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So this the last lecture for our course, and

Â we are going to discuss here the most advanced and

Â modern subject in modern theoretical physics.

Â We're going to discuss homogeneous space times,

Â which solve Einstein equation with Lambda not 0,

Â nonzero cosmological constant.

Â But with for the case when matter energy momentum tensor is 0.

Â And we're going to discuss only the ground states solutions of

Â Einstein equation with nonzero cosmological constant,

Â so-called vacuum solutions of Einstein equation in this case.

Â Or, well, they are homogeneous spaces.

Â We will see what happens.

Â So actually, the case when we have lambda

Â equal to this Einstein equation,

Â basically look as follows, R mu nu minus

Â half of g mu nu R equal to half lambda G mu nu.

Â And in this case, well, up to this factor half,

Â this case actually can be mimicked by the case which we

Â have been considering in the previous lecture.

Â There we have been considering T mu nu of

Â the form rho times u mu u nu plus p times u mu u nu minus g mu nu.

Â And if p is equal minus rho,

Â which corresponds to the case of omega equals to -1,

Â an equation, such a peculiar equation of state,

Â then one can see that energy momentum tensor in this

Â case is reduced to the case rho times g mu nu,

Â which is exactly this case if rho is constant.

Â For the case of rho constant, we encounter exactly this case.

Â And in fact, the rho constant agrees for this case of energy momentum tensor,

Â rho constant agrees with Einstein equation of motion and energy conservation.

Â So we are going to discuss the following situation when G mu nu,

Â which is Einstein tensor, this guy,

Â is equal to plus minus D minus 1 times

Â D minus 2 over 2 times H squared g mu nu,

Â where mu and nu run from 0 to D minus 1.

Â So we're going to consider D-dimensional, D-dimensional,

Â from zero to D minus 1 is D-dimensional spaces of constant curvature,

Â spacetimes of constant curvature.

Â Here, H is Hubble constant, Hubble Constant.

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Hubble constant.

Â In this actual case, it is actually constant,

Â because here, H is independent of anything.

Â And this lambda here is related to this H as follows.

Â It's lambda equal to D minus 1

Â times D minus 2 over 2 H squared.

Â And plus sign corresponds to the positive curvature,

Â while minus sign here, minus sign here,

Â corresponds to negative.

Â So this is de Sitter space,

Â this is anti-de Sitter space.

Â Anyway, why this positive curvature and why this case is negative curvature

Â will become clear in a moment when we'll dig into details of this space.

Â So let me start my consideration of the de Sitter space with a positive curvature.

Â Well, there are, of course, one can do the same way as we did in

Â the previous lecture using this energy momentum tensor, etc, etc.

Â But for many reasons, I prefer another way.

Â Instead of using a matter energy momentum tensor of this form with rho

Â equals to constant, I prefer to address another way such that geometric

Â properties of the spaces that we're discussing become obvious, apparent.

Â So for example, de Sitter space, I am going to explain this, but

Â let me first make it a statement.

Â De Sitter space can be expressed as

Â the following hypersurface.

Â This is a metric tensor, Minkowskian metric tensor,

Â in D plus 1 dimensional spacetime.

Â So D-dimensional de Sitter spacetime

Â can be embedded into D plus 1 dimensional spacetime.

Â So this is a metric in that D plus 1 dimensional space time.

Â Anyways, I'm going to explain in a moment.

Â So this is Minkowskian tensor in D plus 1 dimensions.

Â So the equation which is written

Â here is just X0 squared plus X1

Â squared plus etc plus XD squared.

Â And, well, this is by definition.

Â And equation for the hypersurfaces as follows.

Â It's just H to the minus 2, the same H as here.

Â So this hyperboloid is embedded into the D plus

Â 1 dimensional Minkowskian spacetime with this metric.

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space of constant curvature, a homogeneous space.

Â Well, there are several ways to see that.

Â Well, first way is the following, it's so-called weak rotation.

Â And if we make a change of X0 to

Â i X D+1 both here and here,

Â we obtain that minus sign here

Â is changed to the plus sign

Â here under this change.

Â And this is changed to delta AB, well, with a minus sign.

Â Well, that doesn't matter, because we can omit this sign,

Â it is not relevant for anything.

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So we obtain D plus 1 dimensional Euclidean space and

Â D-dimensional sphere embedded into this space.

Â So what we obtain is just space of,

Â obviously, space of constant curvature of the radius 1 over H,

Â which is apparent from this equation, and

Â which does solve this equation in Euclidean signature.

Â So we can define Einstein's theory in the same

Â manner in space of Euclidian symmetry.

Â Nowhere in the duration of this equation we have used

Â the fact that this metric is of Minkowskian signature.

Â So the same procedures are applicable in Euclidian signature,

Â we can obtain the same equation.

Â And the sphere will solve this equation with plus sign here.

Â And during this change, weak rotation, the curvature of the spacetime is not changed.

Â That's the reason de Sitter space is called

Â the spacetime of positive constant curvature.

Â And that is a way to see that it is, in fact, the space of constant curvature.

Â That's the first way of seeing that.

Â The second way of seeing this is as follows.

Â One can see that isometry of this space

Â contains the following group, SO

Â D minus D minus 1 comma 1.

Â Which is nothing but the Lorentzian boosts and

Â rotations of this spacetime.

Â This group doesn't change this equation.

Â So this group of Lorentzian rotations of this spacetime is the isometry

Â group of the de Sitter spacetime, which is apparent from the equation.

Â At the same time, an arbitrary point,

Â say, for example, the point X0,

Â X1, and so forth, XD, well, X2,

Â let me write X2 also, for obvious reasons, etc.

Â For this point, which is 0,

Â 1 over H0 and etc, 0, everywhere,

Â this point belongs to this hyperboloid.

Â It is moved under this group, but

Â remains unchanged under the action of the SO,

Â I should say that, well, actually,

Â I'm sorry, to say that this is,

Â the isometry group is actually SO

Â D1 without [INAUDIBLE] and the subgroup

Â of this which doesn't change this point

Â is actually SO D-1 S0 D-1 comma 1.

Â So because here D X's and 1 was minus sign X0,

Â that's the reason SO D is isometry of this.

Â And the stabilizer, so-called stabilizer, the subgroup of this group which

Â doesn't move arbitrary point of this spacetime, say, this one, is of this form.

Â Then one can see that this space that we

Â are discussing is homogeneous space SO(D,

Â 1) over SO(D-1, 1).

Â Compare, actually, this with SO(D+1) over SO(D) for the sphere.

Â You see under this weak rotation,

Â we change this to this and this to this.

Â So in fact, all of these homogenous spaces transform into this.

Â