the mention of the
the twins paradox and the McCarthy diagram where one of
the two of the two twins heads off to the some distance and
comes back, I mentioned in passing. I didn't make a big
deal of it at the time. I mentioned in passing that the
the reason that one way of thinking about the reason why
Odyssey is staying at home
go and going to the the the the the rendezvous point. By what
looks like the straight line along the the the T axis,
Odysseus is taking the longest route between those two points.
Whereas Penelope, who's taking their dog leg root,
is taking a shorter route. Although it looks longer on the
diagram
Minkowski space, it's the shorter route. So every every
deviation you take from this, the apparently straight line
and
root
decreases the length and that's that's that's rather strange,
but that's why we talk about duties being a lane of extremal
length rather than minimal lens.
So at this point we now have all of the
the, the, the,
the key ideas to go and talk about general activity and we're
very close to having to getting a bit an insight about about GR.
First of all, last week we talked about inertial frames
and I I said that the the definition of inertial frames in
general activity is basically the same as in special
relativity but with an important generalisation.
Which is an inertial frame is what you are in when you are in
free fall
and freefall means way out in space, or it means and this was
a physical insight that Einstein had. Or freefall is moving
entirely under the influence of gravity. So if you jump up and
down
then you are in freefall False. You're doing that and you are
and and and the quarters tax. You are the coordinate of an
inertial frame
because in that frame
Newton's laws work. While you are falling,
you can let something go and it will fall with you. That was
what Galileo discovered, dropping things from from the
tower of Peters. Allegedly, that everything falls accelerates the
same rate.
If I get something, a nudge while Foster was falling, you're
in this falling lift shaft. You get something, a nudge, and it
would move at a constant speed in your frame until it hit
something. So nuisance laws work in our national frame, and
that's an important link. From the physics before to the
physics of relativity
to national fame are important.
I talked about the equivalence principle
and I gave 2 versions of it.
One would depend on the notion of. If I think I have those
three version, I have two versions,
you know, come back to that one. One which was talking about
uniform motion and uniform gravitational field.
One which generalised that to say that all free falling
inertial frames were equivalent,
completely equivalent for the performance of all physical
experiments. And from that basis we directly deduced that light
must curve the gravitational field.
Who national frames equivalence principle.
I've talked of curvature
in this lecture and the the notion that you can characterise
the geometry
of a space
by making measurements internal to it and using things like the
metric to characterise that geometry and to build your
calculations on top of. And I, and I haven't gone into the
details of that
other than, you know, very sketchily the last time. And
I've talked about geodesics as the way you in a sense discover
what the space that you are in is like. So as I walk forward in
a flat space, I discover I explore the space along that
line. As I take a walk in a straight line in this local
sense on the surface of a sphere, I discovered I I go to
to both poles and and and and.
You want the notion of a straight line. A geodesic
in space is very important,
so
we can now talk about gravity.
So we've been talking about geometry at Raptor now, but now
we're about to talk about gravity.
And the question that gravity is about is about answering is if
you are
near a gravitating object
such as the sun,
how do you move? What? What what, what, what constrains your
emotion? How can you predict where you're gonna move that?
That's the question that a a theory of gravity is trying to
answer. They're trying to answer for, for example, the Earth and
the sun or the moon could run the Earth or or or a ball being
thrown from one place to another. But in all cases, the
question is where does this ball move? And the answer is, well,
let's let's use gravity to work it out
what we can. That then I'm going to jump forward a little bit,
is a third version of the equivalence principle,
which is what we do before we do that.
We knew
how things move. We know we're Newton told us how things move.
And I've, I've referred to that multiple times. So it said, if
you use the three laws, if you just leave something it don't
touch, it doesn't move.
No that's not true under gravity. But if we're inertial
frame that is true, you just leave something and it's falling
with you just if it is. And if you if something is moving the
straight line, it carried on moving in that straight line
until it hits something.
OK,
now the context of special activity.
The space we're talking about
is not just the XY&Z over lift cabin, but it's the space
of because it's Mankowski space.
