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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