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.

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