[MUSIC] So this the last lecture for our course, and we are going to discuss here the most advanced and modern subject in modern theoretical physics. We're going to discuss homogeneous space times, which solve Einstein equation with Lambda not 0, nonzero cosmological constant. But with for the case when matter energy momentum tensor is 0. And we're going to discuss only the ground states solutions of Einstein equation with nonzero cosmological constant, so-called vacuum solutions of Einstein equation in this case. Or, well, they are homogeneous spaces. We will see what happens. So actually, the case when we have lambda equal to this Einstein equation, basically look as follows, R mu nu minus half of g mu nu R equal to half lambda G mu nu. And in this case, well, up to this factor half, this case actually can be mimicked by the case which we have been considering in the previous lecture. There we have been considering T mu nu of the form rho times u mu u nu plus p times u mu u nu minus g mu nu. And if p is equal minus rho, which corresponds to the case of omega equals to -1, an equation, such a peculiar equation of state, then one can see that energy momentum tensor in this case is reduced to the case rho times g mu nu, which is exactly this case if rho is constant. For the case of rho constant, we encounter exactly this case. And in fact, the rho constant agrees for this case of energy momentum tensor, rho constant agrees with Einstein equation of motion and energy conservation. So we are going to discuss the following situation when G mu nu, which is Einstein tensor, this guy, is equal to plus minus D minus 1 times D minus 2 over 2 times H squared g mu nu, where mu and nu run from 0 to D minus 1. So we're going to consider D-dimensional, D-dimensional, from zero to D minus 1 is D-dimensional spaces of constant curvature, spacetimes of constant curvature. Here, H is Hubble constant, Hubble Constant. Hubble constant. In this actual case, it is actually constant, because here, H is independent of anything. And this lambda here is related to this H as follows. It's lambda equal to D minus 1 times D minus 2 over 2 H squared. And plus sign corresponds to the positive curvature, while minus sign here, minus sign here, corresponds to negative. So this is de Sitter space, this is anti-de Sitter space. Anyway, why this positive curvature and why this case is negative curvature will become clear in a moment when we'll dig into details of this space. So let me start my consideration of the de Sitter space with a positive curvature. Well, there are, of course, one can do the same way as we did in the previous lecture using this energy momentum tensor, etc, etc. But for many reasons, I prefer another way. Instead of using a matter energy momentum tensor of this form with rho equals to constant, I prefer to address another way such that geometric properties of the spaces that we're discussing become obvious, apparent. So for example, de Sitter space, I am going to explain this, but let me first make it a statement. De Sitter space can be expressed as the following hypersurface. This is a metric tensor, Minkowskian metric tensor, in D plus 1 dimensional spacetime. So D-dimensional de Sitter spacetime can be embedded into D plus 1 dimensional spacetime. So this is a metric in that D plus 1 dimensional space time. Anyways, I'm going to explain in a moment. So this is Minkowskian tensor in D plus 1 dimensions. So the equation which is written here is just X0 squared plus X1 squared plus etc plus XD squared. And, well, this is by definition. And equation for the hypersurfaces as follows. It's just H to the minus 2, the same H as here. So this hyperboloid is embedded into the D plus 1 dimensional Minkowskian spacetime with this metric. So we're going to explain why this is in fact space of constant curvature, a homogeneous space. Well, there are several ways to see that. Well, first way is the following, it's so-called weak rotation. And if we make a change of X0 to i X D+1 both here and here, we obtain that minus sign here is changed to the plus sign here under this change. And this is changed to delta AB, well, with a minus sign. Well, that doesn't matter, because we can omit this sign, it is not relevant for anything. So we obtain D plus 1 dimensional Euclidean space and D-dimensional sphere embedded into this space. So what we obtain is just space of, obviously, space of constant curvature of the radius 1 over H, which is apparent from this equation, and which does solve this equation in Euclidean signature. So we can define Einstein's theory in the same manner in space of Euclidian symmetry. Nowhere in the duration of this equation we have used the fact that this metric is of Minkowskian signature. So the same procedures are applicable in Euclidian signature, we can obtain the same equation. And the sphere will solve this equation with plus sign here. And during this change, weak rotation, the curvature of the spacetime is not changed. That's the reason de Sitter space is called the spacetime of positive constant curvature. And that is a way to see that it is, in fact, the space of constant curvature. That's the first way of seeing that. The second way of seeing this is as follows. One can see that isometry of this space contains the following group, SO D minus D minus 1 comma 1. Which is nothing but the Lorentzian boosts and rotations of this spacetime. This group doesn't change this equation. So this group of Lorentzian rotations of this spacetime is the isometry group of the de Sitter spacetime, which is apparent from the equation. At the same time, an arbitrary point, say, for example, the point X0, X1, and so forth, XD, well, X2, let me write X2 also, for obvious reasons, etc. For this point, which is 0, 1 over H0 and etc, 0, everywhere, this point belongs to this hyperboloid. It is moved under this group, but remains unchanged under the action of the SO, I should say that, well, actually, I'm sorry, to say that this is, the isometry group is actually SO D1 without [INAUDIBLE] and the subgroup of this which doesn't change this point is actually SO D-1 S0 D-1 comma 1. So because here D X's and 1 was minus sign X0, that's the reason SO D is isometry of this. And the stabilizer, so-called stabilizer, the subgroup of this group which doesn't move arbitrary point of this spacetime, say, this one, is of this form. Then one can see that this space that we are discussing is homogeneous space SO(D, 1) over SO(D-1, 1). Compare, actually, this with SO(D+1) over SO(D) for the sphere. You see under this weak rotation, we change this to this and this to this. So in fact, all of these homogenous spaces transform into this. And we obtain, so, what we see from this consideration. This is a second way of seeing that fact that our space is of constant curvature, a homogeneous space. The fact that there is homogeneous space of this form tells us 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 from every point of the space is equivalent to every other up to the difference between the space like and time like directions. So I explain two ways without calculation to see that we are dealing with a space of constant curvature. [SOUND]