So
if we take a bit of a jump and say, well, what does that
Newton's law of principle tell us about how things move
Minkowski space, It says, well, perhaps they move in a straight
line
and in a straight line. And in Minkowski space the simplest
motion
is not to move at all,
just standing where you are in your frame and you move in a
straight line along the same axis.
And
what if that is true more universally?
And the third version of the College Principle says exactly
that. It says that the that
any physical law
that can be expressed geometrically in special
activity and exactly the same form in a locally inertial frame
of a curved space-time.
There's another of those versions of the course principle
for all the words mean some are important.
So a physical law
we're talking about a a statement of how the universe
works. We're not a mathematical things,
we're expressing the physical and mathematical form. But we
are seeing something which could be, which could be
mathematically otherwise. But in our universe, it's true.
That's what we mean by physical law
that can be expressed geometrically in Sr
And what I said there about when you're standing still, you're
moving in a straight line through Makovsky space. That's
in a sense of physical, a geometrical statement of of how
things move.
If you if you're form momentum is pointing in this direction,
for example along the team member that prime axis that's
going to carry on doing so.
And that's a physical law that's that's in its in the version of
distance laws in expressing Minkowski space. And This is why
we're making such a fuss about geometry when talking about
special activity in the 1st 10 lectures of this course.
And the equivalent of said that has the same form
when expressed in a locally inertial frame, of course
space-time.
In other words, if you are in our
if we could have a reference frame
and it's an inertial reference frame. That is, you're moving
only under the influence of gravity, so we are in space or
you're free fall.
Then
the you can't tell
that you are. You can't tell which of those possibilities it
is,
even when
the space-time you're falling through
is curved,
even when the space-time fallen through is curved. And that's
why we're talking about
this. This local definition of
what geodesic is is important
because it's seeing that you don't have to worry about how
the space around you is curved.
It could be round the twist completely,
but the physical law that we're talking about here is 1, which
says you your velocity vector stays the same
in
the local and national frame. The nation that's local to you,
that's around you,
That's small enough that you can approximate it as being special
activity.
That's why the locality is important.
What this says doesn't happen,
or what this says does not happen is that
the the the the the the statement that you move along
the time axis of your local inertial frame. That doesn't get
any complications from the amount of curvature
in the space you're you're in. So you can imagine
that Newton's laws had an extra term in them which was to do
with how covered the space was, which just happened to be 0 in
the the, the, the, the, the physics we're familiar with
that could that. There's nothing mathematically wrong with that.
That's a perfectly mathematically
good guess to make.
This says that doesn't have.
There's no extra terms because the space you're moving through
is is weirdly curved.
OK.
So that that is the physical content of that. But that seems
to phase
um,
I'm I'm, I'm minimal thing to be confused about.
This thing that you might have thought might have happened
doesn't.
But it's actually terrifically important
because it's seeing. What that means is that if you understand
how things move in special relativity, and we do, then you
understand how things move in general activity as well.
That's the bridge from special activity
to general activity. This statement, the fact there's
nothing it not any more complicated as long as you're
concerned only with your local inertial frame, your your your
local frame, and a locally inertial frame.
But what's most important?
And
the fact that
this
diagram here tells us how things move, they move in a geodesic
along the local time axis
tells us how things move in a curved space. Time things move
along a geodesic of the space.
They could use principles. Let's let's make that jump.
In other words, at this point you have understood half of
general relativity,
which is that
space-time tells matter how to move.
So space-time is curved. Why? It's curved will come to in a
moment,
but it's curved
and we move along geodesics in that space. When we're in
freefall, we move along geodesics in that space,
and so the space curves we will.
You go with it
and that's our path through space-time, through time and
space
and that you well that was have seen videos where people put
weights on on rubber sheets and so on.
What happening there is is that the, the, the, the, the the
weight undersheet makes the initially flat rubber sheet into
a curved space
and then with the with some rules are marvel across that the
marble is instead of going in a straight line it's you know it's
not being being acted on by any breaths of breaths of air. I
mean, it's going as far as concerned, the straight line,
but because it's going in a straight line through a curved
space, it does things like orbiting
and that is the equivalence principle
that once you've worked out how the space is curved, you've
worked.