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Homogeneous three–dimensional spaces

Course video 54 of 66

With this module we start our discussion of the cosmological solutions. We define constant curvature three-dimensional homogeneous spaces. Then we derive Friedman-Robertson-Walker cosmological solutions of the Einstein equations. We describe their properties. We end up this module with the derivation of the vacuum homogeneous but anisotropic cosmological Kasner solution.

Introduction into General Theory of Relativity

National Research University Higher School of Economics

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

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Ministrado por

Emil Akhmedov
Associate Professor

Transcrição

[SOUND].

So with this lecture, we start our discussion of cosmology.

So the idea we're going to use in this lecture and next one is the following.

At the scales of the sun system, we in the space see large empty spaces and

very few bodies separated by large distances.

But at the same time if we go to larger scales like 100 megaparsecs,

then at these scales universe looks rather homogeneous.

We have a very high precision, homogeneous and

isotropic distribution of matter on these scales.

Moreover, observational data show that distant galaxies running

in all directions.

They're running particularly all galaxies that we see are running away from us.

And in this lecture, we're going to discuss

solutions of general [INAUDIBLE] which describes such a behavior.

So let us start with the discussion of the space times which describe homogeneous and

isotropic behavior.

So the most general four-dimensional metric tether which

describes isotropic and homogeneous spatial sections is as follows.

So the metric has this form.

dt squared- a squared (t)

times gamma ij which depends

only on spartial coordinat dx i,

dx j where i and j run from 1,2,3.

So here, dl squared which is

gamma ij(x) dx to the i dx to the j

describe the spatial sections.

And their size is changing in time,

according to the change in time of the scale factor, a(t).

Because of the homogeneity of spatial sections,

all points in this space and all directions

from every point in this space should be equivalent.

That is an idea of homogeneity.

So hence, so there is three dimensional spaces.

They describe something which has constant curvature.

And we going to discuss that in greater detail.

So let us discuss the all sorts of

possibilities for the homogenous spatial sections.

The matrix tensor is space of contrast curvature,

can be written in the falling form.

So we can write this metric in the following form.

dr squared / 1- kr squared +

r squared d omega squared.

[COUGH] Where, as usual,

d omega squared is the metric on the unit two-dimensional sphere.

So it has this form.

Sine squared theta d phi squared.

[COUGH] And there are three options for this parameter K.

K can be equal to either -1, 0 or 1.

So these three cases, we're going to discuss now.

Indeed, let us see that this space describes for these three choices of k,

this space describes [COUGH] homogeneous sedation.

Which corresponds to constant curvature at the same time.

So when k = 0, this is a first case,

when k = 0, in this case we obtain dl squared,

which is dr squared + r squared, d omega squared, which is nothing but

[COUGH] three dimensional flat space written in spherical coordinates.

Obviously, this is a homogeneous space.

Every point of this space is equivalent to every other.

And all directions from every point are equivalent.

And obviously this space corresponds to constant curvature sedation.

The curvature is 0.

So the second case is when k = 1.

In this case, we obtain dl squared,

which is dr squared divided by 1- kr squared.

Sorry, in this case, k = 1.

So here stands 1, + r squared, d omega squared.

Now we can make a change of coordinates.

In this case, we can change to denote

r as sine of kai, of some angle kai.

[COUGH] Obviously for this to be a space, r should run from 1 to -1.

So this is a meaningful change.

And then, if we make such a change in this metric, dl squared becomes familiar to us,

metric d kai squared + sine squared chi d omega squared.

We actually have encountered this metric

in the lecture on Oppenheimer-Synder collapse.

This metric describes three-dimensional sphere.

Indeed, let us see that.

What is three-dimensional sphere?

Three-dimensional sphere is the following surface.

Well, hypersurface.

Three dimensional sphere is this hyper

surface embedded into four dimensional

space with the metric dw squared + dx

squared + dy squared + dz squared.

So, how to see the relation between the metric on this sphere

with the metric written here.

Well, we just have to solve this equation.

The solution of this equation is the following for

Euclidean Four Vector.

It is equivalent to cosine high of high, this high.

Sine of high multiplied by sine

theta and cosine of phi.

Then sine of kai multiplied

by sine of theta sine of phi.

And finally, sine of kai times cosine of theta.

Well obviously, if one square this + square this + square this + square this,

one will obtain that this is equal to 1.

