General Theory of Relativity or the theory of relativistic gravitation is the one which describes black holes, gravitational waves and expanding Universe. The goal of the course is to introduce you into this theory. The introduction is based on the consideration of many practical generic examples in various scopes of the General Relativity. After the completion of the course you will be able to solve basic standard problems of this theory. We assume that you are familiar with the Special Theory of Relativity and Classical Electrodynamics. However, as an aid we have recorded several complementary materials which are supposed to help you understand some of the aspects of the Special Theory of Relativity and Classical Electrodynamics and some of the calculational tools that are used in our course. Also as a complementary material we provide the written form of the lectures at the website: https://math.hse.ru/generalrelativity2015
Penrose–Carter diagrams
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Do curso por National Research University Higher School of Economics
Introduction into General Theory of Relativity
92 classificações
National Research University Higher School of Economics
92 classificações
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Penrose-Carter diagrams
We start with the definition of the Penrose-Carter diagram for flat space-time. On this example we explain the uses of such diagrams. Then we continue with the definition of the Kruskal-Szekeres coordinates which cover the entire black hole space-time. With the use of these coordinates we define Penrose-Carter diagram for the Schwarzschild black hole. This diagram allows us to qualitatively understand the fundamental properties of the black hole.
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Emil AkhmedovAssociate Professor
Faculty of Mathematics
[SOUND] So we are going to discuss
properties of the Schwarzschild
solution that we have obtained during
the previous lecture with the use
of so-called Penrose diagram.
So the idea of Penrose diagram is to select a two
dimensional part of the four dimensional space time.
Such as two dimensional part that is irrelevant for
physics that we are discussing and which includes time.
And to map it to two dimensional compact space.
And let me start with explaining the idea of ordinary stereographic projection.
So the ordinary stereographic projection, one of the ways to do it is as follows.
So we have a non-compact plane.
Non-compact plane, two dimensional plane and we want to map it to a sphere.
So one of the ways to do that is as follows.
So we can see there a sphere that is cutted by this plane by two hemispheres.
We select northern pole of this sphere and map points according to these rules.
So we draw a line which passes through the sphere and through
the plane and the point on the plane is mapped to this point on the sphere.
So the points from outside of this disk of the plane are map to the upper hemisphere.
While the points from inside the disk are mapped to the lower hemisphere
according to the same law.
So this stereographic projection is a conformal map.
Because if we for example, consider the image of
two lines on the plane and
such a projection, the image on the sphere.
One can easily see.
Well, we are not going to spend time on that,
one can easily see that the angles between these lines and the images are the same.
So the angles under this map are respected, so
this is a conformal projection.
And it is done on a two dimensional space.
So the idea behind the conformal map is that
falling any two dimensional metric [INAUDIBLE]
gab is actually a symmetric two by two matrix.
Hence it has three independent components and
using two representations,
xa to xa bar, which is a function of xb.
So two functions.
We can fix two of these functions.
Which is actually was used during the previous lecture
at the very beginning of the previous lecture.
In some form this fact was used.
So we can fix two of these functions.
And any two dimensional metric can be represented hence for example,
in this form in the conformally flat form.
So this a chronical symbol, so
this is a flat Euclidean, well, we are doing this for Euclidean's case.
So this is in Minkowski space, that would be instead like this.
So this is a conformal factor, a so-called conformal factor.
And it is important to fix it coordinate such that this x,
the range of the validity of the new coordinate is compact.
So the reason for that will become clear in a moment.
Probably for mathematical rigor,
I have to stress here that this kind of fixing can be done globally only for
the spaces with spherical topology or spherical topology with punctures.
But this is irrelevant comment for the rest of our lecture.
So anyway, again, any two dimensional metric by use of their two dimensional
general co-transformations can be fixed to be like this with a conformal factor.
The only thing we have to care is that the new coordinates after this fixing should
range in a comfort range.
So to show the idea of Penrose diagram, let me do it for the flat space.
First thing, we going to study the flat space in greater details.
For example, the flat space metric as we know has the following form,
ds squared is dt squared minus
dx squared minus dy squared minus dz squared.
Suppose we are discussing motion along x direction.
So then the relevent part of the space time is this.
So let us select it and maybe the following change of coordinates,
from t and x to new coordinates called psi and psi.
And the coordinate transformation has the following form.
T plus minus x = tangent
of psi plus minus psi over 2.
Notice that while t and x are ranging
from minus infinity to plus infinity for them to range in this region.
Psi and psi has have to range
in the region from minus pi to pi, because we are using tangents here.
As a result, under such a change of coordinates, this metric,
this metric, dt squared minus dx squared is transform to the following form.
One over 2 cosine psi plus psi over 2,
cosine psi minus psi over 2 squared.
D psi squared minus d psi squared.
So we have here conformal, this is conformal factor.
And this is flat metric in compact space.
