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Dt squared- hyperbolic cosine squared
Ht over H squared d omega d- 1 squared,
where this is a metric on d -1 dimensional sphere.
Now we're going to draw Penrose diagram for this space.
Well, the natural transformation is the following.
We can change hyperbolic cosine squared of Ht
by inverse of the cosine squared of theta.
And this theta is ranging between -pi over 2, and pi over 2.
Then, as theta approaches pi over 2 and -pi over 2, this blows up,
which corresponds to the blowing up of cosine of hyperbolic cosine.
So, if we make this coordinate change in this metric,
it acquires the following form.
It becomes 1 over H squared cosine squared theta [d
theta squared- d omega d- 2 squared].
This is a metric in compact space because all the angles here take finite radius.
This conformal factor blows up in these boundaries, and
to draw Penrose diagram, Penrose-Carter diagram, we drop off this.
And now what remains to be done is to
choose one on top of this angle to choose one extra angle from here.
The natural choice, because this metric,
omega d- 1 squared has the form d theta
1 squared- sine squared of theta 1d omega- 2 squared.
So, it doesn't contain any factor here, so here, nothing.
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And then, we, in this and this space, we obtain the following metric,
d theta 1 -d theta 1 squared d theta squared.
So, we can write the matrix on this two dimensional space, and
remember that theta 1 is ranging from -pi over 2 theta 1 is 2pi over 2.
So the result is the following Penrose-Carter diagram,
which is just a square, just a square.
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Where this is theta, this theta 1, this is -pi over 2, this is pi over 2.
This is -pi over 2, this is pi over 2.
So, this is just a square.
The problem with this diagram is that it
doesn't reveal the topology of the de Sitter space,
which is just, so d -1, times R.
This is time, and this is time slice.
Remember that this is time.
And at every given time, we have a sphere of changing radius, of course, but
the topology is like that.
Here, topology is not apparent because these are two
distinct poles of the d -1 dimensional sphere.
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Remember that theta d -1 is ranging from -pi to pi.
And these are not the poles, but the same points.
So, it's like cylindrical topology will be apparent with the use of this angle.
We will see that in a moment.
But the problem with the use of this angle because here, it is here,
and it contains a lot factors of this form, and the metric is not flat there.
So, every position on the diagram would strongly depend on what kind of theta 1,
theta 2, etc., we choose.
So, then we find it is more convenient to consider,
if we draw a Penrose-Carter diagram, to draw it for two dimensional case.
For two dimensional case, remember,
here we have instead of this, it's just theta 1 squared,
one angle, which is ranging, in this case, theta 1 for
d = 2, theta 1 is equal to theta d- 1 for this case.
And we encounter this and theta 1 is arranging from -pi to pi.
And as a result we have to draw d theta squared- d theta- 1 squared but
now, this one is ranging here and this one is ranging here.
As a result, the Penrose-Carter diagram is the following rectangle.
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so while here, it is from 1 to pi over 2, where is theta.
And this theta d -1, in our case this is equal to 1, ranging from -pi to pi.
But it is important that this side of this rectangle is glued to this side,
basically what we obtain is just the cylinder.
And in fact, the cylinder is nothing but
the geographic projection of this hyperbolic, for
which is the form of the two dimensional de Sitter space.
So, this cylinder is just its geographic projection.
Even it can be understood what kind of map is between this and this space.
But we don't have time for discussing this issue.
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So, but it is important now that with the use of this Penrose-Carter
diagram, many properties of the de Sitter space become apparent.
For example, any world line of any
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massive particle has this form.
It's some line which cannot have tangential lines to this line,
cannot have an angle greater than 45 degrees with respect to this.
Because otherwise,
particle will be moving with the speed greater than the speed of light.
Remember that if we have a light-like behavior,
then this is equal to 0, which is equivalent that this is equal to 0.
So, light rays here are just straight lines
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which have 45 degree angle with respect to every axis.
So, this is, in fact, that what we have.
So because this is glued to this, what we have is like this.
This is just a continuation of this line, and also,
we have the following situation that light ray is going like this,
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So, so this is glued to this, and this is glued to this.
And so, what do we see, that in de Sitter space, there is a causal diamond,
this region, which is causally connected to this particle.
Because within this causal diamond, we can exchange light-like signals,
but, for example, this region, from this region we cannot receive.
We can send a signal to this region, but
we cannot receive back light ray from here.
And from this region, we can receive a signal, but
we cannot send a light ray to this region, or to this region.
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So, that's this kind of behavior appears due to the fact
that the de Sitter space is expanding.
Is it, well, not simply expanding.
It's the spatial sections, their volume is changing.
It's from minus infinity to t equals to zero, it shrinks,
and then expands back to infinity.
So, due to this kind of behavior,
we encounter this funny situation with causal properties of the de Sitter space.
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And one last thing that I wanted to explain,
I forgot to say that this line is glued to this one because while here,
we have two opposite poles of this d- 1 dimensional sphere,
here we have the same points.
So, this is not the opposite poles.
So, we have a circle, its opposite poles are here.
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Two generatrix, so this, for example, this and this light ray,
are just two parallel generatrix lines on this light-like generatrix lines
are on this hyper below it.
Emil AkhmedovAssociate Professor
