Remember that before we are using Minkowskian

metric where one plus and the rest are minuses.

Here it's a bit different.

So for z greater than 0, for z greater than 0,

one can see from this expressions that

X0- X1 = 1 over (H z) squared.

And it is actually greater than 0.

So these coordinates,

unlike the previous ones which cover entire into de Sitter space,

these coordinates cover only half of it, given by this condition.

And in fact, if one uses these coordinates,

the induced metric on the anti-de Sitter space

is just ds squared = 1 over (Hz) squared

[dz squared + dx mu dx mu] which is very similar,

very similar to the metric in expanding and

contracting patches.

And remember that we had like this (H

eta plus minus squared) [d eta plus

minus squared- dx squared].

But here, the important crucial difference with respect of this de

Sitter metric here at this time.

But here that is space-like.

Time in anti-de Sitter space is among these coordinates.

It's like dx vector squared- dt squared so time is this.

So there the anti-de Sitter spacetime metric is stationary,

it's actually even static while the de

Sitter spacetime metric is time dependent.

That is a crucial difference between these two spaces.

Actually this coordinates according to this condition one can

easily see what part of the entire, what half of the entire.

In case of two dimensions,

one can see easily what part of the entire anti-de Sitter space it covers.

It's just like this half of the anti-de Sitter space in higher dimensions.

In two it is a bit harder to see.

But it still covers half of the entire anti-de Sitter spacetime.

Well, this ends up our discussion of the anti-de

Sitter space and spaces of constant positive and negative curvature and

actually ends up our introduction to general theory of relativity.

And so that's the end of the story.

Good luck.

Good luck at the exams.

Good luck at the study of this course.

I hope you will enjoy learning this subject and

use it in your further studies of basics of theoretical physics,

quantum field theory, gravity, etc., and etc.

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