Penrose–Carter diagram of the anti–de Sitter space–time

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En provenance du cours de National Research University Higher School of Economics

Introduction into General Theory of Relativity

81 notes

National Research University Higher School of Economics

Introduction into General Theory of Relativity

81 notes

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

À partir de la leçon

Cosmological solutions with non-zero cosmological constant

In this module we derive constant curvature de Sitter and anti de Sitter solutions of the Einstein equations with non-zero cosmological constant. We describe the geometric and causal properties of such space-times and provide their Penrose-Carter diagrams. We provide coordinate systems which cover various patches of these space-times.

De Sitter and anti-de Sitter13:00

Geometry of the de Sitter space–time11:33

Metric of the de Sitter space–time9:19

Penrose–Carter diagram for the de Sitter space–time10:15

Poincare' patches for the de Sitter space–time16:50

Geometry of the anti–de Sitter space–time7:20

Metric of the anti–de Sitter space–time9:16

Penrose–Carter diagram of the anti–de Sitter space–time8:25

Poincare' patch of the anti–de Sitter space–time6:08

Rencontrer les enseignants

Emil Akhmedov
Associate Professor
Faculty of Mathematics

[MUSIC]

So we have obtained this metric on anti-de Sitter space,

which covers entire space time.

Hyperbolic cosine of rho squared- d rho

squared- hyperbolic sign of rho omega d- 2.

D- 2 squared.

Notice that d- 2 coordinates here, 1 here and 1 here, so it's three-dimensional.

Space time and now we want to draw a Penrose-Carter diagram of the space.

So we're going to make the following change: and

we want to tangent of theta will be equal to hyperbolic sine of rho.

So hyperbolic sine of rho will change for the tangents.

And as a result this metric acquires,

after this change here, we get the following

method: h squared times cos squared of theta,

times d tau squared- d theta squared- sine

squared of theta d omega d- 2 squared.

So now one also has to consider,

you see the range of validity again for D = 2,

rho is ranging from- infinity to + infinity and

for D > 2, rho is ranging from 0 to + infinity.

That of course imposes a difference that in case of D = 2.

So as a result if we have D = 2 then theta

is ranging from -pi/2 to pi/2.

And for D > 2, theta is

ranging from 0 to pi/2.

And we have two different situations.

So for D = 2, D = 2, Penrose diagram falling

from here is exactly the same one as we had in case

of de Sitter space which is just this rectangle.

But it's rotated by angle of p over 2.

So this is a timeline, tau.

Remember that, in our case, tau, these two sides are glued to each other and

if we consider universal power, it just becomes a stripe for

tower ranging from- infinity to + infinity.

And theta is ranging from- pi over 2 to pi over 2.

In our case tau is ranging from 0 to 2 pi or from- pi to pi.

Well, anyway, so this for the appropriate antecedent

metric tau is compact and this is exactly what we have and

the fact that time is compact leads to the fact that this is a cylinder.

So that's the same cylinder as we had for de Sitter space by just rotated

by the angle pi of 2 along this axis perpendicular to this plane.

That's what we have for two-dimensional case.

For this case, for the great length two-dimensions,

if we use this, but we have the following situation.

We have the following situation, that, let me draw it like this.

So along theta we have from 0 to pi over 2.

And this is frayed.

And here we have from 0 to 2 pi.

So this is four times longer than this.

And it is important that here it is in two dimensions.

These are two boundaries of the de Sitter space, anti-de Sitter space.

And this is the response to rho = 2 and this corresponds to rho = + infinity.

This is not the boundary, this is the center of the disk.

While this is a boundary of de Sitter, anti-de Sitter space.

And again, if we are dealing with appropriate range of values of tau,

then this is glued to this.

If we deal with universal color, this becomes a stripe.

And now we encounter a very interesting situation for this case.

Notice that light rays again are just straight,

45 angle degrees with respect to this.

And we have the following situation where actually it four times.

This side is four times longer than this side,

so my picture is not quite appropriate.

But it is important that from any point inside the de Sitter space,

light rays reach the boundary, spatial boundary within finite proper time.

And actually the particles are, where massive particles go along these lines.

So, antecedent geometry acts as a, so

it attracts back massive particles and for light rays, it acts as a box.

So we have kind of like as if it a finite distance along this direction.

So the x as a box.

And this led to some peculiar property of the anti-de Sitter space.

If we solve Cauchy problem in space time, we have to solve kind of causal equations,

causal differential equations, then if we want to specify a behaviour

at some point o, we have to consider at causal past of this point.

So we go to the backward light con.

And to obtain something here,

everything depends on the values inside this light con only.

Not on something here because we are dealing with causal situation.

So we specify initial conditions at Cauchy surface.

Then it gives us the value of the field or

solution of differential equation at this point.

That's what happens in say, Minkowski space or in de-Sitter space or

in other regular spaces that we have encountered, Schwartz etc.

But here we encounter a new feature,

that the boundary of the spaces that reach within finite time.

So, this is well our spaces, space time at its partial sections, they're like boxes.

And to the Cauchy problem is, if you specify initial

value with Cauchy surface, it doesn't give you the value of the field here.

The value of solution of the differential equation here.

You have to also specify boundary conditions on top of the initial

conditions.

And this feature,

this property of anti de-Sitter space is called as absence of global hypervelocity.

It's just because related to this fact that you can reach within finite time.

So if you want to solve something here, you have to specify boundary conditions

here, well boundary condition here, better to say because this is not a boundary.

Boundary condition here and initial condition here.

This is absence of global hyperblicity of anti de-Sitter space.

That's how it renews itself.

And we going to continue our discussion with the Poincar coordinates and

Poincar patch of anti-de Sitter space.

>> [MUSIC]

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