And the conformal factor blows up when
psi plus minus psi is equal to pi.
So it becomes infinite,
which allows one to map non compact
space time to a compact space time.
That is exactly the point.
Now and one can see that dt squared minus dx squared =0 implies or
equivalent to the fact that d psi squared minus d psi squared = 0.
So light rays here and here have basically the same properties,
the same angles, etc., etc.
Let us draw this psi psi space.
Let us draw it.
So psi
Psi and psi and then we draw its path.
So no, it's not
totally this space,
it is square.
So these are straight angles.
This is minus pi,
this is pi, minus pi, pi.
The boundary of this square is exactly where this conformal factor blows up.
Blows up.
So one can check that the whole space,
the whole space, so the whole line minus to plus infinity.
The whole line x from minus to plus infinity,
at t equals to minus infinity, at t equals to minus
infinity mapped here to this single point.
At the same time the whole x, the whole
space line at t equals to plus infinity mapped here.
The whole timeline, t equal t belonging from minus t to plus infinity.
At x equals to minus infinity
is mapped to this point.
At the same time,
the whole timeline at x equal to plus infinity mapped to this point.
At the same time, so this is called I minus,
this is called I plus, this is called I0, I0.
At the same time, these lines are light-like.
Because they consistent, make an angle 45 degrees with respect to the axises.
And these are the lines at infinity from which light rays originate and terminate.
So they originate from this or from this and terminate here and
here, so as time goes by, these light rays go as force.
And they make according to this condition.
They make 45 degrees with respect to the axises.
So these are just straight lines, this are light rays.
So this scri plus light like future infinity, this is scri minus.
And also this is scri minus and this is scri plus.
So this is exactly the point that we can draw on the Penrose diagram,
we can draw explicitly.
So we mapped non-compact spacetime into compact square.
The reason why here, the whole plane, Euclidean plane is mapped to the sphere.
Is because the whole points of this plane at infinity have the same properties.
So they are indistinguishable basically from each other and
map to a single point on the sphere to the northern pole.
So point of infinity is mapped basically, as we go to infinity, the image
of the point, of the very far away point, gets closer and closer to the pole.
Here the situation is different because different points at infinity
have different properties.
One of them are space-like, so these points are space-like at infinity.
These points are also space-like at infinity.
These points are time-like, time-like at infinity, so
time-like points are mapped to these points.
Space like to this and light like to these points.
So that's difference of that map of the conformal map
in Euclidian space and in Minkowski, so this is called stereographic projection.
This is called Penrose diagram.
So we have mapped noncompact
Minkowski space to compact space and
dropped off conformal factor.
And so t and x range from minus
infinity to plus infinity.
While psi and psi are ranging from minus pi to pi.
And as a result we have mapped the noncompact two dimensional spacetime
to the compact square in this.
Let me draw it more carefully.
So this is psi, this is psi,
so the square is like this.
So this is minus sub pi and I zero.
So the whole time like infinity at x equals
minus to infinity is not through this single point.
So this is pi, which is the same I0 but
at x equals to plus infinity.
So this is minus pi, I minus.
So this is t, whole past infinity
plus infinity is mapped to here to this single point.
This is pi I plus.
Which corresponds to future infinity equals to [INAUDIBLE].
And these are scri plus, this is scri minus plus light like infinity.
And this is scri plus, scri plus future light like infinity.
So light rays on this diagram, as I explained,
the angles do not change under this conformal map.
So the light rays here are given by the equation like this.
And would correspond to light rays, which are given by
the equation like this, so they look like this here on this line.
So, a world line of a particle that travels,
of a massive particle or
a massive observer has the same falling form.
Of course, explicit map from a world line in this coordinates to a world
line in this coordinate is quite hard exercise.
Depends on every concrete curve, but in principle we know that
the properties of any world line which is here.
It should start from past infinity, not light like,
but past infinity and at future infinity.
And it has to have the property that all tangential
lines to this curve should have angles less than 45 degrees.
This is important.
Anyway, so this is a world line and
it goes like this as time goes by, as time goes to future.
So light rays also have the direction from past to future.
Any cushy surface in this space
depicted have this property that
theyare like mostly space like.
So they start from this point and at this point and
directive such that they are more or less space-like.
So their angle with respect to this axis is less than 45.
So now this is Penrose diagram of flat Minkowski space.
And now one can see that any observer in
the Minkowski space has access to whole space.
Because as time goes by, this guy sees more and
more of the [INAUDIBLE] surface in this Minkowski space.
So these are [INAUDIBLE] surfaces, so
sees more and more bigger fraction of them.
So as it reaches this point, it sees all of it.
So there are no regions in this space time
which are closely disconnected from each other.
Closely disconnected from each other.
Now soon, we're going to see that in black hole case,
the situation is quite different.
So we are ready to move on with Penrose cut diagrams for black holes.
[SOUND]
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