Poincare' patch of the anti–de Sitter space–time

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Из курса от партнера National Research University Higher School of Economics

Introduction into General Theory of Relativity

81 оценки

National Research University Higher School of Economics

Introduction into General Theory of Relativity

81 оценки

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

Из урока

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

Познакомьтесь с преподавателями

Emil Akhmedov
Associate Professor
Faculty of Mathematics

[MUSIC]

To conclude our discussion of the anti-de Sitter spacetime,

let me just introduce Poincare patch and

Poincare coordinates of the anti-de Sitter spacetime.

Well to remind you, anti-de Sitter spacetime is given by this equation.

-X 0 squared + sum over

j from 1 to D- 1 X j squared-

XD squared = -1 over H squared.

And it is embedded into the space which has

signature +, -, -, -, and +.

Now one can solve this equation,

this equation, as follows by giving X0

= z over 2 [1 + 1 over z squared (1

over H squared- x mu times x mu)].

And z here is greater than or equal to 0.

X1 = z over 2 [1- 1 over

z squared (1 over H squared

+ x mu times x mu)].

And X mu, big X,

X mu = x mu over H,

where mu = 2, ..., D.

And one last point is that x mu times x mu =

eta mu nu times x mu times x nu.

But eta mu nu has a signature

which is a bit different from the one that we are using.

It has pluses, all pluses and one

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.

[SOUND]

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