Then if one uses this in this metric,

flags into here, one obtains exactly this

metric on the sphere, on the sphere.

Well obviously, the sphere is a homogeneous space.

In fact, this surface together with

this metric is invariant on the SO(4) rotations,

Rotations in four dimensional space.

And every point of this at the same time, every point of this space

has a stabilizer, such as transformation which doesn't move it.

For example, if W = 1, this is equal to 0, 0, which solves this equation.

So this point, it is quite a generic point, quite a generic point.

This point is stable under SO(3) rotations in this part.

So the space under discussion is the falling homogenous space,

SO(4) / SO(3).

So by the action of this group,

we can move from any point of this space to any other point.

And the action of this group doesn't change that point.

That's a reason, it's a homogenous space.

And that explains that every point of this space is equivalent to every other,

because it can be obtained by the action of this group.

And every direction of this space is equivalent to every other direction.

And obviously, this is a space of constant positive curvature, and

that was the space of the constant zero curvature.

So finally we have to discuss this metric dl squared for

dr squared, 1- kr squared + r squared

d omega squared for the case that k = -1.

So in this case, obviously we encounter this iteration,

dr squared, 1 + r squared + r squared d omega squared.

Now if one will make a change that r is

equal to hyperbolic sine, sinh of kai,

then this metric acquires the following form.

d kai squared + hyperbolic sign of

kai squared, d omega squared.

What is this space?

This space is actually [INAUDIBLE] hyperplane, as we will see in a moment.

So it's a space of constant negative curvature.

So how to see that?.

The space of constant negative curvature is given by the following equation.

So it's hypersurface defined by this equation in the four

dimensional Minkowski space of the following form

embedded into Minkowski space with the following metric.

So, how to see before showing the relation between

induced metric on this space with this metric.

Let us discuss why this space is homogeneous.

First of all one can straightforwardly see,

rewriting this equation in the form that X squared

+ Y squared + Z squared is W squared- 1.

So for each value of W greater than 1 or lees than -1,

we have a sphere, three dimensional sphere.

So one can picture the space as follows.

So it's a two shaded hyperboloid whereas upper shape starting,

this is W and this is X, Y, Z etc.

And there is another shape of this hyperboloid which starts with,

this is 1, this is -1.

And as W increases by

absolute value from 1 or

-1, the size of the sections increasing.

So basically, this picture corresponds to the case when we neglect Z.

So say, put that equals to 0.

So this is a first sign that we're dealing with a space of constant curvature but

we are going to give it bit more explanations.

One can see that this space has obvious isometries which is Lauren's group.

Which contains Lauren's group, SO(3,1).

And this group respects this equation,

hence respects the space that we are discussing.

And second thing.

One can see that there is a point like

W,X,Y,Z such that 1,0,0,0.

This point solves this equation.

So it lays inside the space.

And the stabilizer of this point is rotation in this direction SO(3).

Hence the space that we are discussing is homogeneous space of the form

SO(3,1) divided by SO(3).

So every point of this space can be obtained from

every other point by the action of this group, but

generic point of this space has the stabilizer of this form.

So this sub-group of this group doesn't move an arbitrary point in this space.

So, that's the reason it's homogenous space.

Every point of it is equivalent to every other, and

every direction from every point of it is equivalent to every other direction.

So that's the reason to expect this space to have a constant curvature,

to have a constant curvature.

And finally if one will solve this equation,

this equation as follows.

(W,X, Y,Z) = hyperbolic

sign of Chi, hyperbolic sign

of Chi times Sine of Theta,

ordinary Sine times Cosine.

Of phi times hyperbolic sine of kai,

times sine of theta times sine of phi.

And finally, hyperbolic sine of chi times ordinary cosine of theta.

So obviously, if one will plot square of this minus square of this minus

square of this minus square of this, one will have this equation as the consequence

of the fact that cosine squared + sine squared is equal to 1.

While hyperbolic cosine squared minus hyperbolic sine squared is equal to 1.

So, this does solve this equation.

And, at the same time, if one will use this form of W, X, Y,

and Z in this metric, here, one will obtain this metric.

Notice that this space, it's a space,

it's not space-time.

Metric in it is spatial, purely spatial.

Well, it is embedded into Minkowskian signature space time.

It is because this hyperbolic while whole space has Minkowskian signature.

This whole hyperboloid is space like in this Minkowskian signature space time.

[MUSIC]

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